Potential Mathematical Models for the Western Musical Scale
A Historical and Empirical Comparison

By Daniel White - Copyright © 03/05/2007

(This is an updated version of the dissertation as required for my course at the University of East Anglia. I thought I'd put it online (06/05/2007) for anyone who might be interested. In the future, I believe all scholarly texts will be primarily online, because it is easier for the author to design, maintain and reference, and better for the reader to seek references and information quickly (hyperlinks, wildcard search and no book-hunting)).

Phenomena 1: First order beating of partials
Phenomena 2: Two types of consonance
Phenomena 3: The 'mind's eye'
Phenomena 4: Instruments such as the organ can exhibit acoustic distortion in ET
Phenomena 5: Just intonation may allow for 'key colour'
Phenomena 6: Easier to play/sing just intonation
Phenomena 7: ‘Nice looking’ numbers
Phenomena 8: ‘Pull’ of notes
Phenomena 9: Larger tolerance level for some listeners
Summary

Psychoacoustical / empirical studies

Summary

Final conclusions

A more 'natural' tuning?
Future experimentation

Endnotes

Appendices

Appendix 1 - 5-limit Just intonation lattice
Appendix 2 - Tuning preferences throughout the ages
Appendix 3 - Deviation from pure intervals in each key of just intonation

Introduction and claims

For this dissertation, I will be reviewing the issue of tuning and temperament, and research the foundation of what the human ear perceives as 'in tune'. More specifically, I intend to investigate three inter-related and partially overlapping ideas:

Argument 1: That the type of consonance often associated with just ratios (the droning, lack of 'beating', 'pure' sound) may be a separate type of consonance associated with the twelve intervals of the scale (minor/major second, major third, major fourth etc.), despite the close mathematical proximity of said systems. Obviously for some people, these may coincide (where the droning intervals are also exactly mapped to the twelve intervals of the scale for their auditory system), and hence it would be easy to conflate these consonance types.

Argument 2: That the differences between preferences in tuning may be down to the possibility that for the second consonance type (stated previously), two people/cultures could perceive the same physical interval differently in the 'mind's eye' (or conversely, a physically different interval, they could perceive the same). As an extreme example, can a tritone to group A, sound like a perfect fifth to group B?

Argument 3: That despite Argument 2, there may be a 'most' natural tuning system for the second consonance type stated in Argument 1 (the twelve intervals). Whether this is represented by equal temperament, just intonation, Pythagorean intonation, or another model completely, is very hard to resolve.

Why care?

What's the point in researching what the 'right tuning' is for music (if such a thing could be said) ? After all, most people hear interval inaccuracies on the order of 25 cents as 'on-tune', or at least are not noticeably disturbed by such intervals [1][2].

Apart from the benefit for the purists, and intrinsic value, part of the reason is so we can form a foundation upon which other theories can be based. The analysis of harmony for example may be influenced according to whether the twelve intervals can be explained by pure ratios, or logarithmically equally spaced intervals as defined by equal temperament. On the other hand, the intervals may have no absolute mathematical pitch, instead being dependant on the observer. In this case, one may need to take lean further towards psychoacoustic research and techniques.

Historical context

For Western music, tuning has always been an issue of contention throughout the centuries, with the history as foggy as the science of it is messy. Rarely will you find a topic so rich with complexity, laced with numerology, sprawling with pitfalls, and coloured with controversy. We have the ancient Greeks to thank for starting it all, who in their culture of philosophy, cosmology, mathematics and music were the first in the Western world to relate the musical scale to ratios of simple numbers.

In particular, one man, named Pythagoras, is often reported to have discovered the potentially secret relation between the musical intervals and mathematics in the sixth century B.C.E [3]. These proportions, which can be heard through dividing a musical string and plucking, were derived from the first four positive integers (tetraktys) - one, two, three and four - to create what he thought were 'all' of the possible notes of the scale. Indeed, we can derive the octave, perfect fourth, and perfect fifth immediately from these numbers, and ultimately all of the twelve notes by using successive powers of three, to create the canonical 'scale (or circle) of fifths'. Mystical significance was assigned to these numbers, and the music using such consonant intervals (in a melodic context, as harmony as we know it (simultaneous sounding of notes) was not employed until around the 9th century [4]) was supposed to cure all ills, and relax the body, mind and spirit.

Even in the times of the ancient Greeks, theories were varied though, and theorists such as Ptolemy put their own spin on the original scale, using the next prime number - 5 - to increase or maybe refine the existing spectrum of musical intervals, and what could be the first realization of "just intonation" (perhaps Pythagoras avoided this prime number, because it wasn't part of the tetraktys [5]). On the other hand, Aristoxenus held that the consonances of music were arbitrary and subjective to the listener - perhaps just cultural constructs [6].

Nevertheless, it was Pythagorean intonation which entrenched itself in Western thought, and was commonly believed to be the theoretical foundation of the scale up until around 1500 [7]. However, between 900 and 1300 c.e., certain intervals such as the 2nd, 7th, 6th and most notably the major 3rd were not used, perhaps because it was banned from use, or maybe they were just not accustomed to such an interval. Alternatively, maybe the third was not usually appropriate in the music of the time. Finally, it’s just possible the Pythagorean major third was too sharp for their tastes [4].

Even in later centuries with the advent of tonality, the Pythagorean "Wolf" interval commonly limited music to only two or three sharps/flats in the key signature [8]. In practise though, we also hear how many organists were effectively tuning to equal temperament ("participata") according to Gaf(f)urius' report from 1496 [9] (or was it 1483? [10], also see [11][12] for the early use of ET in general), though it would seem that Pythagorean intonation was dominant in theory and practise before 1400 [13][14].

The story on Just intonation (5-limit) is somewhat different though. Despite the writings and thoughts of theorists such as Benedetti, Helmholtz, Descartes, Lloyd, and Francesco Galeazzi [15], we find that Barbour exclaims [16] (also see [17]):
"There was no golden age of just intonation, coinciding with the golden age of polyphony. In fact it is difficult to believe that just intonation was ever practised anywhere.".

However, at the least, it seems clear that various 'flavours' of just intonation have been in common use from Baroquian times [18][19][20][21][22], and leading up to a "good part" of the 18th century in harpsichords and especially organs [23]. Choirs however may have used either just intonation or equal temperament, or both, according to who you believe [24][25].

Taken from "How Music Became a Battleground for the Great Minds of Western Civilization", Page 19

So raged a storm throughout the centuries [26][27][28][29][30], and temperaments and tunings were (and still are [31]) fought against and defended (see Appendix 2 for an extensive list of individual preferences). Instrument makers tried to appease both sides of the fence by incorporating extra keys to accommodate both just intonation and equal temperament [32], by 'splitting' the black key into two [33][34] (such instruments were controversial, as they were cumbersome to make and play).

For various reasons given below, equal temperament finally 'won' in the end, after growing rapidly in acceptance from around 1750, with even organ tuners eventually succumbing, by replacing it over the old meantone system after around 1850 [35][36] (Duffin argues ET generally came much later though - around 1917 [37]). Despite massive resistance at first [38], the swing from just intonation (Pythagorean or the 5-limit variety) to equal temperament was a necessary step forward in Classical and Romantic music to allow for complex harmony and key modulation. Theorists such as Vincenzo Galilei (1520-1591) [39] and Rameau (1683-1764) played a big part in promoting equal temperament too [40].

However, although equal temperament has probably been ubiquitous around the globe since 1850 [41], the majority consensus, is that equal temperament - the tuning used predominantly in most instrumental and vocal music throughout the modern world - is a compromise when compared to just intonation [42][43][44][45][46][47][48][49][50] (exceptions are very rare [51]). In fact, from my research so far, it would appear that the only theorist explicitly recognizing equal temperament as some kind of ideal would be Simon Stevin, (who also was one of the first to formalize the mathematics of the equal tempered scale to a precise degree). I quote:

To illustrate the necessity for equal temperament, Galilei composed a 'song' - the "Discorso particolare intorno all'unisono" - which if sung with the intervals of just intonation, would sound wrong and out of tune [53]. Interestingly, this demonstration of equal temperament predates Bach's 'Well Tempered Clavier' (1722) by over 100 years.

"His contemporaries apologetically presented the equal system as a practical, but Stevin held that its ratios, irrational though they may be, were "true" and that the simple, rational values such as 3:2 for the fifth were the approximations!" [52].

But for the most part, you'll find famous theorists (including Zarlino, Mersenne, and Rameau) seeing equal temperament as a compromise [42]. Also from Lloyd's "The Myth of Equal Temperament" [43], I quote:

"All musicians know that equal temperament is an acoustical compromise, tolerated by many ears on the piano, and designed to satisfy as completely as possible three incompatible requirements - true intonation, complete freedom of modulation and convenience in practical use in keyed instruments - and that it sacrifices the first of these to the second and third."

Therefore, despite the obvious modulation problems, even today, you'll find clusters of just intonation advocates around the world who insist on using pure intervals for modern day composition, as well as existing repertoire [54].

Historical summary and implications for tuning

From a historical perspective, it seems clear that nothing for certain can be said about universal tuning preference, but that doesn’t necessarily forbid such a theory if there are many variables to take into account. Before I argue the various merits and shortcomings of each tuning, it will be appropriate to first detail some of the maths involved.

Tuning models and maths

Curiously, there is more than one way of representing the twelve notes of the Western scale (it can be termed 'Western', but at least a subset of the scale at least partially generalizes to at least Indian, Chinese and Arab-Persian music as well [55][56][57][58] (also see [59][193][195][61][62])).

In fact, some models will point at other notes outside of the familiar twelve notes, even if they're hinted at to a lesser degree (according to how the mathematics is interpreted). Suffice to say that most of the pitches produced by these models sound so very similar, but are subtly different by a fraction of a percent in pitch according to the model. For example, the major third in just intonation is very slightly lower than the equal tempered version (by about 1/7th of a semitone).

Below I will detail and contrast the three different models I mentioned in the first paragraph of this thesis; equal temperament, just intonation, and Pythagorean intonation. It should be noted that these are only models, and although somewhat endearing in their simplicity, they do not necessarily represent the intervals the mind 'wants to hear' [63]. They obviously come close at least, but it seems clear that what counts as 'in-tune' varies from person to person and from era to era. However, I will attempt to find the 'nearest match' later in this dissertation, based on large-scale experimental studies.

Equal temperament

("12-tet", "12-eq", "12edo", "12-et", "12-equal", "Even temperament")

Equal temperament intervals can be expressed by the formula: 2^n/12, where n is a number from 0-12 for a single octave. This spaces all of the pitches (geometrically) evenly inside the octave, so no semitone interval is bigger than any other. Consequentally, notes outside the major mode become enharmonically equivalent, so D# although notated and functioning differently to Eb in theory, is heard as the same pitch.

Historically, equal temperament was devised to remove the Pythagorean comma found in Pythagorean intonation, to allow for modulation to any key without sounding too off-tune. If you traverse up the scale of fifths from a tonic of C, it turns out that the twelfth interval - B# - is around a quarter of a semitone (23.46 cents) sharp of the original C [64]. To compensate for this, equal temperament provides the solution by compressing each note, so that each n^th interval along the scale of fifths is lowered by a factor of 1.001129ⁿ (1.955 cents or one 50th of a semitone, for the first fifth - G). This way, the twelfth fifth is exactly the same note as the original (assuming octave equivalence).

Possibly because of the prevalence of equal temperament in Western music, most people don't hear its intervals as 'off-tune' (regardless of theoretical judgments about its validity or compromise). Some claim though that some of its intervals, especially the major third, are off-tune, since they deviate from certain pure ratios, creating "beats" and therefore "dissonance".

