12et expression

The 12 Golden notes is all it takes...

  • Music scales & theory - The Basics.
  • Which 12-note Tuning is correct?
  • The big question - Is the equal tempered 12 tone scale really a compromise?
  • Quick reference to tuning systems
  • Are more than 12 notes in music valid?
  • Extra notes and observations
  • Survey - submit your votes!

  • If you know anything about music at all, you'll know that the
    standard (Chromatic) scale is made up of 12 notes or tones - each
    note unique and with its own flavour when in harmony. You probably also
    know that these very notes are the basis for every chord and melody in most Western
    music - and that they can be represented by numbers (or 'frequency').
    Although there are few references on this page, my dissertation has almost 200 of them, so visit there if you need sources, quotes, and verification.
    You might even know that these very notes (all 12 of 'em!) are
    usually spread smoothly in nice logarithmic steps from 1 to 2 (an octave).
    In other words, a pitch played an octave higher is twice as high in pitch as the original, and all 12 notes are spaced evenly inside this octave.

    However, note that I have highlighted the word 'usually'. What you may not know is that there are other ways of producing these notes/pitches (each approach different and giving near but not quite the same results) and that there is debate over which exact frequencies should be used for music!

    What you also don't know (and nobody really
    knows) is why there are 12 - full stop.

    Why not 5, 17, 66 or 59127 ?? Perhaps this has something to do with our brain's psychoacoustical response, with a bit of maths thrown in for good measure - but it's not really exactly clear why there are twelve notes. Here are some theories anyway (starting with the most respected/common, and finishing with the most speculative or least respected):

    Theory 1: Some have argued that the importance of the number 12 in music is thanks to the fact that the 12 equal tempered pitches approximate many simple ratios such as 4/3 or 5/4 very closely, but this surely isn't the full story, since other numbers (such as 19 notes to the octave) are quite good at this too. Also, you can use progressively ever more complex ratios - thus theoretically producing an infinite amount of possible pitches (most of which do not approximate the 12 golden notes very well at all). Finally, there are even 'rival' ratios for certain intervals such as the minor seventh (which could be 9/5, 16/9, or 7/4). Actually, pure ratios in general are ideal for timbre (the basis behind the harmonic series), but I don't think they should necessarily stretch to melody and chords.

    Theory 2: Others have argued how successive powers of 3 will 'complete' the scale after 12 iterations (the basis behind Pythagorean tuning), but this can't be the full story, since the twelfth iteration (312 or 3-12) - known infamously as the 'wolf' note - is a fraction over (or under) the octave. As a result, you could quite easily iterate further, and divide the octave nicely into 53 notes, 306 notes and beyond. Actually, even if 312 directly intersected the octave, this wouldn't be 100% proof, but it would be a good sign of 'mathematical evidence'.

    Theory 3: Alternatively, the 12-note scale could just be an arbitrary cultural construct, with no special reason to choose 12 above 5 or 50 note scales. This goes against common sense and a lot of scientific evidence, but one can't rule out the possibility.

    Theory 4: It's always a possibility that there may be no mathematical explanation why there are 12 notes. In the same way that science can't explain what it 'feels' like to see the color 'red', perhaps the notes of the scale are beyond mathematics, and reach into the qualia/metaphysical/spiritual realm (if it exists).

    Theory 5: A neat honeycomb lattice appears to fit around the major/minor 12 note system. This seems an interesting coincidence until you realize a similar pattern can be achieved with the 16 note scale and beyond. There still might be something unique about the 12-lattice though.

    Theory 6: Moving on to very speculative territory, you can surround a single sphere perfectly with 12 identically sized spheres - with each sphere perfectly touching its neighbouring spheres (this forms the points of the cuboctahedron, or the faces of its dual - the 'rhombic dodecahedron'). Twelve also has the exclusive property of being the Gravitational Symmetry Limit - another sphere arrangement based on the icosahedron (See this site for more information). It seems inherently dodgy to relate musical pitch to geometry, but it's a tiny possibility. While we're in geometrical territory, there also appears to be an interesting relation to the 4 dimensional 24-cell as explained here.

    Theory 7: As dodgy as the last one - a number of curious relationships exist between simple ratios and 12. For example, 37/213 * 5 is very close to the equal tempered perfect fourth (1.000000739402 off).

    Theory 8: If the last two are dodgy, then this ones takes the cake. According to Schoenberg (who promoted the idea of atonal 12-tone style music), there are 12 notes because if you take the sum of its digits: 1 and 2, they make 3. Wow, what a revelation. Assuming the number 3 is special in this context, now we just need to explain the magic behind the arbitrary 10-based decimal system :-) Seriously, with that sort of illogical reasoning, if someone was determined enough, you could find something 'special' about practically *any* number.

    Actually, there may be no single 'reason why' there are 12 notes, but instead, lots of 'little' coincidences - all of which might appear to hint at twelve. Either way, it would be easy to believe the number 12 is a universal, almost 'magic' number. Obviously, others still, could argue that music scales with more than 12 notes are indeed valid. In fact, there are a couple of reasons for tailoring music to use scales comprised of more than 12 notes (Just Intonation (JI) approximation and the 'unusual' style of Microtonal/Xenharmonic music). Later on in this article, I will explain all this - and why it is my opinion that such scales are mostly spurious - or at least not necessary (apart from for timbre purposes, as I'll also explain later).

    What exact 12 pitches make up the Chromatic scale?

    So, I've already said there's debate over which frequencies should be used for 12 notes of the chromatic scale. It turns out that these are obtained by using very different mathematical methods, but all of them with the goal of producing a scale that can be universally used for all music (or at the very least - a genre of music). Later, I'll give a few examples of these different methods and their resulting pitches. Suffice to say that most of these sound so very similar, but are subtly different by a fraction of a percent in pitch according to what method you choose.

    Part of the aim of this article is to shed some light on the what the best 'scale-building' method actually is. What exact pitch frequencies sound best for melodical purposes and together in harmony?

    I have researched the subject quite heavily (see my dissertation), and have spoken about it to many people. However, I'm still not 100% certain what pitches make up the 'perfect' scale, or even whether these notes should be fixed (or perhaps vary fractionally throughout a tune as it plays). There's also the possibility that each tuning has its 'advantages' and 'disadvantages' - in which case, the tuning would be appropriately tailored according to the content of the music. On the other hand, there's a remote chance that no tuning is a 'perfect fit' for a given tune - even if the pitches are dynamically altered in real-time by computer while the music's playing. Finally, we have to consider the case where tuning preference is observer dependent. In which case, one person/culture may hear in their 'mind's eye' something different to another person/culture, even given the same interval.

    If we take a look at the messy history of tuning, or the experimental studies, we'll find conflicting preferences and results all over the place. In fact, rarely will you find a topic so rich with complexity, laced with numerology, sprawling with pitfalls, and coloured with controversy.

    Before going any further, I should make it clear that all sound is based on mathematical frequency. Whenever you hear a musical pitch, this can be represented by a number. For the purposes of this document, I am going to give a default value of 1 for the pitch of C. An octave above this (still C), would therefore be 2, and an octave above that would have a frequency of 4 (also, halving would produce lower octaves of C (i.e. 0.5, 0.25, 0.125 etc. etc.)).
    Another example is the Major 2nd (D for now - assume a tonic of C). The default value for D is approx. 1.1224 (200 cents). Suffice to say that the approximate values: 2.245 or 4.49 are still D, but octaves higher. Also 0.561 or 0.281 are still D, but octaves lower. Sound confusing? See the diagram below which should help to clarify things.
    The measurement is in 12-equal temperament and uses an accuracy of two decimal places for the frequency.

     [12 notes]

    All of the important pitches for one octave can be represented inside the values of 1 and 2 (making 12 pitches).
    Later on in this article, I'll be using the term 'normalise'. This simply means using multiples or divisions of two to change the original number to a fraction between 1 and 2 (e.g. if we wanted to normalise the number 7, we would half it until it was lower than 2 and higher than 1 (i.e. 7, then 3.5, then 1.75 (bingo))). To the ear, this is the same 'note' as 7, only a couple of octaves lower of course.

    Anyway, we all know what a major triad sounds like. It's that nice warm chord comprising the notes: C, E and G. But what is the exact frequency for the note E? Shown below are three possibilities. They're not just theoretical, as they've been used throughout history and are still sometimes used today, but does that make equally as valid?

    Let's make C = 1.0 (0 cents)

    Well, E can either be:
    1.25 (which is 5/4 or 386.31 cents)..... or it could be:
    1.25992 (24/12 (Two to the power of (4 divided by 12) ) or 400 cents)..... or it could even be:
    1.26562 (81/64 or 407.82 cents)

    All of these pitches aren't chosen randomly, but come from various 'scale building' techniques of which I have described further below. As you can see, they are very close to each other, and also /sound/ very close, so it is very hard to tell which of these is the definitive E (Major third) that should make its way into every tune.
    Finally, there's also the possibility that these intervals are good in different ways.
    Actually, it's worth mentioning again that the pure third (1.25) clearly has a purpose in music, since it forms part of the harmonic series - and gives instruments their particular timbre. But in my opinion, timbre isn't necessarily related to melody or chords (the twelve intervals).

    Likewise, the pitch 'G' (perfect fifth) has possibilities. Here are the two most likely candidates:

    1.5 (701.95 cents) and 1.49831 (700 cents)

    1.5 is three divided by two obviously. And 1.49831 seems an awkward number but is actually (27/12). By mathematical 'coincidence', these two are amazingly close.
    Also incredibly close - are the different possible pitches for F (perfect fourth):

    1.3333 (498.045 cents) and 1.33484 (500 cents)

    This time, 1.333 is 4/3 and 1.33484 is 25/12
    But which is the real (best sounding) perfect fourth? Or are they both melodically useful in different ways?