5-limit just intonation

("just intonation", "JI", "pure temperament", consisting of "pure", "true", "just", or "justly intoned" chords, ratios or intervals)

Just intervals can be expressed by the formula 2^m x 3ⁿ x 5^o, where 2^m is used to optionally keep the interval limited to a single octave. 5-limit just intonation is actually a superset of Pythagorean intonation (3-limit JI), so any that are found there can be found here. By a curious mathematical 'coincidence', many of the intervals (especially the simpler ones) approximate the equal tempered ones very closely. In Appendix 1, you'll be able to see a two dimensional lattice, with additional data given for cents, and its proximity to the nearest 12-et interval.

Because ever higher exponents can be used for more complex ratios, there are theoretically an infinite amount of just intervals. However, only the nearest exponents are usually used (low powers of three (-2 to +2), and powers of 5 in the range of -1 to +1). Outside this range, intervals tend to sound increasingly off-tune. It turns out that many seemingly simple ratios compete for some intervals (for example, the minor 7th could be expressed as 9:16 (996.1 cents), or 5:9 (1017.6 cents), or even 4:7 (968.8 cents) if we allow the next prime as used in 7-limit just intonation).

Typical example of just intonation scale.

5-limit Just Intonation lends itself to a 2 dimensional matrix representation, but for practical purposes, this is often reduced to a single 1 dimensional subset. This table to the right is one of the more commonly used just intonation scales (correlates with Barbour’s [65]).

Perceptually, just intoned intervals have sometimes been called "pure", "droning", and "weird" by various people [66][67]. That just intoned intervals have any effect at all is taken foregranted in this dissertation (whether or not JI forms the basis of the musical scale). Apart from the obvious relation to the harmonic series, and the characteristic drone-like sound, there’s also neurological evidence [68].

When creating instruments directly with a synthesizer (especially simple sounds such as saw and square waves), sometimes they can sound more interesting with a very slightly smaller or bigger ratio than Just [69], possibly to avoid the 'static-ness' of the sound being played. Due to the phenomenon of virtual pitch, just intervals can also be perceived as having an extra 'phantom' tone - the implication of the fundamental (see later).

Unfortunately, there are internal inconsistencies inside just intonation [70]. Due to the logarithmically unequal spacings between the notes in the octave (for example, the two different types of whole tone: 8:9 and 9:10), it's possible for some chords to contain different intervals on fixed-pitch instruments, or 'travelling pitch' even on non-fixed pitch instruments. (See "Mathematical inconsistencies inside Just Intonation" for a more detailed explanation). In a single key (say C major), it turns out that the tonic, dominant, and subdominant have pure triads (4:5:6) [71]. However, keys distant from this key (approximately greater than 3 keys sharp or 2 keys flat [72]) will also have many off-tune intervals (also see Appendix 3).

Because of these issues, instruments will usually be tuned to slightly less restrictive 'pseudo' just intonation scales. These include various flavours of just intonation (such as tuning to Pythagorean, except decreasing each note by a full syntonic comma every 4th fifth) and tunings such as meantone (tuning to Pythagorean, except decreasing each note by 1/4 of a syntonic comma every fifth [73][23]), Well temperament, or some other compromise. These generally allow for more (not total [74]) freedom in modulation without too many intervals sounding off-tune, whilst maintaining some of JI’s unique properties, such as the pure major third as used in meantone, or the unequal intervals to allow for 'key colours' (see "Why are Just intervals often preferred?" -> "Phenomena 5").

3-limit just intonation

("Pythagorean intonation", "circle of fifths", "cycle of fifths", "spiral of fifths")

3-limit just intonation is actually a subset of 5-limit just intonation, and is naturally one dimensional, rather than two as in 5-limit. Once again, it is curious mathematically how many of the ratios coincide closely with the intervals taken from equal temperament or 5-limit just intonation. Pythagorean intervals can be expressed very simply by the formula 2^m x 3ⁿ, where 2^m is used to optionally keep the interval limited to a single octave. Despite this simple use of multiplications using 3 and 2, seemingly complex ratios can result (such as 531441 / 524288 for B#).

Equivalently, it is possible to create the scale by tuning consecutive fifths (each at 701.96 cent intervals) up and down, using the previous fifth as a reference point for the current fifth.

Because fixed pitch instruments such as the piano can only have a finite number of keys, it is impossible to obtain all of the pitches that the never-ending series in the scale produces. Technically, this is thanks to the twelfth fifth not intersecting with the octave, forming a discrepancy known as the "Pythagorean comma" - equal to 3¹²/2¹⁹, 1.0136..., or about 23.46 cents.

Thus, instead of a neverending "line" or "spiral" of fifths, a fixed pitch instrument would use a truncated version - the "circle of fifths", perhaps by tuning six fifths up and five fifths down (creating C G D A E B F# forwards, and C F Bb Eb Ab Db backwards). With this arrangement, the interval between F# and Db is known as the 'wolf' note or interval, being slightly flatter than a true fifth (1.4798, or 678.49 cents wide). Suffice to say that other nearby intervals (such as Ab to B in this arrangement) also feel the knock-on effect. The 'better' intervals would be the ones which don't travel so far from the 'flat side' to the 'sharp side'.

With non-fixed pitch instruments however, or dynamic 3-limit just intonation, the effect is lessened somewhat, since a true spiral of fifths can be obtained, allowing double flats and sharps if needed.

Mathematical inconsistencies inside Just Intonation

As previously said, there are internal contradictions inside just intonation. For example, to take the following progression:

CEG - 1.0, 1.25, 1.5
ACE - 1.666, 1.0, 1.25
DFA - 1.125, 1.333, 1.666 (intervals are 1.185 and 1.481, not 1.2 and 1.5)
GBD - 1.5, 1.875, 1.125
(pitches normalized to between 1 and 2)

The D minor chord has JI-relative dissonant intervals of 1.185 (294 cents, and 22 cents away from the pure minor third) and 1.481 (680 cents and 22 cents away from the pure fifth). It is possible to make the major second 9:10 instead of 8:9, thus solving the issue of the D minor chord above, but then the G major chord will have problematic intervals instead.

To attempt a solution, we can try to alter the pitches during the music for fretless, non-fixed pitch instruments such as the human voice, trombone, violin, or synthesizer. This is sometimes called "dynamic just intonation". Alas, in this case, although the chords are consistent if taken individually, what often happens is that the music ends on a different pitch to the one it originally started in. For example, take the following progression:

CEG - 1.0, 1.25, 1.5
ACE - 1.666, 1.0, 1.25
DFA - 1.111, 1.333, 1.666
GBD - 1.481, 1.852, 1.111
CEG - 0.987, 1.234, 1.481
(pitches normalized to between 1 and 2, apart from the fifth chord to show deviation)

In each case, a note in the previous chord is the same pitch as the same note in the current chord. As we can see though, the C in the last chord is about 23 cents lower in pitch (over a fifth of a semitone) than the C in the first chord. During the course of a long piece of music, this could be increased to many semitones - sharp or flat!

Also see J. Murray Barbour's "Just Intonation Confuted" which also points out this kind of contradiction [70]. Could any of this count as evidence against just intonation? It certainly hints that way, but of course it's far from proof.

Things get somewhat more interesting for the black notes. Barbour also points out four possible interpretations of the diminished seventh in just intonation [75]:

Ellis       10:12:14:17     (1   1.2   1.4	   1.7)
Paule White 25:30:35:42     (1   1.2   1.4	   1.68)
Poole       25:30:36:42     (1   1.2   1.44	   1.68)
Helmholtz   225:270:320:384 (1   1.2   1.4222  1.70666)

Needless to say, there are the various permutations which can be derived from these four (and other intervals like 1.666 for the diminished seventh). Which of these (if any) is the real diminished seventh? Or are they good in different ways? Or does it vary from person to person and/or according to the musical context?

Also, there are two main candidates for the minor third interval in Just Intonation:

a: 6/5 = 1.2 (more usual)
b: 7/6 = 1.1666~

JI-a (1.2) is 15.6 cents above the equal tempered minor third, and JI-b (1.166) is even further away at 33.2 cents below ET. Despite the relative simplicity of the ratios, to my ears, it is clear that neither of them sound as ‘in-tune’ as the equal tempered version (at least for melodical/tonal purposes - i.e. "chromatic consonance" - see later). It is interesting to compare these ratios to the Pythagorean minor third which although looks more complex (32/27 = 1.1851... = 25/33), is in some sense simpler, because it only uses prime numbers 2 and 3 to build its scale, rather than the prime numbers 5 and 7 as required by 5-limit and 7-limit JI in (a) and (b) respectively.

The Harmonic series, and consonance

Now that we've detailed the tuning systems under review, it will also be helpful to look at various acoustical phenomena which may directly or indirectly influence the perception of intervals. The most obvious being the harmonic series. Next, I will cover various definitions of consonance, and define two custom types which I will use for the purposes of this dissertation - "Rational" and "Chromatic" consonance.

Related to just intonation, the harmonic series, discovered roughly in the seventeenth century [76], forms the basis for an instrument's timbre, 'tone quality' or 'texture'. It is unclear whether the phenomenon is learnt or ‘in-built’ into our auditory system [77], but using sine waves as the basic building block of sound [78], intervals which are at integer multiples of the fundamental feel as if they camouflage into the fundamental to form a single clear 'note' or 'pitch' [79]. For example if the fundamental is 256 Hz, or the pitch of C, then the following harmonic series will be produced:

C (fundamental - 256 Hz, normalize to 1)
C (2nd harmonic - 512 Hz, normalize to 1)
G (3rd harmonic - 768 Hz, normalize to 1.5)
C (4th harmonic - 1024 Hz, normalize to 1)
E (near?) (5th harmonic - 1280 Hz, normalize to 1.25)
G (6th harmonic - 1536 Hz, normalize to 1.5)
Bb (near) (7th harmonic - 1792 Hz, normalize to 1.75)
C (8th harmonic - 2048 Hz, normalize to 1)
D (9th harmonic - 2304 Hz, normalize to 1.125)
E (near?) (10th harmonic - 2560 Hz, normalize to 1.25)
F/F# (pseudo) (11th harmonic - 2816 Hz, normalize to 1.375)

...and so on indefinitely. (generally, real instruments will have each successive harmonic tail off in intensity). With effort/practise, it’s possible to hear these harmonics individually, despite their camouflaging properties [80], although this is a much harder feat in short or rapidly decaying tones [81].

(People such as Rameau even considered these pitches as the basis for consonance [76], although intervals such as the major fourth and sixth aren't noticeably represented). The amplitude and range of these harmonics will define the sound's distinct timbre. For example, a bright-sounding instrument (such as a trumpet) will have many such harmonics. A flute on the other hand will generally only emphasize the first few harmonics - something closer to a pure sine wave.

Interestingly, as long as some of the higher harmonics exist (the most influential being in the range of 500 Hz to 2000 Hz [82], and the first 6 to 8 harmonics [83]), the fundamental harmonic doesn't even need to be present for its effect to be 'heard' (known as "virtual pitch", "periodicity pitch", "subjective pitch", or "residue tone" [84], and possibly discovered in (or at least by) 1841 by Seebeck [85]). This can best be exemplified by an old radio or phone line which cuts out all of the bass frequencies, but where the virtual pitch can still be perceived. Virtual pitch is a phenomenon of the second order, so its effects are created in the neural pathways closer to the brain, rather than inside the ear [84], and is therefore perceived even if the interval is played dichotically in each ear [84].