    Anyway, it turns out that there are nigh-on (but not quite) identical versions for each of the 12 pitches - depending on which formula you use to reach these numbers.

    At this point, you might want to try the survey at the end of this page, and come back here after you've heard what tuning you prefer.

    Where do all these numbers come from?

    All of these nearly identical versions are 'okay'. This is firstly because they're quite close to each other anyway, but also maybe because the brain has a certain amount of 'tolerance' and can get conditioned to slightly different pitches (if they're been heard enough times, it's possible the original pitches can then even sound off! [reference]). However, this doesn't mean that there isn't a perfect scale from which to deviate. Before you see the tables, I'll try to explain a little bit about each tuning.
    Reminder note: If I start giving pitch names (such as "Eb" or "C#"), [12 note wheel] assume I'm relating these to the note C (so Eb would be a minor 3rd etc.)

    "In the red corner..."
    12-Tone Equal Temperament
    ("12-teT", "12-eq", "12edo", "12-eT", "12-equal", "Even temperament")
    The scale which is used predominantly throughout most of the world (piano/guitar/keyboard/midi/etc.) is the '12 tone even (or equal) tempered scale' based on the twelfth root of two. It differs from many other scales in the sense that all 12 pitches are 'fixed' and equally spaced apart from each other - enabling modulation through any tonal key without sounding 'off tune'. It is made up thus:
    2x/12 - where any note in the first octave can be produced by defining x from 0 to 12.

    For example, 24/12 (1.25992...) is closest to the Major Third or E (well... is the real Major third in my opinion) ...or try 29/12 (A or the Major 6th or 1.68179...)) etc. etc. All of the 12 notes and corresponding frequencies are shown in the colourful diagram over to the right (click the picture for a larger view).

     [JI diagram]
    Click here
    for a bigger, better
    and more complete version.
    "In the blue corner..."
    5-limit Just Intonation

    ("Just Intonation", "JI", "Pure temperament", consisting of "Pure" or "Just" ratios or intervals)

    Just Intonation aims to build its scale by multiplying or dividing whole numbers - thus allowing for 'beat-less' chords (see later). It is called 5-limit, because only the prime numbers up to 5 are used (that's the first 3 prime numbers - 2, 3 and 5). These are then used in combinations as demonstrated in the diagram to the left (such as 3/5 = 0.6 = 1.2 when normalised = Minor third or Eb) or on their own (e.g. 5 = Major Third = E = 1.25 when normalised between 1 & 2), or even a combination of three or more (such as: 5*3*3 = 45 = 1.40625 when normalised = augmented 4th) ). Theoretically, you could have a more complicated ratio such as: 3*3*5*5 = 225 (normalised to 1.7578125 = between a major sixth
    Well temperament
    'Unfortunately', due to the nature of Just Intonation (JI) temperament, any piece of music would sound best played in one key - and without straying too far from this key (unless you want 'clashing' notes to creep in). Because of this, Just Intonation isn't a system well suited to traditional instruments which contain only fixed pitches (such as the keyboard or flute).
    A reasonable compromise is to use a tuning such as 'Well' temperament. Without delving too much into the technical theory, this attempts to strike a compromise between the supposedly 'pure' sound of Just Intonation (JI), and the practicality and ease of use of 12-equal. Personally, I prefer the sound of the various Well Temperaments over Just Intonation, as it's closer to 12-eT (although not quite close enough for me... ;-)
    Visit Wikipedia's definition for more information.
    and minor seventh - kinda 'grates' in my opinion)
    , but on the whole, more complicated ratios will tend to produce more dissonant (even 'off-tune') intervals. This makes the system suspiciously arbitrary for melodical purposes in my opinion, and it turns out that many such calculations even produce similar numbers, causing debate over (for example) the best major 2nd or minor 7th in the tuning. It's important to say that 5-limit encompasses everything that the 3-limit system is - and more (see next paragraph).

    Lastly, there are internal inconsistencies (which I have detailed later on this page here) and practical problems too (see yellow box out for further info). Because of these reasons, most Just Intonation enthusiasts would instead use something like 1/4 or 1/6 comma meantone - which is sort of in between Just Intonation and Equal Temperament.

    For a more detailed description of 5-limit Just intonation, see here.

    "In the green corner..."
    3-limit Just Intonation
    ("Pythagorean Intonation", "circle of fifths", "cycle of fifths", "Just Intonation")

    A subset of 5-limit JI, this tuning was discovered by Pythagoras over 2500 years ago, and is often known as the 'circle of fifths' (diagram shown below). Instead of using the first three prime numbers (2, 3 and 5) as 5-limit does, it only uses the first two prime numbers - 2 and 3. Because of this, it is necessary to produce all of the important 12 pitches by multiplying the same numbers over and over again (example: the Minor 2nd (C#) is produced by calculating 3 to the power of 7 (that's 3 * 3 * 3 * 3 * 3 * 3 * 3) ). Compare this to 5-limit - which can get to a (nearly) identical pitch more quickly by using less multiplications or even single numbers. For example, to get closest to the major third in 5-limit, all you would need is the number 5 (normalise to 1.25 by dividing by 4 (2*2)). On the other hand, to get closest to the major third in 3-limit (maybe some would argue it is the real major third for some or all occasions - I certainly prefer it over that particular 5-limit version), you would need to calculate: 3*3*3*3 = 81 ( = 1.265625 when normalised to within 1 and 2 by constant halving). Just like 5-limit Just Intonation, there are practical problems if you try to play in a key other than the key that the instrument was tailored for.