Because the ear/mind (human auditory system) can take so much tolerance, even near-integer relationships will hint at the fundamental too, and this is demonstrated by real instruments, which usually closely approximate the harmonic series [86][87][88]. Even the piano's greater degree of inharmonicity will strongly hint at a single pitch (though to a lesser degree on the lower notes). Roughly speaking, to determine the fundamental pitch of any numbers of partials, we can use the place theory of pitch perception [89], where we can find the highest common factor [90] (e.g: partials 1f & 1.25f indicate a fundamental of 0.25, and partials 1f, 3f, 5f indicate a fundamental of 1).

Timbres with frequencies such as the following:
1f, 2f, 3f, 4f, 5f (correlates approximately to the piano intervals)
1f, 2.01f, 2.99f, 4f, 5f
3f, 5f, 6f, 8f, 9f

...all strongly hint at f being the fundamental. However, timbres containing partials such as 1f, 2.42f, 3.6f, 4.91f will tend to form perceptually separate pitches, and less of any distinct single pitch. This a large simplification though, and the mind often attempts to search for a single virtual pitch (even from a choice of many if the partials are sufficiently ambiguous to suggest multiple virtual pitches) [91][92][93]). One such ambiguous virtual pitch is created from the partials 900 and 1100 Hz. In this case, either 180 or 220 Hz virtual pitches can be perceived depending on context [94].

Helmholtz and Plomp/Levelt

Helmholtz is often attributed to the claim that the level of dissonance in an interval was created by the 'beating' of all possible pairs of sine wave partials [95] (though Sorge in the 18th century was probably the first to consider this idea [76]). For just two tones, the beating effect or interference starts to occur when they are so close in frequency that the wave will 'tremolo', slower at first and pleasant [69], but becoming quicker and more dissonant as the interval widens. As the interval further widens, the tremolo starts to transform into a perception of two tones, and past around an interval of a whole tone, the perceived consonance starts to return again. This type of consonance has been dubbed "Tonal" or "Sensory" consonance [96].

The beats created by such nearby intervals are also known as "first order" beats [97], and they are created because of the limitations inside the ear (see later). Interestingly, the effect disappears as soon as each tone is played in the left and right ear dichotically [98]. Because the ear creates these first order beats rather than the mind, it is known as a phenomenon of the "first order" (unlike virtual pitch for example, which is a phenomenon of the "second order" - whose beats are 'created' inside the 'mind').

Helmholtz's results were given extra credence by Plomp and Levelt who in 1965 (and later in 1969 by Kameoka and Kuriyagawa [95]), performed an experiment on many musically untrained people [99][95][100] (who would not necessarily have preconceived ideas about consonant intervals [101]). The diagrams below demonstrate the resulting perceptual consonance of two sine waves at various base frequencies:

From these graphs [102][103], we can see that the most consonant interval is at unison, and the most dissonant interval is roughly 1-4 semitones according to pitch range [104][102], or more accurately, about 25% of the critical bandwidth [101][105]. After the high dissonance peak, the interval widens, and the interval starts becoming consonant again.

(Based on the ear's basilar membrane, the "critical bandwidth" interval is near where the human ear starts to recognize it as two separate (pure sine) tones instead of one (Plomp 1964; Plomp and Mimpen 1968) [106][107][108]; If the interval present is finer than the critical band, then the usual two activation areas on the basilar membrane start to overlap, and merge into ‘one’ - hence the perception of a single tone [109].)

Another approximate figure of 32Hz beating was given earlier by Helmholtz as the maximum roughness for beating [95] (beginning to set in around 15 Hz [110]).

But turning our attention back to the graphs; This result is surprising, because the standard intervals aren't represented - not even the fifth or octave [102]! It's clear that the 'musicality' aspect doesn't factor in (or is drowned out in this experiment) [111], and maybe sine waves are best at avoiding the musicality aspect [112]. Indeed, a minor third (for example) in the higher frequency range is supposedly more tonally consonant than a minor third in the lower frequencies [113][114]. Obviously, they have the same kind of interval quality (despite the rougher sound in the lower range), and that should be evidence enough to distinguish tonal consonance from the type of consonance associated with the twelve musical intervals. Terhardt seems to agree also, with his two-component model of musical consonance [115][116].

However, Plomp and Levelt then went on to use develop a model based on a harmonically rich tone, where all pairs of partials were tested for their dissonance. Based on six harmonics, the following graph resulted [117]:

Here we can see that some of the intervals in the chromatic scale are represented, although the relatively flat landscape between the minor third and major third seems to undermine any supposedly special properties of these intervals (the same goes for the major third and major fourth). However, other graphs dotted around the place have much more noticeable peaks and troughs [118] (whilst others still don’t seem to even distinguish the minor third [119]). Perhaps more importantly, many of the dark notes such as Db, F# or Ab aren’t noticeably represented in any of them.

"Rational consonance"

Throughout this dissertation, I will use the term "rationally consonant" (or dissonant) to refer to this phenomenon. Very roughly, it will refer to just intoned intervals, particularly the simpler ones [120]; however I am not necessarily claiming that such intervals exactly form the musical scale. As already said, such intervals will produce a droning, pure, or even ‘mechanized’ sound. If the just interval is part of the harmonic series of another fundamental pitch, then the JI interval may also fuse more easily into other timbres. Here is an excerpt which seem to embody the just intonation viewpoint:

" ...The difference is astonishing it seems almost impossible for two notes to clash, and it is quite possible for a bouquet of notes to blend, mix and tumble together in a way that is pleasing to the ear." [121].

[124]

Psychophysiologically, this is a "second order" effect, and the beats created by near rational intervals are called "second order" beats (also called "subjective beats"), which are the result of neural processing [122], and are in contrast to the (easier to measure) first order beats created by the physical ear as discussed in the Helmholtz and Plomp/Levelt section. If played dichotically, then unlike first order beating played dichotically, the beats are still noticeable [123].

"Chromatic consonance"

Often, traditional theory points to the idea of the twelve notes of Western tonal music as being derived from the harmonic series, or from the Plomp/Levelt results, or from basic ratios such as 6/5 or 5/3 (although modern psychoacoustical research takes a rather different stance in that the intervals may at least partially be learned [125] (also see later)). However, despite the somewhat close correlation with pure ratios, I contest that the twelve notes could well form another type of consonance, which I will refer to as "chromatic consonance" in this dissertation. Furthermore, these perceptual intervals may not be fixed, but could vary from person to person, or from time to time for a single person. I will give further justification for this theory later, but for now, I invite the reader to listen to these sounds:

1A: ET version - C, E, G, Bb - 1.0 (0 cents)..., 1.25992 (400c)..., 1.498 (700c)..., 1.7818 (1000c) - Saw-wave timbre (10 harmonics)
http://www.skytopia.com/project/scale/dom7-1A-ET.mp3

2A: JI version - C, E, G, Bb - 1.0 (0 cents)..., 1.2495 (385.6c)..., 1.499 (700.8c)..., 1.7485 (967.3c) - Saw-wave timbre (10 harmonics) - nigh-on pure for slight ‘phasing’ effect.
http://www.skytopia.com/project/scale/dom7-2A-JI.mp3

1B: ET version - C, Bb - 1.0 (0 cents)..., 1.7818 (1000c) - Saw-wave timbre (10 harmonics)
http://www.skytopia.com/project/scale/dom7-1B-ET.mp3

2B: JI version - C, Bb - 1.0 (0 cents)..., 1.7485 (967.3c) - Saw-wave timbre (10 harmonics)
http://www.skytopia.com/project/scale/dom7-2B-JI.mp3

In each case, the JI version has a much flatter minor seventh than the ET version (31 cents or over ¼ of a semitone!). However, to my ears, they both sound consonant in different ways. The ET version feels as if it can be more easily used for general tonality and melodical purposes (though the JI version isn’t necessarily forbidden from that). However, the JI version sounds more fusing, droning, and consonant in a completely different way. Therefore, the ET version is ‘chromatically consonant’, and the JI version is ‘rationally consonant’ (see previous section). In the same way, the major third can sound good both in JI and ET. However, it’s also conceivable that due to acculturation, the JI version of the major third (14 cents flatter than the ET M3rd), for some people may sound more chromatically consonant, as well as being more rationally consonant. Therefore, for this group, it is easy to see how one could conflate chromatic consonance with rational consonance. Interestingly, one study of barbershop singing has found a preference towards the ET interval for the major third, and the much flatter 7-limit JI interval (7/4 = 1.75) for the minor seventh [126].

Roughly though, we can use equal temperament as a guide to indicating "chromatic consonance". High spikes of consonance would appear at each of the equal tempered intervals, or close to them.

Since starting this dissertation, I’ve found in my research that something similar to ‘my’ idea has been suggested by Revesz (1912), and Idson & Massaro (1978). Coincidentally, they have used a similar term to my own - "Chroma", denoting the "c-ness", "d-ness", etc. of an interval [127] (also see [128]). Mieczyslaw Kolinski on the other hand uses the word "tint" and points to twelve of these tints in the Western musical scale [129]. Also, Arvindh Krishnaswamy hints at a hybrid tuning scheme, mentioning just intonation and equal temperament in the very same paragraph [63]. Finally, Terhardt may come closest to the theory, except he calls chromatic consonance - "musical consonance", and rational consonance - "psychoacoustic consonance" [130].

Other types of consonance

Other types of consonance include those relevant in a musical context, where certain intervals may also be consonant or dissonant according to surrounding chords and key, along with other factors. They will not be explored in this dissertation, apart from where such intervals directly influence the intonation (such as a sharper leading note).

Ear/brain idiosyncrasies which could affect intonation

Given below are phenomena one should take into account when comparing tuning systems.

Octave stretching

When octave stretching is spoken of, there are usually two kinds of stretching that could be referred to.

The first is how instruments such as the piano have many tones which contain naturally stretched (slightly wide) harmonics [131][132]. Because of this, the perceived pitch of the sound is somewhat higher than the physical fundamental would suggest. Therefore, it is a good idea to stretch all of the notes in the scale to compensate for the stretching for the lower notes. William Sethares’ research relates to this type of octave stretch, where experimentation has been done using increasingly stretched partials [143]. For this dissertation, I will ignore this type, since it will only apply to piano type timbres, rather than 'ideal' timbres.

The second type of octave stretching is what I'll refer to as "octave stretch" for this dissertation. It's the kind present even if the timbre is perfectly harmonic. It has been found through experiment that the ear usually ([133]) perceives the subjective, perceptual octave as slightly sharper than the physical octave. Therefore, one would need to stretch all of the intervals by approximately +10 to +20 cents per physical octave to compensate (around frequency^1.0125 for any arbitrary interval) [134][135][131][136][137] (pure tone intervals have somewhat greater deviations than even this [138], perhaps by about 35 cents [131] and as high as 50 cents per octave [139]). In the context of tuning comparison for less than an octave, it may not sound like very much, but it's still large enough to potentially muddy the waters when determining the 'sweetness' or 'consonance' of various intervals.

For real music, or even simple chords, the ideal degree of octave stretching depends on the overall spectral content [140], though a positive amount for orchestral performance does seem to be most common [88] (also see [142] (3 cents per octave)).

First/second order beating

See earlier ("Rational consonance"), and later ("Phenomena 2").

Pure versus complex tone

It's true that in many cases, the perceived pitch of a complex tone is slightly lower than the same pitch of a pure tone [144][145]. Whether this phenomenon affects the width of an interval is questionable. However, it's worth saying how real the effect can be, even in actual performance [146].