    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.

    Pythagoras' line of fifths. Frequencies are accurate to 3 decimal places. Click picture for larger view
     [Pythgoras diagram]

    Incidentally, if you were to have a theoretical 2-limit version of Just Intonation, the only pitches you'd be able to get for your scale - are pure octaves, since only multiplication or divisions of the number 2 would be allowed.
    Finally, I'll very briefly mention a few other Just Intonation variants. There's a more complicated 7-limit version of Just Intonation. This goes one further and now uses the prime numbers: 2, 3, 5 and 7. Naturally, it encompasses both 5-limit and 3-limit, so anything that can be produced in those can be done in 7-limit. The main reason for the use of 7-limit is to obtain ratios such as 7/6, 7/5, 7/4 etc. In my opinion though, these tones are slightly off-tune, and I don't think should be used as melodical fundamentals. However, don't get me wrong - I love pure intervals such as 7/4 when they're used properly as a direct relation to one of the twelve 12-eT fundamentals. In fact, it goes without saying that this interval is invariably present as the seventh harmonic in the timbre of many synth and orchestral instruments. In particular, bells tend to emphasize (make louder) intervals like 5/4, 7/4 and 11/8 (see later in this article for more information regarding the harmonic series).
    Also, there's a branch off Just Intonation called Meantone temperament. Similar to Well Temperament, each pitch is usually a 'fixed' compromise so that it can be used for traditional instruments such as the piano. The main aim is to attain 'pure' thirds (at the cost of the pure fifths) by pitching the 'whole tone' between the two most commonly used whole tone intervals from pure JI (10/9 & 9/8) - producing a pitch of 1.118. See Wikipedia's Meantone definition or Microtonal.co.uk or this page for more information.
    Just Intonation could be said to be in its purest form when each pitch and interval is tailored during the course of the music. This is called 'Dynamic' or 'Adaptive' Just Intonation. A computer is mostly necessary to handle this, though non fixed-pitched instruments (such as the human voice!) can handle the job with practise.
    Apart from the arguable sweetness of the JI intervals, you'll find that there are inherent problems with D-JI too. For example, you might find that the tonic changes throughout the tune, and by the time you've reached the end, the whole piece might have transposed by 1 or more semitones! Also, it is often mathematically impossible to keep certain chords and melodies internally consistent with each other. See this site for more information.

    So, here are the tunings side by side
    (C is the tonic for each scale)

    For the below math, here are some general jargon busters:
  • "Pitch" is analogous to frequency or hertz. So for example, if C represents 1000 Hz, then E at Pitch 1.25 is basically 1000*1.25 = 1250 Hertz. (Concert pitch C is actually approximately 523.25 Hz. Relate all math to that).
  • For "The Math" column in the "5-limit Just Intonation" and "Pythagoras Intonation" tables, you may need to constantly half or double the number to reach the pitch which is between 1 and 2 (normalizing).
  • A "cent" is basically 1/100th of a semitone, so if you want to convert to semitones, just divide by 100.
  • Pythagoras Intonation
    (3-limit Just Intonation)
    NoteRatioPitch & math'Error'
    from ET
    B# 531441/524288 (1.01364) 31223.46 23.46
    E# 177147/131072 (1.35152) 31121.505521.505
    A# 59049/32768 (1.80203) 31019.551019.55
    D# 19683/16384 (1.20135) 3917.595 317.595
    G# 6561/4096 (1.60180) 3815.64 815.64
    C# 2187/2048 (1.06787) 3713.685 113.685
    F# 729/512 (1.42382) 3611.73 611.73
    B 243/128 (1.89843) 359.775 1109.775
    E 81/64 (1.26562) 347.82 407.82
    A 27/16(1.6875) 335.865 905.865
    D 9/8(1.125) 323.91 203.91
    G 3/2(1.5) 311.955 701.955
    C1(1) 30same 0
    F4/3(1.33333) 3-1-1.955 498.045
    Bb 16/9(1.77777) 3-2-3.91 996.09
    Eb32/27(1.18518) 3-3-5.865 294.135
    Ab128/81(1.58024) 3-4-7.82 792.18
    Db256/243(1.05349) 3-5-9.775 90.225
    Gb1024/729(1.40466) 3-6-11.73 588.27
    Cb2048/2187(0.93644) 3-7-13.685 1086.315
    Fb8192/6561(1.24859) 3-8-15.64 384.36
    Bbb32768/19683(1.66478) 3-9-17.595 882.405
    Ebb 65536/59049 (1.10985) 3-10-19.55 180.45
    Abb 262144/177147 (1.47981) 3-11-21.505 678.495
    Dbb 524288/531441 (0.98654) 3-12-23.46 1176.54
    12-Equal Temperament
    C (Unison) 1.02 (0/12)0
    C#/Db (Minor 2nd) 1.05952 (1/12)100
    D (Major 2nd) 1.12252 (2/12)200
    D#/Eb (Minor 3rd)1.18922 (3/12)300
    E (Major 3rd)1.25992 (4/12)400
    F (Perfect 4th) 1.33482 (5/12)500
    F#/Gb (Aug. 4th)1.41422 (6/12)600
    G (Perfect 5th)1.49832 (7/12)700
    G#/Ab (Minor 6th) 1.58742 (8/12)800
    A (Major 6th) 1.68182 (9/12)900
    A#/Bb (Minor 7th)1.78182 (10/12)1000
    B (Major 7th) 1.88772 (11/12)1100
    C (Octave) 2.02 (12/12)1200

    Equal temperament (12-teT).
    This scale is special because not only does it include every important pitch, but also there aren't any 'dud' notes generated. I think ultimately, that these are the only notes needed for all the best and most satisfying music (apart from for timbre purposes, or when a pitch 'slides', but this is usually done in-between beats, so it's more of an 'effect' than tonal deliberation).
    5-limit Just Intonation
    PitchThe math 'Error'
    from ET
    111 = C 0
    16/151.066661/5*1/312c > C# 111.7
    9/81.1253*34c > D 203.9
    6/51.23/516c > D# 315.6
    5/41.25514c < E 386.3
    4/31.333331/32c < F 498
    45/321.406255*3*310 < F# 590.2
    64/451.422221/5*1/3*1/310c > F# 609.8
    3/21.532c > G 702
    8/51.61/514c > G# 813.7
    5/31.66665/316c < A 884.4
    16/91.77771/3*1/34c < A# 996
    15/81.8755*312c < B 1088.3
    2/1220c = C 1200

    'Just Intoned' pitches: There are many I haven't included in this table, but it includes all 12 notes and most of the simplest ratios. Ideally, the array of 5-limit JI intervals should be represented in two dimensions, rather than in a linear chain as shown above. See this picture for the full version.

    A lot of these notes sound quite reasonable, and of course they also have the alleged 'advantage' of being beat-less (that is, they don't have extra interference producing low frequency waves). But in my opinion, notes such as Eb, E, Ab, A and even F# are simply out of tune with the root, and should only be used for timbre in the form of the harmonic series.
    To convert from cents to interval ratio, use: 2 ^ (c / 1200), where c is the number of cents. A cent is 1/100th of a semitone.
    To convert from semitones to interval ratio, use: 2 ^ (c / 12), where c is the number of cents.
    To convert from interval ratio (analogous to hertz or frequency), to cents, use: 1200 * (log(i) / log(2)), or simply: 1200 * logBase2(i), where i is the interval.
    To convert from interval ratio, to semitones, use: 12 * (log(i) / log(2)), or simply: 12 * logBase2(i), where i is the interval.

    Which tuning is the winner? - The big question

    The general consensus is that the equal tempered 12 note scale has always been an admittedly good but imperfect 'compromise' when put up against 'Just interval', 'Well tempered', 'Mean-tone' or 'Pythagorean' tunings. Intervals defined by simple ratios (via 3 and 5-limit JI) are said to produce the 'sweetest' or 'purest' chords in music, but I don't think this is necessarily the case.

    If you haven't already guessed, I think the 'winner' is almost certainly 12-Tone Equal Temperament, with Pythagorean Intonation in a close second place. To my ear, Just Intonation (or meantone) melodies and chords sound slightly off. If you've read this far, you might well know that traditional Pythagorean notation will distinguish Eb and D# as slightly different pitches, but I think that for all intents and purposes, they are (or should be) the same note. Also, ratio defining pitches such as 3/2, 4/3 and 5/4 look mathematically neat, and are fine for timbre purposes, but I think that most likely, they are not appropriate for melody and chords in the traditional sense.

    So, my theory of the equal tempered 12 note scale being more inherently 'on tune' than other tunings is certainly controversial... and in fact I may even be wrong ;-) However, the philosophy behind the idea didn't just come to me overnight. Amongst other comparisons, I've spent hours trying to 'get used to' the tone of 1.25 (5/4) over 1.259 (24/12) against Root C by comparing them in a chord or melody. In my opinion, a major third tuned at the logarithmic pitch of ~1.2599 (12-eT) pitch was 'sweeter' and better.
    There are proper music listening tests to come, but for now - try hearing this comparison, and see which one you prefer:
    12 equal version: ripple12et.mp3     5-limit Just Intonation version: rippleji.mp3

    And that's not all... In the same way that the major third pitched at 5/4 is claimed to be the sweetest sounding third, the minor third (Eb) also happens to be pitched amazingly close to a particular Just interval ratio: 6/5 (1.2). This is also claimed by many as the ideal minor third, but in my opinion, it's surely too sharp to be considered the real McCoy, so once again the 12 equal version (23/12) or 1.1892 is the proper interval in my opinion. In fact, my opinion stretches across all 12 notes and their nearest JI equivalents.