Difference tones [147]

These extra ‘pollution’ tones occur in the ear [148], but need high volume for any perceivable effect. They are "caused by a non-linear distortion of the primary wave from stimulus in the cochlea" [149]. The main difference tone can defined as:

New tone 1 = f2 - f1

Other lesser tones include:
New tone 2 = 2f1 - f2
New tone 3 = 3f1 - 2f2

As said, these won’t have much effect on testing intonation, at least not if the musical tones present are relatively quiet.

Aural harmonics

Even more than difference tones, these need to be played very loud, but pollution harmonics can be perceived, even from a single tone [149]. Thus if f1 is heard, then f2, f3, f4 etc. can also be perceived.

Features of additional tone

The perception of an interval is dependant upon the timbre and intensity of both components in the interval. If this is a factor as is claimed, then it’s easy to see how this could distort the original mathematical models [145].

Left versus right ear

It's negligible for most people, but the right and left ear may experience slightly different subjective pitches, given the same physical frequency. This is known as binaural diplacusis (or Diplacusis binauralis, or interaural pitch difference (IPD)) [150][151][145], and it occurs mostly with pure tones, and probably much less with complex tones [152].

Loud tone versus quiet tone

Volume too can affect the perceived pitch (second-order effect) of a tone [145]. Pure sine wave tones are more susceptible to deviation, and it's been reported that pitch can drop by as much as a minor third for some people. Pitches greater than 2 KHz will rise in pitch as the volume increases, and drop in pitch if originally lower than about 1 KHz [153].

Complex tones may exhibit a similar effect, but only to about a fifth of the degree [154], or not all [151] according to who you listen to.

Ear/brain idiosyncrasy summary

Apart from octave stretch and first/second order beating, in general, most of the above idiosyncrasies have little effect on the music’s intonation [151], partly because each effect is so small, but also because music tends to utilize complex rather than pure sine waves. Additionally, the categorical nature of pitch or interval perception will mean that the mind will often interpret the pitch according to the nearest 'expected' one [127][128][125][2]. According to the source, this may be one of the twelve notes of (or near) equal temperament, or perhaps take into account enharmonic subtleties [155] for an extra twelve or more notes (as is present in Pythagorean intonation or 5-limit just intonation).

Why are Just intervals often preferred?

So why do so many see equal temperament as imperfect, and just intonation as the more ‘natural’ scale? Is it because it really could be the basis of the musical intervals?

Naturally, other questions arise; can tuning systems from one period in history be used to adapt to the music from another period or culture? Are tunings specific to each person so that their favourite tuning can be used for any style of music? Many early-music enthusiasts today believe that music should be performed in the same tuning it was originally performed (in my opinion, ignoring the fact the original composer may have been 'mistaken' or acculturated to start off with).

Bach demonstrated the variety of intervals with his "Well-Tempered Clavier", which recent research (2005) has informed us was probably not intended for equal temperament after all [18]. Though we can by no means necessarily trust Bach's ear 100% (even for his own compositions), does his use of just intonation provide validation, or even preference over equal temperament, at least for the Baroque music genre?

Maybe, but before jumping to any conclusions, the preference of just intoned intervals by composers and theorists, past and present, may be explained by various phenomena, including, but not limited to:

Phenomena 1: First order beating of partials

One of the most commonly cited reasons is how the 'beats' (first or second order) found more often in equal tempered intervals indicate the partials do not properly coincide, and therefore represent the 'off-tuneness' that we want to avoid [156][47]. Perhaps the example which epitomizes first order beating most is the following experiment (also see [157]):

Sound 1a: Major third in just intonation (f1.0 and f1.25 (0 cents and 386 cents)) - first 5 harmonics.
http://www.skytopia.com/project/scale/JIthird-h5.mp3 - (download to hear)

Sound 1b: Major third in equal temperament (f1.0 and f1.25992 (0 cents and 400 cents)) - first 5 harmonics.
http://www.skytopia.com/project/scale/ETthird-h5.mp3 - (download to hear)

Compared to the JI version, the ET version has a slight tremolo effect (quiet-loud-quiet-loud).

However, if we exclude the fourth harmonic from the major third in both cases:

Sound 2a: Major third in just intonation (f1.0 and f1.25) - first 5 harmonics, excluding 4th harmonic in major third.
http://www.skytopia.com/project/scale/JIthird-h5-missing4.mp3 - (download to hear)

Sound 2b: Major third in equal temperament (f1.0 and f1.25992) - first 5 harmonics, excluding 4th harmonic in major third.
http://www.skytopia.com/project/scale/ETthird-h5-missing4.mp3 - (download to hear)

These two sound much more similar to each other than before! Therefore, it should be much easier to compare the raw pitch of the two dyads if we exclude the 4th harmonic of the M3rd in this way. Obviously, any comparison between equal tempered and just intervals to find the ‘superior’ tuning should take this effect into account. It’s interesting to note that the critical band and limit for pitch discrimination varies from person to person too [158], so that could affect the results too.

The tremolo effect in 1b is down to first order beating of the 5th harmonic of the tonic with the 4th harmonic of the major third. We can thank the limitations of the basilar membrane in the ear to cause this effect, as explained earlier (Helmholtz and Plomp/Levelt). To see more clearly the clash, observe the harmonic series

	Harmonic:	1	2	3	4	5	6	7	8
Tonic		1	2	3	4	5	6	7	8
ET M3rd		1.2599	2.5198	3.7797	5.0397	6.2996	7.5595	8.8194	10.0793
JI M3rd		1.25	2.5	3.75	5	6.25	7.5	8.75	10

As you can see, the 4th harmonic of the ET M3rd (5.0397) is very close to the 5th harmonic of the root (5.0), which creates the tremolo.

There are still subtle second order beating differences between 2a and 2b (creating the drone in the JI version), but these are much more subtle, and so it’s possible that any alleged ‘off-tuneness’ may be accommodated by the first order beating more than any second order beating (though this is not definite, and I don’t want to speak for others). If we really have to include the 4th harmonic of the major third, then the only 'solution' is to improve the design of our ear (!) to allow finer graduations of basilar membrane activation areas, or alternatively, by pitching each tone separately in the left ear and right ear, where the tremolo effect completely disappears [98].

Personally, I prefer the ET major third in one way (melodically, tonally, more major-3rd-ally), but it goes without saying that the Just third sounds more like a fused sound (therefore being more consonant in another way). On top of tonal consonance (associated with Plomp/Levelt), these two major types of consonance form my 'own' theory (Phenomena 2). See Phenomena 3 to see how the first consonance type ("Chromatic") could vary from person to person though.

Phenomena 2: Two types of consonance.

This phenomena addresses Argument 1 as pointed out near the beginning of the dissertation.
As demonstrated above with sounds 1a and 1b (and even sounds 2a and 2b), simple number ratios of partials create consonance of the second order. This correlates to the "rational consonance" type as defined earlier in this dissertation. Just intoned intervals may always be more rationally consonant, but not necessarily more "chromatically consonant", where such a measure is calculated according to the fundamentals (perceived pitch) only. In other words, I think the (lack of the) number and strength of the 'beating' of the partials does not look at the important factor of Chromatic consonance, even though that they provide timbres which more easily fuse, or create droning or pure intervals.

Evidence for this theory can be summarized as follows:

a: Listen to the sounds detailed in Phenomena 1 or the pure minor seventh (4:7) earlier in this dissertation under "Chromatic consonance". The pure just ratio is not just less (or more) on tune than the equal tempered version - it also has an extra quality of 'drone-ness' to it due to the periodicity of the partials (lack of beating). Perhaps this is a separate quality than the ‘on-tuneness’ of an interval (for the major second, major third, perfect fourth etc.).
b: The lack of 'beats', and periodical pattern of coinciding wavelengths both imply that two tones would need to be played simultaneously (i.e. as a chord) for the perceptual feeling of harmony. However, as we all know, even if these intervals are played linearly (as a melody, or broken chord), the feeling of the interval is still omnipresent. This is despite the fact that no periodical pattern, or any form of timbre interaction of the two tones is taking place.
c: The sensation of a true 'octave' is slightly higher than what just intonation (or equal temperament) would predict, despite the physical interval being the simplest ratio of 2:1 [159]. Something like 2.01:1 may be preferable instead. However, if this ‘true perceptual octave’ interval is played simultaneously, obviously unwanted beats occur [146]. Again, this indicates the two types of consonance.
d: Revesz (1912), Idson & Massaro (1978) have also suggested the use of the term ‘chroma’ to represent a "certain musical-categorical value (chroma), e.g. "c-ness", "d-ness", etc." [127][128].
e: According to a study by Lawrence Borden, equal tempered thirds were preferable in the lower range, and just intoned major thirds were preferable in the higher range [160]. Is it possible that in the higher range, the M3rd combined with the root was treated more as creating a fusing sound (rational consonance), whilst in the lower range, the ear finds it much harder for them to fuse, and thus the sense of chromatic consonance becomes prominent instead?

Along with my own subjective dual perception of the interval (see heading: "Chromatic consonance"), none of these points conclusively prove there are two consonance types instead of one, but it's surely a possibility.

Phenomena 3: People may hear in their 'mind's eye' a different interval to someone else.

This phenomena addresses Argument 2 as pointed out near the beginning of the dissertation.
Phenomena 1 and Phenomena 2 may account for some of the reason why just intoned intervals may be preferred by many people, but it's hard to imagine this accounts for every case. Alternatively (or additionally), the reason why some may prefer the just intoned interval, and some the equal tempered interval - is that they are really hearing the same thing in their 'mind's eye'! In this sense, the musical intervals are 'learned', with certain perceptual intervals mapping to certain physical intervals (Helmholtz would probably frown at the idea, since he believed the "natural intervals are only natural for uncorrupted ears" [161]). Anyway, if this is the case, then it can take one of at least two forms:

1: They literally hear the interval differently (apart from beating effects). For example, someone who hears a dim. fourth, may hear it as a fifth. To provide a more subtle example, an interval of 390 cents could be heard as 400 cents (or vice versa).

2: They hear the same 'pitch height difference', so in one sense, they are still hearing a '390 cents' interval. However, if we assume that the brain abstractly maps to 12 intervals, and that these intervals are being represented by different neuron clusters in the brain, then it may be the case that they are receiving no minor 3rd neuron pollution, and only major third neurons are being excited. This is in contrast to people like myself, where the 390 cents interval would excite not many, but a few neurons for the minor third as well as the major third, and thus cause interval ambiguity.

In both of the cases above, the timbre (or feeling of the beats created by an interval) would more or less remain the same for all people. However, as stated in Phenomena 2, I count this is a separate type of consonance to the consonance type created by the 12 intervals.

Obviously, all of this doesn't necessarily mean there's something special about any particular number (such as 1.25992 (2^4/12) or 1.25 for the major third). The thing that would be special is the major third sensation in the mind's eye. It's just that different people would need different input from the outside world to activate this interval as sweetly/exactly as possible inside their mind.