    However, a conclusion based on what my ear likes isn't necessarily going to suffice ;-)
    Let's dive into some theory... There are supposedly two main 'reasons' why 12-eT is considered a 'compromise' - which I'll explain now.
    1: Splitting the atom: Just like in physics, when you zoom down to the atomic level of music, unexpected things start to happen, and things go crazy :-) This section assumes some level of acoustical knowledge upon the reader, but I'll try to make it as readable as possible.
    Firstly, it is true, that 12-eT has 'problems' due to inherent complexities such as harmonic partial clashing, and periodic 'beating' - which produces extra unwanted tones. Although these make the 12-eT scale seemingly 'imperfect' (and therefore music itself - or at least the transmission mechanism of sound and what our brain does with it), the truth is that Just Intonation, Meantone and every other type of chromatic scale has these exact problems too.

    2: The false assumption that Just Intonation is inherently 'sweeter' that 12-eT
    Above, I've discussed how the tunings compare when you go down to the basics of sound. I've demonstrated that in some cases, Just Intonation appears to have slight 'advantages' when it comes to harmony at the timbral level. However, this is often taken by many to signify that the JI fundamentals themselves are aesthetically 'on-tune' with each other. Of course, in reality, there's no way to know for sure, apart from an ear test - and even then, that will be subjective, and hard to tell since they're so close in frequency. So why then do many music theorists think that the equal tempered scale is a compromise?

    Here are some more reasons that would seemingly appear to give evidence to Just Intonation (my rebuttals are in blue):

    This above issue intrigued me. Does the harpsichord really sound better tuned to Just Intonation - and if it does, in what way does it sound better? This following Youtube video created by Bradley Lehman may shed some light on the issue. I have put the SAME VIDEO side by side, so you can compare the differences more easily. Start the left at 0:06 seconds, and start the right version at 1:19 seconds. Play a bit of the "equal tempered" left one (say 5-10 secs), then stop it. Then play the "Well tempered" right version for the same number of seconds, and then stop that. Keep doing this so you get a feel for how they are different.

    Also do the same process for the C-major prelude of the Well Tempered Clavier at 2:35 (equal temperament) on the left, and 4:42 (well temperament) on the right.

    After listening myself, I do indeed find that Well tempered version sounds 'smoother' albeit very slightly less 'on tune'. It is smoother in the sense that some of the notes in the equal tempered version are disproportionately louder than others, Whilst the well tempered version has a more consistent volume. However, I believe this is largely down to the complex acoustical interference of the instrument rather than any problem inherent in equal temperament.

    Visit the Youtube page of this video to read other comments on this video including the author's.

    I also composed a short piece to specially compare equal temperament with the less subtle Just intonation tuning (the above videos used Well temperament, which is a compromise between ET and JI, and therefore it was more subtle). It uses a synthetic harpsichord style instrument with lots of upper harmonics. For a fair comparison, I tailored the piece to stick mostly to the key of C. See which one you prefer! (both files are just 254 k each).
    12 equal version: fugue12et.mp3     Just Intonation version: fugueji.mp3

    As expected, I found the 12-eT version definitely to sound more on tune, which would indicate that along with the harmonic series, the 12 equal 'tempered' notes /do/ seem to represent an ideal. There was nothing that was really better about the JI version. It just sounded a little more 'jangly', with maybe a few notes quieter than 12et - and that was it.

    Curiously, even a lot of sites mostly dedicated to standard music theory (12-equal based) seem to 'concede' that 12-eT is a 'necessary compromise' born out of convenience due to the practical difficulties of Just Intonation. This is often said as though they were 'taught' this for historical reasons, rather than an objective comparison via a listening test (though I suppose this isn't always the case). Of course, in my opinion, I think the practicality aspect is simply a bonus on top.

    This is one of the reasons why I compiled a survey at the end of this page so you can decide for yourself :-)
    Judging from the results so far, 12-eT is ahead in two of the three experiments (chord, ripple and tune). Make sure you cast your vote.

    Mathematically, I suppose the biggest 'evidence' I can claim against the validity of Just intonation is the way it's so internally inconsistent. Melody is often at odds with harmony - even in Dynamic JI. Also, it can produce an 'infinite' number of pitches - thus causing debate over whether (for example) the real Major 3rd is taken from 3-limit JI (81/64 or perhaps 8192/6561), or 5-limit JI (5/4). Likewise, it's arguable whether the Minor Seventh is represented in 7-limit or 5-limit or 3-limit. Also, where does one draw the line? You could easily draw upon multitudes of new pitches using ever higher limits (11-limit, 13-limit, 17-limit...). The harmonic series goes on 'forever' too, but at least they all merge with the fundamental to form a proper timbre.