We can be by no means be certain that people hear different intervals - given the same physical frequency. However, evidence for the above is as follows:

a: If people can hear a different subjective pitch in each ear (diplacusis), then it follows that the sense of pitch is not absolute for one person, let alone different people. This phenomenon may extrapolate to intervals, and thus to the perception of how sweet they are for various people.
b: There is evidence to suggest that the musical intervals are learned, rather than innate. In fact, Cazden claims that the "individual judgment of consonance can be modified by training, and so cannot be due entirely to natural causes" [162] (also see [163][164]). If this is indeed true, perhaps someone who initially prefers the sound of an equal tempered third can get used to the just intoned major third (and vice versa).
c: Supporting b, my own subjective experience shows that the sweetness of an interval (such as the major third) can subjectively change from one moment to the next. In other words, it may sound better in ET after accustomization over a short period (5 seconds), where straight afterwards, JI sounds ‘off-tune’. The reverse happens too; I can accustom myself to the JI third, and then the ET version will sound off-tune.
d: Anecdotal evidence this time, but following an email conversation in 2003, I quote:

"We spent 2 or 3 weeks at the Oberlin Baroque Performance Institute. We were surrounded by historic temperaments and awash in beautifully pure intervals and chords. When we left, we went directly to the Aspen Choral Institute, and on our first evening in Aspen we went to hear the Chamber Orchestra, an orchestra made up of some of the finest young players in the country.
When our ears, which had been purified at Oberlin, heard that orchestra, we heard it as being terribly out of tune and could hardly stand it. Our ears readapted quickly, of course, which must prove something, and we no longer heard ET and its false intervals as out of tune, but it was a very illuminating experience. And ever since, I have enjoyed acoustically pure playing much more than ET playing." (John Howell).

If it is possible to recondition oneself like this, it follows that people could - given enough time - get semi-permanently conditioned to appreciate different physical intervals to maximize perceptional sweetness. It would be interesting to see what he would now make of the equal tempered major third minus the 4th harmonic.

Phenomena 4: Instruments such as the organ can exhibit acoustic distortion if tuned to ET [165][166][167?] (also see [34][168]).

This means that in this case, the reason why many might tune the organ to just intonation or meantone is due to inherent acoustical/reverberation problems, rather than an inherent flaw in equal temperament itself! The same problem happens to a greater extent with the harmonica and harmonium [166]. Personally I prefer the sound of equal temperament in general, but from experience, I know that the harmonica will often sound ‘fluttery’ if it's tuned to equal temperament (one can imagine playing each note on a separate harmonica, or a hypothetical 'superior' model of the harmonica that produces the same sound, but doesn't suffer the problems when tuned to equal temperament).

Phenomena 5: Just intonation may allow for 'key colour' [169][170][171][172].

But in this case, equal temperament (or Pythagorean intonation) may still be the blue print from which to detune from. It's true that the pitches of just intonation could may create different moods for each key because there are different sized spacings between the intervals according to the key. See Appendix 3 for a breakdown of these intervals.

Phenomena 6: Easier to play/sing just intonation

Just because a just intonated interval is easier for performers to ‘lock’ on to, that doesn’t necessarily make it better sounding. If this is true, then experiments should be based on the perceived effect on the listener, rather than necessarily what happens in a performance. Even Helmholtz said: "correct singing by natural intervals is much easier than singing in tempered intervals" [173]. Other than that, I have little evidence for this idea - it’s just a possibility. But one should try to eliminate as many variables as possible.

Phenomena 7: ‘Nice looking’ numbers

Interval comparison should be based on actual hearing tests, rather than solely on numerological assertions. Pitches represented by just intoned numbers like 1.5 and 1.3333 may 'look' better than irrational numbers like 1.498307076877 or 1.33483985417. It goes without saying that many important numbers in mathematics (such as PI) are irrational.

Phenomena 8: ‘Pull’ of notes

One other factor we’ve ignored up until now is the pull of the tonic on its leading note, which has also been shown by various studies [174][175][176][177]. Could this be explained by Pythagorean intonation, or maybe the stretch of the scale? It’s probably not down to whole-range or single-octave scale stretch, because the tonic is optionally only one semitone away from the leading tone. Also, a minor seventh leading down to the submediant is actually flattened due to the supposed ‘pull’ of the submediant, and a minor second is reported to be flatter than what equal temperament would suggest.

The above three examples hint at Pythagorean intonation though, since (assuming a key of C), Bb is lower down the series than A, and B is higher up the series of fifths than C, and Db is lower down the series of fifths than C. The immediate question arises though - how about if the progression is from B to Bb, or from Bb to B, or from Db to D? If the ‘pull’ acts anything like the previous examples, then it can’t be explained by Pythagorean intonation, and must instead be another variable to take into account on top of whatever tuning is supposed to be the right one.

Other problems with Pythagorean intonation may exist too. It could be argued that some of the 'wolf' is still retained in the intervals - the most noticeable being the interval from C to F#/Gb. In other words, if C to B# is a wolf interval (12 fifths away, and 23.5 cents sharp of 12-et), and C to A# has 'most of the wolf' (10 fifths away, and 19.5 cents sharp from 12-et), then it stands to reason that the interval from C to F# (or Gb) in Pythagorean tuning is a "half-wolf" interval (F# or Gb being six fifths away, and 12 cents sharp or flat of the 12-et equivalent).

Before we leave Pythagorean intonation, here is a piece which may demonstrate the idea of tuning flats lower than sharps. Apparently, the Gb in the first bar may sound more appropriate if it’s higher than the F# in the second bar [198]. Obviously, just intoned scales can do something similar.

Phenomena 9: Larger tolerance level for some listeners

Many may prefer the just intoned intervals simply because they can't really distinguish between the fine differences between equal and just intonation. However, it's relatively easy to hear the beating effect, so they go by that instead, and assume that's the sole measure of consonance. This phenomena assumes that there are two types of consonance as explained in Phenomena 2.

Summary

Because of these phenomena, it’s possible (but probably naive) to speculate that just intonation enthusiasts may like equal temperament after all. Most likely, the preference of simple just intoned intervals will still hold for many people, and that for this group, the equal tempered intervals would still sound worse in every way - even when appropriate harmonics are omitted to avoid first order beating. In which case, Phenomena 3 may be true, but if that also turns out to be false, and it's a big if, then 5-limit just intoned intervals could be the basis of the scale. Pythagorean intonation may also form the scale, but the resulting waveform is often far less periodic than 5-limit versions, and thus doesn’t have many of the pure intervals that 5-limit JI possesses. Also, because Pythagorean pitches are spaced apart equally (apart from the wolf obviously, but let’s assume a spiral of fifths, not a circle), and the major third is even sharper than the ET third, these properties actually bring Pythagorean intonation closer to equal temperament than to just intonation conceptually.

Now may be appropriate to detail many of the studies to compare intonation experimentally.

Psychoacoustical / empirical studies

To further help make up our minds, I now present around 20 studies gathered from many sources, which try to determine whether listeners and performers show any inclination towards a particular tuning.

1875: Just intonation preference - "'The Sensations of Tone', Eng. trans. of 1875, p. 506
"The performers of the highest rank to really play in just intonation has been directly proved by the very interesting and exact results of Delezenne." " - Helmholtz

1937: Pythagorean preference - "String players, both in solo performance and in ensemble, tend towards the Pythagorean intervals rather than the just intervals" [60][178].

1960: Pythagorean or ET preference? - "It has been found that choral groups sing the major thirds sharp and the minor thirds flat, contrary to the opinions of those who claim the good choral groups sing in just intonation." [60].

1963: Pseudo-Just intonation preference - "Boomsliter and Creel (1963) found that, within small groups of melodic notes, simple ratios are preferred, although the `reference' point may vary as the melody proceeds." [179]

1975: Not just intonation - "A reciprocal effect exists: just melodic intervals are consistently judged to sound flat (Terhardt and Zick, 1975)." [180]

1976: Equal temperament preference - "an investigation in which 17 musically experienced subjects were asked to adjust the frequency of a variable synthetic sung tone so that it constituted certain dyads with another synthetic sung tone." [181]

1980: Equal temperament (M3rd), and harmonic 7 = 4:7 (7-limit JI) preference (Min7th) - "two barbershop quartets, repeatedly rendering a cadence a great number of times. (Measurements by Hagerman and Sundberg, 1980.)" [126]

1986 - Equal temperament preference - "Moreover, professional musicians appear to prefer equally tempered intervals to their just counterparts. See the results of Vos (1986)." [128]

1989-1991: Equal temperament followed by just intonation - "...rule Mixed intonation (Sundberg et al. 1989; Friberg, 1991). Here each new note is first intonated following the melodic rule. Then the intonation is slowly changed towards the harmonic intonation with the root of the current chord as the reference. In this way relatively short notes are intonated melodically while relatively long notes will gradually move towards less audible beats within the current chord." [182]

1990: No simple answer - (Rakowski [179])

1993: No simple answer - (Loosen [179])

1995: Equal temperament AND Just intonation preference - "Fyk (1995) makes a division of intonation in diatonic music into four classes, based on a number of experiments with violinists: 1. Harmonic tuning. Many of the deviations from equal temperament take place in trying to tune in just intonation (small integer ratios) with the underlying harmony." [183]

1995: Just intonation preference - " 3. Corrective tuning. When a performer perceives a small deviation in playing a melody, he/she corrects it by adjusting the note itself or the note that follows immediately. The former kind of adjustment occurs frequently in string quartet playing in order to create a given chord in just intonation." [183]

1998: Equal temperament preference - "Subjects (N = 16) were experienced wind instrumentalists (8 professionals, 8 advanced students). Subjects recorded a duet, first playing the melody with a synthesized harmony line, and then vice versa. Target intervals were analysed, converted to cent distance, and compared. Results indicated that deviation was greatest when compared to just tuning and least when compared to equal tempered tuning." [184].

1999: Not just intonation - "However, others have concluded in contrast that there is no evidence that performers tend to play intervals corresponding to small integer ratios for either melodic or harmonic situations (Burns 1999)." [179]

1999: Pseudo equal temperament preference - "There is a tendency for small intervals to be tuned smaller and for large intervals to be tuned larger than equal temperament (Burns 1999)." [179]

2003: No simple answer - (Kopiez [179])

????: Between equal temperament and Pythagorean intonation - "tuning of melodic intervals adjusted by a professional violinist." [185].

????: Equal temperament (lower range) and Just intonation (higher range) - "Instead, preference for temperament was found to be directly related to pitch height. Preference for Equal Temperament was found in the lowest tested octave and a preference for Just Intonation was found in the highest tested octave." [160].

Summary

As can be seen, results are as mixed as a Haitian cocktail. No particular tuning stands out, but it’s possible that there could be certain variables which may explain the somewhat contradictory results. Some of these are detailed in the "Why are Just intervals often preferred?" section earlier. For example, it’s possible of course that the disagreement could at least be partially explained by the two different types of consonance detailed earlier (or three if we include ‘tonal consonance’). In which case, the varying instrument types could lend more to one type of consonance than another.

It's also interesting to note from the studies that just intonation may have no more claim to the basis of the scale than equal temperament or Pythagorean intonation. Therefore, at least for chromatic consonance, it is misleading to claim that equal temperament is necessarily more of a 'compromise' than Just intonation. As Simon Stevin proposed, it may be Just Intonation, after all, which is the 'compromise'.

Final conclusions

Thus ends our journey through the mine-ridden multi-dimensional maze of tuning. If there’s anything we’ve found out, it’s that:

The history points to ambiguous tuning results

The mathematics points to ambiguous tuning results

The experimental studies point to ambiguous tuning results

Famous theorists’ preferences point to ambiguous tuning results

It also seems clear that intonation depends on many factors including:

Listener

Instrument idiosyncrasies (organ or harmonica for example)

Octave stretch (15 cents per octave, or less for real music)

Timbre 1a - First order beating of partials (some may find them unpleasant)

Timbre 1b - Second order beating of partials (some may find them unpleasant)

Timbre 2 (single harmonic (fundamental) - higher, or many harmonics (complex tone) - lower)

Timbre 3 (harmonic or inharmonic/stretched)

Timbre 4 (pitch salience; noisy or pure)

Context in music? ('pull' of notes - leading tone etc.)

Consonance type aimed for? ('chromatic consonance', or 'rational consonance')

Acculturation?

Type of music?

Mood of listener?

Volume of sound? (Affects pitch, but does it affect interval?)