    Are more than 12 notes in music valid?

    In my opinion, the short answer is not really, unless you count the pure ratio frequencies used in timbre as being 'extra' notes :-)
    Before I continue with this section, I'd like to say that used properly in the timbral context, emphasizing frequencies such as 5/4, 7/4 & 11/8 really can sound great in music. If I had to describe the effect of the ratio 7/4, I'd say it has an 'eerie', 'mechanical' or 'droning' almost bell-style sound to it. In fact I made a short mp3 to demonstrate these exotic pitches, and how good they can sound. You can download and listen to it here.
    For my own tastes (possibly ET biased), I have no qualms about strongly emphasizing these frequencies - as long as the real or implied fundamental from these harmonic/s are:
  • A: ...A 12-eT note ...and...
  • B: ...Possibly only a fundamental that is desired for that particular chord in the tune (and going even further - maybe even just the tonal key or bass/root of that chord).

    Anyway, back to why people use more than 12 notes in the scale. There are a couple of reasons for tailoring music to use more than 12 notes....
    The first of these reasons - is to form a tuning scale with pitches close to the Just Intervals (more notes in the scale will be more likely to do this, especially if the piece is modulating to other keys). It turns out that some of the best equal tempered scales to accomplish this use 19, 31 or 72 notes per octave. In fact, its goal is similar to Well or Meantone temperament and just like those, it's now possible to change key much more easily without any clashes. On the downside, it's not so easy to learn due to the many notes, so not really very practical for many people. In essence, I suppose one could call these scales Chromatic, because on the whole, only the 12 notes are actually used (the ones closest to the supposedly 'ideal' Just Intervals). In a given key, all of the rest are mostly scrapped - with the exception of using them for timbral purposes (like 7/4).
    To sum up my views on this system... As you might guess, it is my opinion there's a chance that the original ratio pitches (or 'Just intervals') are ultimately flawed for the chromatic fundamentals, so I think it may be a false objective to try and get close to them. However, there is the slight advantage that you can use some of these notes to build a 'new timbre' when combined with the fundamental or lower overtones.

    Take a look at this (off-site) URL for all the scales with more than 12 notes: List of unorthodox equal tempered scales

    The second reason for using more than 12 notes is an attempt to use all of these notes in a 'type' of music called 'Microtonal' or 'Xenharmonic' music (nb: the exact definition of 'Microtonal' is still debated, and it can confusingly also refer to the first reason). Once again, any number of notes can be chosen - 10, 13, 19, 31, 43 etc. etc. (you name it) are used per octave, but unlike the first reason mentioned in the previous paragraph, most or all of the notes are used.

    These unorthodox scales used in Xenharmonic music sound very strange to most people (at least in the Western hemisphere), but there are quite a few people who compose music like this. I guess many supporters of this style of microtonal music would argue that it's only because of 'cultural conditioning' that music with more than 12 (equally spaced) notes sounds so strange to the majority of people.
    On the contrary, I think personally that microtonal/xenharmonic music is essentially 'atonal' music - where the emphasis is on the 'textural' sound and rhythm - rather than any actual harmony. If the intervals derived from these scales actually sound any good, I think it's either because that those particular notes are close to the 12 'golden' pitches anyway (which is perhaps why I should have said that it's a hybrid of atonal and tonal in the first sentence of this paragraph), or because of the timbres such scales can produce. I'm willing to admit there's a chance I'm wrong, but somehow, I doubt it. This is the perfect kind of subject for discussion, so visit the Skytopia Forum if you would like to debate =)

    If you would like to listen to and find out more about microtonal music, then visit this web site and see what you think: Microtonal music

    If the info on this site has been of sufficient interest, a small donation would be appreciated:
    Amount you wish to contribute:

    ...but don't just take my word for it.
    The 12-eT Vs. JI survey.

    Decide for yourself what tuning you prefer. I have compiled a poll for you to select your favourite. In each sub-section, you'll hear two versions. All you have to do is select which one you prefer. The two downloads will be shown to the right of each poll. Try and listen a few times to each one before making up your mind.

    Each sample is in MP3 format (if anyone wants me to upload in WAV or AIFF, email me).
    Firstly, before you visited this site, could you recognise the subtle difference between the sound of Just Intonation and Equal Temperament?
    How discerning do you think your ear is? (in terms of minute differences in pitch - not music taste).
    Excellent - I believe my ear to be very sharp
    Good to Very Good
    Poor - I often can't easily tell if something sounds off-tune
    Not too sure to be honest
    Ripple options
    I prefer version A
    I prefer version A by a whisker
    They're both about as good
    I prefer version B by a whisker
    I prefer version B
    I hear no difference between the two
    I think version A and B are about as good in different ways, so both are valid tunings
    The Ripple test (vote to the left)
    Download A 12-equal version
    Download B Just Intonation version