At the moment, I still think the model which best explains the diverse range of tuning results would be one containing two consonance types - "chromatic consonance" and "rational consonance", where the former is partially determined by acculturation (Phenomena 3 or Argument 2), and the latter applies universally. The model would also take into account octave stretch, and maybe note ‘pulling’ effects and/or sharper sharps and flatter flats.

A more 'natural' tuning?

This addresses Argument 3 stated near the beginning of the dissertation. Notice the word 'partially' in the last paragraph's sentence: "where the former is partially determined by acculturation". Despite the idea that two different physical intervals could sound the same to the mind's eye for two different people, there's still the chance that one tuning may be easier to get used to, and perhaps therefore in a sense provides the 'truer' intervals for chromatic consonance. An analogy could be likened to riding a bike with the handle bars turned by 30 degrees. Over time, the mind compensates, and then returning to 'normal' positioning of the handle bars would feel strange (for a short while). However, riding a bike with the handle bars twisted by 30 degrees may include costs, such as taking longer to get used to, and may ultimately provide slightly less accurate control of the bike.

As an extreme example, one may try to recondition oneself to prefer the perfect fifth at 40 cents flatter than the ET or JI tuned fifth. As one might imagine, this would take a long time to condition anyone to! It therefore follows that even if it's by fractional amounts, perhaps the just intoned interval is quicker to condition people to than the other models on average. Or maybe it's the ET or Pythagorean models that would work out quicker - who knows. In summary, there may still be ideal pitch values for the twelve intervals, whether they are JI, ET, Pythagorean or whatever else.

Future experimentation

Ultimately, I would focus on two areas of study - octave stretch, and model preference for intervals. For both studies, I would propose the following properties:

1: Test on hundreds of subjects.
2: Ask listeners to move a semi-large, long, accurate, weighted, sensitive and accurate slider with their hand until they think it sounds perfectly on tune. Each experiment would be repeated say, 20 times.
3: Use synthetically produced complex tones (around 10 harmonics) to reduce variables. The first type would be similar to a saw wave, and the second type would be similar to a pulse wave (for full harmonic content).
4: Include versions without and also with a little vibrato so as to put the emphasis off the timbre, and concentrate on the raw pitch.
5: Experiment across the whole pitch range from low frequencies to high.

The first area of study would look for the preferred amounts of octave stretch for each person. In each interval, I would test the exact preferred interval width for each of the following intervals (the note in [*] represents the note that the listener would adjust to suit their preference).

1: C1+[C2], C1+[C3] and C1+[C4] separately (octave, double octave, and triple octave).
2: C1+[E1], C1+[E2], C1+[E3].
3: C0+G0+C1+[C2], C0+G0+C1+[E2], C0+G0+C1+[C3], C0+G0+C1+[E3] (...test to see if spectral content below the bass note affects the interval being tested for.)
4: C1+C2+[C3], C1+F1+G1+Bb1+C2+[C3] (the chosen intervals have been picked to maximize agreement between the mathematical tuning models so as to reduce muddying variables).
5: Repeat tests 2 and 3 but replacing the major third in each case with the minor third.

Unfortunately, it's tricky to tell in chords with more than two notes whether the listener may be adjusting the note relative to the bass note, or rather to any one of the components (particularly the nearest pitch to the pitch being tested for). This is why tests 3 and 4 are so important.

The second area of study would focus on the intervals themselves, so we can look for possible mathematical models:

1: For the major third interval, test with and without the 4th (and maybe 1st) harmonic in the third to see if it makes a difference.
2: Experiment on the major and minor third, fifth, major and minor sixth, major and minor seventh. All of those again, but invert the interval to see if that makes a difference.
3: Find the most consistent people, and highlight their results as being potentially more valid data.

It would be interesting to see to what degree tuning preference relates with preferred degrees of octave stretch. For example, if the 'ideal' model is equal temperament, but the amount of octave stretch is 15 cents, then for the major third, this would falsely indicate a preference closer to Pythagorean intonation. On the other hand, a false indication towards equal temperament may occur for the minor third, because octave stretch (about ^1.0125) could raise the interval from the Pythagorean minor third (1.1852) to near the equal tempered version (1.1892).

Only after basic research for intervals is complete would it be wise to test intonation for music and/or real instruments, both of which introduce many more variables.

Endnotes:

[1] http://www.music-cog.ohio-state.edu/Music829B/tuning.html : "First, in the 1930s, Carl Seashore measured the pitch accuracy of real performers and showed that singers and violinists are remarkably inaccurate. For non-fixed-pitch instruments, the pitch accuracy is on the order of 25 cents. Yet Western listeners (and musicians) are not noticeably disturbed by the pitch intonation of professional performers."
[2] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p177.
[3] Dave Benson, "Music: A Mathematical Offering" (2006), p138 (available online).
[4] Dave Benson, "Music: A Mathematical Offering" (2006), p139-140 (available online).
[5] Robert Asmussen, "Periodicity of sinusoidal frequencies as a basis for the analysis of Baroque and Classical harmony: a computer based study", paper (2001), p8.
[6] "The Dictionary of the History of Ideas", Volume 3, p262.
[7] James Murray Barbour, "Tuning and Temperament: A historical Survey", Da Capo Press, New York (1972) ; originally published: East Lansing (1951), p4
[8] Walter Piston and Mark Devoto, "Harmony", Victor Gollancz Ltd (1978), p451: "At the time of JS Bach's first compositions (about 1703), 12et had not achieved wide acceptance. Music at the time had at most two or three flats or sharps in the key signature...."
[9] James Murray Barbour, "Tuning and Temperament: A historical Survey", Da Capo Press, New York (1972) ; originally published: East Lansing (1951), p5-6
[10] Stuart Isacoff, "Temperament: How Music Became a Battleground for the Great Minds of Western Civilization", Alfred A. Knopf, a division of Random House, Inc., New York (2001), p101
[11] Paul Guy, "A (very) abbreviated history of tuning theory, 550 BCE - 1999 CE" http://www.guyguitars.com/eng/handbook/Tuning/history.html "It seems likely that makers of fretted instruments had used some similar formula long before Galilei revealed the secret to the whole world, as many 15th century authorities stated with confidence that the fretted instruments - lutes, viols, etc - "had always used equal temperament."
[12] Fritz A. Kuttner, "Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory", Ethnomusicology, Vol. 19, No. 2. (May, 1975), p174: "the conclusion is that a temperament based on slightly lowered fifths must have existed in Europe decades earlier than 1496. This may easily have been equal temperament (with its fifths lowered by 2 cents), an assumption well supported by..."
[13] James Murray Barbour, "Tuning and Temperament: A historical Survey", Da Capo Press, New York (1972) ; originally published: East Lansing (1951), p25
[14] J. Murray Barbour, "Irregular Systems of Temperament", Journal of the American Musicological Society, Vol. 1, No. 3. (Autumn, 1948), p20: Pythagorean intonation used universally in middle ages
[15] Patrizio Barbieri, "Violin intonation: a historical survey", p71: "Francesco Galeazzi--who spent many years in Rome as a violinist at the Teatro Valle--stated in 1791 that the best performers changed the position of the major and minor tones according to the key of the composition. He included a fingering chart containing such alterations, by commas. Furthermore, he suggests that his chart would make the less precise players laugh heartily, especially the ordinary ones who play merely for the practise."
[16] J. Murray Barbour, "Just Intonation Confuted", Music & Letters, Vol. 19, No. 1. (Jan., 1938), p59. Also: "After the time of Fogliano, no outstanding sixteenth-century theorist mentioned just intonation as theory, who did not also discuss temperament as tuning practise--equal temperament or fretted instruments and some variety of mean-tone tuning for keyboard instruments."
[17] J. Murray Barbour, "Just Intonation Confuted", Music & Letters, Vol. 19, No. 1. (Jan., 1938), p48: "Choral societies and string quartets <in general>, are supposed to use just intervals, thus interpreting music as intended by the composer. Such a conception is essentially false. There is no system of tuning that has the virtues popularly ascribed to just intonation. Neither singers nor violinists use just intonation. Furthermore, as it is usually defined, just intonation is a very limited, cumbersome and unsatisfactory tuning system. If these statements seem startling, it may be well to examine the evidence."
[18] Bradley Lehman, "Bach‘s Extraordinary Temperament: Our Rosetta Stone", Early Music 33 (2005), p3-23, p211-231. Selection of responses (some for, some against) appears in p545-548.
[19] Patrizio Barbieri, "Violin intonation: a historical survey", p74: "In conclusion, we cannot state that Baroque violinists played strictly with 'just' or mean-tone intonation, but we can at least be sure that they used a tuning of the syntonic-mean-tone-type."
[20] Patrizio Barbieri, "Violin intonation: a historical survey", p74: "Vincenzo Galilei had already noticed that major 3rds were sung at least approximately in just intonation: his statement is fully reliable, because as a reference instrument he had adopted the lute, whose equal-tempered major 3rds were much larger than syntonic ones."
[21] J. Murray Barbour, "Irregular Systems of Temperament", Journal of the American Musicological Society, Vol. 1, No. 3. (Autumn, 1948), p20. "<just intonation was> perhaps used to limited extent in choral music in 1500s."
[22] Violin intonation: a historical survey, p69: "They show that violinists of all schools, at least until the middle of the 18th century, played in just or in mean-tone intonation; moreover, the Italians, especially during Corelli's time, enjoyed playing quarter-tones."
[23] Patrizio Barbieri, "Violin intonation: a historical survey", p72
[24] J. Murray Barbour, "Just Intonation Confuted", Music & Letters, Vol. 19, No. 1. (Jan., 1938), p52: "Although Helmholtz was ready to admit that ears might be corrupted by contact with equal temperament, N. Lindsay Norden, contemporary Philadelphia organist and choirmaster, admits no such possibility: "As we shall show, no singer can sing a cappella in any temperament.... A cappella music, therefore, is always sung in just or untempered intonation". Forty odd years ago, however, Max Plank, later to win world fame for his quantum theory, voiced the opinion that, with negligible exceptions, all the unaccompanied choirs he has observed in Berlin used equal temperament."
[25] Alexander Wood, "The Physics of Music", Methuen & Co. Ltd (1944), p193.
[26] Violin intonation: a historical survey, p69: "Because it poses many practical problems in performance, some scholars believe that just intonation is only a myth; however, such an opinion contradicts much historical evidence."
[27] Ruth Franklin, a review of: "Temperament: The Idea That Solved Music's Greatest Riddle" (2001), http://www.powells.com/review/2001_12_13.html
"Galilei published a treatise in 1589 attacking almost everything his teacher held dear:
"Such contortions are nonsense, Galilei countered. All scales are man-made, with no basis in nature whatsoever. Indeed, just intonation is itself only an ideal; in practice singers automatically temper their intervals for the sake of overall harmony. Zarlino responded by calling Galilei's treatise "an assault on God's plan." "
[28] Llewelyn S Lloyd, "The myth of equal temperament", Music & Letters, Vol. 21, No. 4. (Oct., 1940), p357: "Do you know that Spohr maintains that the singer should learn intonation from a piano in equal temperament?--!--?--:--; what marks of admiration shall I use? The fitting exclamations have yet to be invented.... And why should the singer cultivate temperament?... Thanks to the indestructibility of natural organization it cannot be learnt."
[29] Patrizio Barbieri, "Violin intonation: a historical survey" (1991), p87: "In any case, even in the middle of the 19th century the struggle between syntonic and Pythagorean had not completely faded, as a report of Delezenne clearly shows; <Ch.-E -J. Delezenne, 'Sur les prmcipes fondamentaux de la musique', Memoires de la Societe des sciences, de I'argiculture et des arts de Lille, xxvi (1848), p39-128. See also the separately published version of the same memoir (Lille, 1848), p31> in 1869 Cornu and Mercadier checked experimentally that syntonic was preferred in harmonic contexts and Pythagorean in melodic ones."
[30] Review author: J. Murray Barbour, Notes, 2nd Ser., Vol. 22, No. 2. (Winter, 1965 - Winter, 1966), p897-898. Book reviewed: "Intervals, Scales and Temperaments", Ll. S. Lloyd; Hugh Boyle. "However, in rejecting equal temperament, Lloyd would substitute for it his own just (or, as he would call it, "true") system, a system which is wholly rigid and which skilled choral singers and string-players do not use, despite Lloyd's assertions and those of other belated followers of Helmholtz, such as Stanford and Tovey"
[31] Ross W. Duffin, "How Equal Temperament Ruined Harmony", W. W. Norton & Company (2007), p16: "Not long ago someone wrote a popular book on the history of musical temperament and concluded that Rameau discovered equal temperament (ET) in 1737, and basically we all lived happily ever after <Stuart Isacoff, "Temperament: The idea That Solved Music’s Greatest Riddle", (New York, 2001)> "
[32] Stuart Isacoff, "Temperament: How Music Became a Battleground for the Great Minds of Western Civilization", Alfred A. Knopf, a division of Random House, Inc., New York (2001), p18
[33] Llewelyn S Lloyd, "The myth of equal temperament", Music & Letters, Vol. 21, No. 4. (Oct., 1940), p347.
[34] Peter Williams, "Equal Temperament and the English Organ, 1675-1825", Acta Musicologica, Vol. 40, Fasc. 1. (Jan. - Mar., 1968), p64-65: "It is astonishing that people felt so strongly against Equal Temperament to try to realize these impractical <more than 12 note per octave organ> systems, especially since it was so necessary to the music on the early 1800s. At the same time, sympathy can be felt for those idealists who hated sharp thirds and flat fifths on the sustained tones of the organs; their sensitivity was not mere English conservatism."
[35] "Sol-Fa - The Key to Temperament" http://www.bbc.co.uk/dna/h2g2/A1339076
[36] Peter Williams, "Equal Temperament and the English Organ, 1675-1825", Acta Musicologica, Vol. 40, Fasc. 1. (Jan. - Mar., 1968), p61: "Organ tuning habits around 1814 are uncertain". p62: "Cathedrals such as Lichfield maintained unequally tuned organs until the last quarter of the nineteenth century."
[37] Ross W. Duffin, "How Equal Temperament Ruined Harmony", W. W. Norton & Company (2007), p138: "<specialists> are convinced that ET "took over" around 1800. In fact, as I hope I‘ve demonstrated, ET did take over, but not until around 1917. Before that there was lip service to ET as a standard from about 1850".
[38] Stuart Isacoff, "Temperament: How Music Became a Battleground for the Great Minds of Western Civilization", Alfred A. Knopf, a division of Random House, Inc., New York (2001), p5
[39] Suzannah Clark, Alexander Rehding, "Music Theory and Natural Order from the Renaissance to the Early Twentieth Century", Cambridge University Press (2001), p24
[40] Stuart Isacoff, "Temperament: How Music Became a Battleground for the Great Minds of Western Civilization", Alfred A. Knopf, a division of Random House, Inc., New York (2001), p224
[41] "Sol-Fa - The Key to Temperament" http://www.bbc.co.uk/dna/h2g2/A1339076
[42] James Murray Barbour, "Tuning and Temperament: A historical Survey", Da Capo Press, New York (1972) ; originally published: East Lansing (1951), p11
[43] Llewelyn S Lloyd, "The myth of equal temperament", Music & Letters, Vol. 21, No. 4. (Oct., 1940), p348
[44] Robert Asmussen, "Periodicity of sinusoidal frequencies as a basis for the analysis of Baroque and Classical harmony: a computer based study", paper (2001), p2
[45] Walter Piston and Mark Devoto, "Harmony", Victor Gollancz Ltd (1978), p451: "ET made every interval (except the octave) out of tune by an equal but tolerable proportion"
[46] David M. Howard and James Angus, "Acoustics and Psychoacoustics - Second edition", Focal Press (2001), p114-115.
[47] Arthur H. Benade, "Fundamentals of Musical Acoustics", Oxford University Press (1976), p273: "Everyone notices the resulting beats, and all the musicians in the group will say that an out-of-tune (sharp) major third is being sounded".
[48] John Backus, "The Acoustical Foundations of Music", W. W. Norton & Company, Inc (1969), p128.
[49] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p176.
[50] Charles A. Culver, "Musical Acoustics - fourth edition", McGraw-Hill Book Company, Inc. (1956), p137.
[51] John Backus, "The Acoustical Foundations of Music", W. W. Norton & Company, Inc (1969), p127: "Because of these difficulties, the just scale has never been of any practical use. Its theoretical attraction to individuals with numerological inclinations is extremely strong, however, so much so that it has even been called the "natural" scale, as though it had some fundamental basis in nature not possessed by other scales. It appears in practically every book dealing with the acoustics of music, where it has been given an emphasis it does not deserve." …….p130: "<Helmholtz's> authority prevailed for some time; even today, many musicians believe that good string players performing by themselves, without the piano, play the just intervals rather than the tempered."
[52] James Murray Barbour, "Tuning and Temperament: A historical Survey", Da Capo Press, New York (1972) ; originally published: East Lansing (1951), p7
[53] Suzannah Clark, Alexander Rehding, "Music Theory and Natural Order from the Renaissance to the Early Twentieth Century", Cambridge University Press (2001), p24.
[54] Stuart Isacoff, "Temperament: How Music Became a Battleground for the Great Minds of Western Civilization", Alfred A. Knopf, a division of Random House, Inc., New York (2001), p230.
[55] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p13: "Explanations of sensory consonance are concerned with the fact that common musical intervals correspond - at least in Western, Indian, Chinese and Arab-Persian music (see Burns 1999) - to relatively simple ratios of frequencies, although some of these explanations do not require exact integer tunings, only approximations."
[56] Arvindh Krishnaswamy, "On the Twelve Basic Intervals in South Indian Classical Music", (2003), p2: "Recently we argued that there are only 12 distinct constant-pitch intervals used in present-day Carnatic music, and similar to Western music, the 12 main tones seem to be roughly semitonal in nature. However, we also noted that pitch inflexions, which are an integral part of this music system, may lead to the perception of additional "microtonal" intervals. But even in the case of inflected notes, the 12 basic intervals serve as anchor points, which makes them all the more important."
[57] Marc D Hauser & Josh MdDermott, "The evolution of the music faculty: a comparative perspective", Nature Neuroscience - Volume 6 - Number 7 (July, 2003), p663-668 http://www.nature.com/cgi-taf/DynaPage.taf?file=/neuro/journal/v6/n7/full/nn. Monkeys seem to favour the intervals of the diatonic scale.
[58] Arthur H. Benade, "Fundamentals of Musical Acoustics", Oxford University Press (1976), p297: "Listeners brought up in the European tradition of music often get the impression that Indian music abounds in all sorts of microtonal intervals, although the notation system would tend to show the lack of substantive existence of such notes. (It is true that certain notes in certain ragas are to be played slightly sharp of flat to increase the emotional effect or the mood of the raga. However, in present-day musical usage such alternatives are becoming rare.)"
[59] Johan Sundberg, "The Science of Musical Sounds", Academic Press, Inc (1991), p96. Notable exceptions include the 7-tone Pélog and 5-tone Sléndro scales::
7-tone Pélog scale: 250, 370, 520, 790, 940, 1055, 1220 cents
5-tone Sléndro scale A: 240, 480, 720, 960, 1200 cents
5-tone Sléndro scale B: 240, 500, 755, 1010, 1200 cents
[60] John Backus, "The Acoustical Foundations of Music", W. W. Norton & Company, Inc (1969), p130-131: "<With modern acoustical equipment>, it is found that string players, both in solo performance and in ensemble, tend towards the Pythagorean intervals rather than the just intervals <P. C. Greene, Violin Intonation, in JASA, IX (1937), 43-44; J.F. Nickerson, Intonation of Solo and Ensemble Performance of the Same Melody, in JASA, XXI (1949), 593-95>. The same thing occurs in choral singing <W. Lottermoser and J. Meyer, Frequenzmessungen an Gesungen Akkorden, in Acustica, X (1960), 181-84>. It has been found that choral groups sing the major thirds sharp and the minor thirds flat, contrary to the opinions of those who claim the good choral groups sing in just intonation."
[61] James Jeans, "Science and Music", Cambridge University Press (1953), p161: "Even primitive races whose music is polyphonic use scales in which most intervals are consonant".
[62] James Jeans, "Science and Music", Cambridge University Press (1953), p164-165: Discovery of two Egyptian flutes dated from 2000 B.C.E. - which use the Lydian scale.
[63] Arvindh Krishnaswamy, "On the Twelve Basic Intervals in South Indian Classical Music", (2003), p13: "One should perhaps be willing to accept a hybrid tuning scheme rather than try to fit Carnatic music to a preconceived scheme like JI. That said, Table 8 lists the most promising ratios for each note, selected for simplicity from Table 6. In addition to these ratios, the Western ET values should also be considered, especially for the more complex intervals. In the end, it is even possible that certain notes do not conform to any known tuning scheme or even allow themselves to be represented by a single numerical value."
[64] John Backus, "The Acoustical Foundations of Music", W. W. Norton & Company, Inc (1969), p120.
[65] J. Murray Barbour, "Just Intonation Confuted", Music & Letters, Vol. 19, No. 1. (Jan., 1938), p48:
24 27 30 32 36 40 45 48
c d e f g a b c
[66] Dave Benson, "Music: A Mathematical Offering", (2005), p145: "The second sounds consonant but weird, and after a while begins to sound almost normal."
[67] "Some Notes Regarding Tuning and Temperament" http://www.music-cog.ohio-state.edu/Music829B/tuning.html : "Thirdly, listeners seemingly adapt to whatever system they have been exposed to. Most Western listeners find just intonation "weird" sounding rather than "better"."
[68] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p14: "Modern expositions of this idea exist as well (Boomsliter and Creel 1961; Partch 1974), in which consonance is viewed in terms of the period of the combined sound. If two frequencies form an interval of a small integer ratio, the period of the combined sound is shorter which should be more pleasant for the ear. This is in fact a testable hypothesis and these so-called periodicity theories of consonance assume some time-based detector in the ear. Neurological evidence for such temporal models exist. Cariani (2004) provides evidence to ground pitch-based theories of tonal consonance in inter-spike interval representations. He finds that for both pure and complex tones, maximal salience is highest for unison and the octave separations and lowest for separations near one semitone". Tramo, Cariani, Delgutte, and Braida (2001) claim that 1) pitch relationships among tones in the vertical direction influence consonance perception and 2) consonance cannot be explained solely by the absence of roughness (see further below). They provide neurophysiological, neurological and psychoacoustic evidence to support these claims."
[69] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p15: "Slow beating is generally perceived as being pleasant"
[70] J. Murray Barbour, "Just Intonation Confuted", Music & Letters, Vol. 19, No. 1. (Jan., 1938), p50: "Paradoxically, if all the chords are made pure, the pitch if the key as a whole will probably fluctuate. On the assumption that the pitch of a repeated note remains constant in successive chords, the pitch of the key will not vary so long as the roots of chords move by fourths or by fifths ; but if a root falls a minor third or rises a major third, the pitch is lowered by a comma ; in the reverse progressions (root falling a major third or rising a minor third) the pitch is raised by a comma. If throughout a composition the upward tendencies are exactly balanced by the downward tendencies, the final pitch will be the same. But these tendencies seldom do balance."
[71] David M. Howard and James Angus, "Acoustics and Psychoacoustics - Second edition", Focal Press (2001), p147.
[72] J. Murray Barbour, "Irregular Systems of Temperament", Journal of the American Musicological Society, Vol. 1, No. 3. (Autumn, 1948), p20.
[73] Meantone is tuned so that each fifth in Pythagorean intonation is decreased by a quarter of a "syntonic comma" to make the major thirds more in tune, at the cost of the fifth. A Syntonic comma is defined as a Pythagorean major third divided by a just major third, or (81/64)/(5/4), or 81/80, or 1.0125, or 21.506 cents, or near a fifth of a semitone.
[74] John Backus, "The Acoustical Foundations of Music", W. W. Norton & Company, Inc (1969), p127: Even meantone is too restrictive for remote keys.
[75] J. Murray Barbour, "Just Intonation Confuted", Music & Letters, Vol. 19, No. 1. (Jan., 1938), p49
[76] Dave Benson, "Music: A Mathematical Offering", (2005), p140: "In the seventeenth century, it was discovered that a simple note from a conventional stringed or wind instrument had partials at integer multiples of the fundamental. The eighteenth century theoretician and musician Rameau (chapter 3) regarded this as already being enough explanation for the consonance of these intervals, but Sorge <G. A. Sorge, Vorgemach der musicalischen Composition, Verlag des Autoris, Lobenstein, 1745–1747> was the first to consider roughness caused by close partials as the explanation of dissonance."
[77] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p60:
Built in: Wightman 1973 & Goldstein 1973
Learnt: Terhardt 1972 & 1974
[78] Guy Oldham, Murray Campbell, "Harmonics - 1. General.", Grove Music Online ed. L. Macy (Accessed 03/05/2007) http://www.grovemusic.com "In 1822 the French mathematician Fourier showed that any waveform, however complex, could be decomposed into a set of simple sine wave components."
[79] Guy Oldham, Murray Campbell, "Harmonics - 1. General.", Grove Music Online ed. L. Macy (Accessed 03/05/2007) http://www.grovemusic.com "If the waveform is periodic, corresponding to a regularly repeating pattern of pressure variation, then its sine wave components are members of a harmonic series. In this case it is difficult to perceive the components separately; they are fused into a single sound with a definite musical pitch. In contrast, a sound which has a set of components which are not harmonics (or close approximations to harmonics) will not normally be perceived as having a clear pitch, and the components can be heard separately."
[80] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p149.
[81] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p150.
[82] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p5: "Experiments have pointed to a dominance region which goes from roughly 500 Hz to 2000 Hz (Plomp 1967; Ritsma 1967). The partials that are falling in this region have a bigger influence on the pitch than other partials. Smoorenburg (1970) showed that it is possible to create a virtual pitch with only two partials in the dominance region"
[83] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p148: "The crucial tone components that activate the central pitch processor responsible for this recognition process <a single perceived pitch> are the first six to eight harmonics."
[84] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p45.
[85] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p6
[86] Murray Campbell, "Inharmonicity", Grove Music Online ed. L. Macy (Accessed 03/05/2007), http://www.grovemusic.com "For the relatively supple strings normally used on bowed and plucked instruments, the inharmonicity coefficient is very small, and such strings can be treated for most musical purposes as having exactly harmonic natural mode frequencies."
[87] Guy Oldham, Murray Campbell, "Harmonics - 3. Strings.", Grove Music Online ed. L. Macy (Accessed 03/05/2007), http://www.grovemusic.com "Bowing a string in the normal manner gives a periodic vibration of the string, and the sound therefore has a frequency spectrum containing exact harmonics (neglecting some minor transient effects)."
[88] Ernst Terhardt, "Stretch of the musical tone scale" (2000), http://www.mmk.ei.tum.de/persons/ter/top/scalestretch.html "So, on first sight - and if the above explanation were the whole story - it would appear that - where intonation is concerned - the piano were exceptional among the musical instruments. However, it turns out that stretching of the tone scale is very common in musical performance. Solo instruments such as the wind and string families, as well as singers, tend to play sharp in the high pitch region. And in the orchestra the bass string players are often advised to avoid tuning their instruments sharp but instead rather to tune slightly flat. These tendencies are clearly visible in the results of statistical frequency measurements on solo performances by expert players on the violin, flute, and oboe (Fransson et al. 1970a). Although these instruments produce truly harmonic complex tones, a stretch of the tone scale was found that resembles that of the piano - with the only exception that even the middle octaves were not unstretched. So, in fact, the aforementioned non-stretched keyboard instruments (the organs) turn out to be the exception rather than the rule."
[89] David M. Howard and James Angus, "Acoustics and Psychoacoustics - Second edition", Focal Press (2001), p128, p133. Problems on modelling pitch perception - the 'place' theory versus the 'temporal' theory of pitch perception.
[90] David M. Howard and James Angus, "Acoustics and Psychoacoustics - Second edition", Focal Press (2001), p120-121.
[91] Dave Benson, "Music: A Mathematical Offering", (2005), p148: "There is also a related concept of virtual pitch for a complex tone. If a tone has a complicated set of partials, we seem to assign a pitch to a composite tone by very complicated methods which are not well understood."
[92] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p6: "Tones with inharmonic partials can produce a virtual pitch which will be the fundamental of the harmonic series which is the closest to the inharmonic partials in the sound (Rasch and Plomp 1999). When there is ambiguity about which harmonic series the partials of a sound belong to, more than one virtual pitch is possible and can be perceived (but not at the same time) depending on the context (see Schulte, Knief, Seither-Preisler, and Pantev 2001)."
[93] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p148.
[94] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p49.
[95] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p15-16: "They are related to critical bandwidth with the rule of thumb that maximal tonal dissonance is produced by intervals subtending 25% of the critical bandwidth, and that maximal tonal consonance is reached for intervals greater than about 100% of the critical bandwidth (see fig. 1.7). This is a modification of Helmholtz's 32Hz criterion for maximum roughness because the critical bandwidth is not equally wide at all frequencies."
[96] http://eceserv0.ece.wisc.edu/~sethares/consemi.html
[97] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p30-31.
[98] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p35: "An illustrative experiment is the following: feeding each one of the two tones f1 and g2 dichotic ally into a different ear, the primary beat or roughness sensation disappears at once, both tones can be discriminated even if the frequency difference is way below delta(fD), and their combined effect sounds smooth at all times! The moment we switch back to a monaural input, the beats or roughness come back. Of course, what happens in the dichotic case is that there is only one activated region on each basilar membrane with no chance for overlapping signals in the cochlea ; hence no beats or roughness <There is, however, an overlap of neural signals in the upper stages of the neural pathway, giving rise to "second order" effects>."
[99] Dave Benson, "Music: A Mathematical Offering", (2005), p141
[100] William Sethares, "Tuning, timbre, spectrum, scale", Springer (1998), p47.
[101] David M. Howard and James Angus, "Acoustics and Psychoacoustics - Second edition", Focal Press (2001), p139.
[102] William Sethares, "Local Consonance and the relationship between timbre and scale", paper (1993), p1-2
[103] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p16
[104] David M. Howard and James Angus, "Acoustics and Psychoacoustics - Second edition", Focal Press (2001), p125.
[105] Johan Sundberg, "The Science of Musical Sounds", Academic Press, Inc (1991), p73.
[106] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p4
[107] Johan Sundberg, "The Science of Musical Sounds", Academic Press, Inc (1991), p71.
[108] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p34:

[109] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p35: "The existence of a finite limit for tone discrimination is an indication that the activated region on the basilar membrane corresponding to a pure tone must have a finite spatial extension. Otherwise, if it were perfectly "sharp," two superposed tones would always be heard as two separate tones as long as their frequencies differed from each other - no matter how small that difference - and no beat sensation would ever arise. Actually, the fact that the roughness sensation persists even beyond the discrimination limit, is an indication that the two activated regions still overlap or interact to a certain degree, at least until the critical band frequency difference is reached."
[110] David M. Howard and James Angus, "Acoustics and Psychoacoustics - Second edition", Focal Press (2001), p75.
[111] E. Terhardt, "Definition of pitch" (2000), http://www.mmk.ei.tum.de/persons/ter/top/senscons.html "As these results were shown to be not appreciably different when the experiments were carried out with musically trained listeners as opposed to musically naive listeners, the first important conclusion is that even when asked to estimate "consonance", listeners do not estimate consonance in the sense of music theory, but do estimate sensory consonance, where sensory consonance is something different from affinity of tones. The second conclusion then is, that tone affinity must be a distinctly more subtle phenomenon than sensory consonance, because, obviously, in the experimental competition between tone affinity and sensory consonance, the latter dominates."
[112] Dave Benson, "Music: A Mathematical Offering", (2005), p141: "Anyone with any musical training can recognize an interval of an octave or a fifth, but for pure sine waves, these intervals sound no more nor less consonant than nearby frequency ratios."
[113] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p15: "Due to the dependency of the critical bandwidth on frequency, intervals (like minor thirds) that are consonant at high frequencies, can be dissonant at low frequencies. (This is consistent with musical practice where small intervals appear in the treble parts and the larger intervals like octaves and fifths appear in the bass part.)"
[114] Juan G. Roederer, "The Physics and Psychophysics of Music - Third edition", Springer-Verlag New York Inc. (1995), p168.
[115] Aline Honingh, "The Origin and Well-Formedness of Tonal Pitch Structures" (2006), p18: "Terhardt (1977) developed a two-component model of musical consonance. He argues that the concept of consonance obviously implies the aspect of pleasantness, but that pleasantness is not confined to musical sounds. Therefore, he has termed this aspect of consonance 'sensory consonance'. He argues that, as sensory consonance was not conceptualised to explain the essential features of musical sounds, there must be another component to account for this. This other component was termed 'harmony'. Thus, musical consonance consists of sensory consonance and harmony."

Potential Mathematical Models for the Western Musical ScaleA Historical and Empirical Comparison

Contents:

Introduction and claims

Why care?

Historical context

Historical summary and implications for tuning

Tuning models and maths

Equal temperament

5-limit just intonation

3-limit just intonation

Mathematical inconsistencies inside Just Intonation

The Harmonic series, and consonance

Helmholtz and Plomp/Levelt

"Rational consonance"

"Chromatic consonance"

Other types of consonance

Ear/brain idiosyncrasies which could affect intonation

Octave stretching

First/second order beating

Pure versus complex tone

Difference tones [147]

Aural harmonics

Features of additional tone

Left versus right ear

Loud tone versus quiet tone

Ear/brain idiosyncrasy summary

Why are Just intervals often preferred?

Phenomena 1: First order beating of partials

Phenomena 2: Two types of consonance.

Phenomena 3: People may hear in their 'mind's eye' a different interval to someone else.

Phenomena 4: Instruments such as the organ can exhibit acoustic distortion if tuned to ET [165][166][167?] (also see [34][168]).

Phenomena 5: Just intonation may allow for 'key colour' [169][170][171][172].

Phenomena 6: Easier to play/sing just intonation

Phenomena 7: ‘Nice looking’ numbers

Phenomena 8: ‘Pull’ of notes

Phenomena 9: Larger tolerance level for some listeners

Summary

Psychoacoustical / empirical studies

Summary

Final conclusions

A more 'natural' tuning?

Future experimentation

Endnotes:

Potential Mathematical Models for the Western Musical Scale
A Historical and Empirical Comparison