    What exactly will you be hearing?
    The following upward ripple based on the pentatonic scale:
    C, D, E, G, A, C, D, E, G, A......
    The timbre of the waveform morphs between a 'ramp' and 'sine' wave. As you can see, I've specifically targeted the notes E & A, as I believe these are the most melodically off-tune in the Just Intonation version. But see what you think!
    Triad options
    I prefer version A
    I prefer version A by a whisker
    They're both about as good
    I prefer version B by a whisker
    I prefer version B
    I hear no difference between the two
    I think version A and B are about as good in different ways, so both are valid tunings
    The Triad test (vote to the left)
    Download Sample version A
    Download Sample version B

    What exactly will you be hearing?
    This one's also a blind test. In each sample, you'll hear a Minor triad, followed by a Major triad. The timbre of the waveform is a pure 'ramp' wave - each with a slight vibrato (5.2 cents either way). This is mainly because without this tiny (sinusoidal) vibrato, the Just intervals (5/4 and 6/5) would sound too 'plain' thanks to lack of timbre 'phasing'. Basically, it takes the emphasis off the timbre, and concentrates on the raw pitch. Also, the wavelength (frequency) of vibrato is slightly different for each pitch in the triad, so as to limit the amount of constructive interference.
    Harpsichord tune options
    I prefer version A
    I prefer version A by a whisker
    They're both about as good
    I prefer version B by a whisker
    I prefer version B
    I hear no difference between the two
    I think version A and B are about as good in different ways, so both are valid tunings
    The Harpsichord tune test (vote on left)
    Download A 12 equal version
    Download B Just Intonation version

    What exactly will you be hearing?
    It's sometimes said how the harpsichord is better tuned to JI. So I knocked up this tune in about an hour - made specially to compare JI and 12-eT :-)
    Each mp3 file is just 254k each. To avoid key problems, most of the tune sticks to the home key (C), and only once modulates to the key of D. Again, try to listen to each version carefully before deciding which you prefer.

    Additional comments
    Anything you might want to add goes here.
    Why and how were your preferred selections better than the alternatives?
    How much can you 'trust' your ear?
    Any comments on this site in general?

    Or see results!

    Also see people's individual comments!

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    Other web sites comparing music in different temperaments:

    The "Warped Canon" Page - Contains 'Pachelbel's Canon' in many different tunings (midi format).
    http://nowitzky.hypermart.net/justint/: Features Moussorgsky's "Promenade" in JI and 12-equal temperaments for comparison. Also, if you're looking for another survey, try out this page from the same site. It's a test to see your favourite Dominant Seventh Chord. No prizes for guessing which version was my favourite ;-)

    Other web sites concerning tuning and scales in general:

    Scientific American - Exploring the Musical Brain - A look at how certain animals (such as the humpback whale, canyon wren and wood thrush) base their 'song' on the diatonic and chromatic scale.
    Music of Sacred Temperament - So far, about the only site I've seen that also favours 12-eT as some kind of ideal. Site is also dedicated to the works of Bach.
    The dissonant tritone and Just Tuning theory - Examples of how the Just intonation ratios can be 'rigged' to justify preconceptions of consonance.
    Relating Hearing, Language and Music - Research by David Schwartz, Catherine Howe and Dale Purves from Duke University demonstrate how everyday speech appears to correlate with the 12 tones of the musical scale. Also see this site
    Combinational Music Theory - Excellent paper by Andrew Duncan, explaining some very interesting patterns in the diatonic and pentatonic scales.
    Bach's understanding of key character - The first part of this open debate spoke about the virtues of meantone and whether enharmonic notes (such as Ab and G#) should really be different. The second half argues whether Bach intended equal temperament or well temperament for his "Well-tempered Clavier".
    The Elements of Acoustics and Psychoacoustics - Information on the objective and subjective response to loudness and pitch.
    Pitch shifts and pitch deviations - Explains how subjective pitch is not entirely independent from timbre, intensity etc. - due to slight errors in the ear/brain system.
    http://www.casaninja.com/christi/academic/classicaltuning.html - Detailing some of the convoluted and messy history behind tuning.
    Tuning for Beginners - Contains an introduction to Microtonal tuning.
    Well vs. equal temperament - A controversial thesis arguing how the Well-tempered Clavier by Bach wasn't intended to demonstrate 12-Equal temperament - and that 12-eT only came into use in the 20th century.
    Wikipedia.org - Musical_tuning - Information and links relating to tuning in general.
    Tonotopic Interference - An interesting theory of consonance.
    http://www.justonic.com - More about Just Intonation.
    Brian McLaren - Lots of interesting and controversial posts on microtonality from Brian McLaren (taken from the tuning forum at Mill College).
    The Creation of Musical Scales, part II - A look into the history of world tunings by Thomas Hightower. Site explores both Eastern and Western music scales.
    Just Intonation vs. Equal Temperament - A video showing the apparent superiority of JI. I disagree fully due to the reasons given on this page, but it's a good comparison none-the-less.

    Message your comments to the Skytopia Forum
    or email me

    Visit the Soundburst Shrine for some intricate and catchy music!
    Also visit the Aesthetics of music page to see if music can be rated outside of human opinion.

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