The 12 Golden notes is all it takes...

  • Music scale theory - The Basics.
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  • Which 12-note Scale is correct?
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  • The big debate - Is the equal tempered 12 tone scale really a compromise?
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  • Are more than 12 notes in music valid?
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  • Extra notes and observations
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  • 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. 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').
    Also visit the:
    Music theory and Art aesthetics page to see if music can be rated and evaluated objectively
    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.

    However, what you probably don't know is that there are numerous ways of defining these notes/pitches (each approach different and giving near but not /quite/ the same results) and that there is debate over which 12 notes or pitches should be used for music!! (In fact, about the only thing truly agreed upon is the octave. All cultures and tastes have adopted a doubling of frequency as the base for their scale.)

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

    Why not 5, 17, 66 or 59127 ?? I myself can't know for sure if the only sensible scale should be made up from 12 notes, but never-the-less, I have a hunch that only the universal 12 'magic' notes are needed for the best music. Obviously, some could argue that 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 (JI Approximation & the ummm... 'unusual' Microtonal music).     Later on in this article, I will explain all this - and why it is my opinion that such scales are ultimately spurious.

    Download and hear the music that inspired me to write this page! (music in mp3 format)

    Which 12-note scale is correct?

    To add even more confusion, there are different scales even within the 12-note framework! By a pseudo coincidence of maths, there are a few techniques of determining the exact pitches that these very 12 notes should be.
    Most of these sound similar, but are subtly different by a fraction of a percent in pitch according to what scale you choose.

    There are loads of apparent inconsistancies and contradictions in most of these 'tailored' scales and I have researched the scales, and spoken about it to many people. However, I have still yet to conclude what notes 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 possiblity that each scale has its 'advantages' and 'disadvantages' - in which case, each scale would appropriately be tailored according to the content of the tune.

    Anyway, we all know what a major triad sounds like. It's that nice warm chord comprising the notes: C, E and G. Who would have thought that once again, it's unclear about what pitches should be used.

    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 (2^(4/12) 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...

    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 (2^(7/12)). 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 2^(5/12)
    But which is the real (best sounding) perfect fourth?

    Where do all these numbers come from?

    NEW! Take the pitch test at the end of this page, and see which interval you prefer!
    Will it be the 12 Equal C major chord - or the Just Interval equivalent?
    All of these nearly identical 12-tone scales are 'OK'. 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 used to pitches if they sound slightly off (similar to the way when you're born - how the eye adapts an upside-down image over time and reverses it to the right way up!). However, this doesn't mean that there isn't a perfect scale from which to /deviate from/.

    The scale which is used predominantly throughout most of the world (piano/keyboard/midi/etc.) is the '12 tone even (or equal) tempered scale' and it is made up thus: 2^(x/12) where x is from 0 to 12. These pitches are shown in the table in the middle.
    Next up is the scale discovered by Pythagoras based on the 'circle of fifths'. These are shown on the left.
    I have also covered the "Just Intonation" (JI) set of pitches - which are apparently very good for chords as opposed to 'melody'. These are shown on the right. All scales assume a root of C.

    Pythagoras tuning system
    B# 531441/524288 (1.01364) 1.5^12
    E# 177147/131072 (1.35152) 1.5^11
    A# 59049/32768 (1.80203) 1.5^10
    D# 19683/16384 (1.20135) 1.5^9
    G# 6561/4096 (1.60180) 1.5^8
    C# 2187/2048 (1.06787) 1.5^7
    F# 729/512 (1.42382) 1.5^6
    B 243/128 (1.89843) 1.5^5
    E 81/64 (1.26562) 1.5^4
    A 27/16(1.6875) 1.5^3
    D 9/8(1.125) 1.5^2
    G 3/2(1.5) 1.5^1
    C1(1) 1.5^0
    F4/3(1.33333) 1.5^-1
    Bb 16/9(1.77777) 1.5^-2
    Eb32/27(1.18518) 1.5^-3
    Ab128/81(1.58024) 1.5^-4
    Db256/243(1.05349) 1.5^-5
    Gb1024/729(1.40466) 1.5^-6
    Cb2048/2187(0.93644) 1.5^-7
    Fb8192/6561(1.24859) 1.5^-8
    Bbb32768/19683(1.66478) 1.5^-9
    Ebb 65536/59049 (1.10985) 1.5^-10
    Abb 262144/177147 (1.47981) 1.5^-11
    Dbb 524288/531441 (0.98654) 1.5^-12
    Equal Temperament
    C (root) = 1.0 = 2^(0/12)
    C#/Db = 1.05946 = 2^(1/12)
    D = 1.12246 = 2^(2/12)
    D#/Eb = 1.18921 = 2^(3/12)
    E (third) = 1.25992 = 2^(4/12)
    F = 1.33484 = 2^(5/12)
    F#/Gb = 1.41421 = 2^(6/12)
    G (fifth) = 1.49831 = 2^(7/12)
    G#/Ab = 1.58740 = 2^(8/12)
    A = 1.68189 = 2^(9/12)
    A#/Bb = 1.78179 = 2^(10/12)
    B = 1.88775 = 2^(11/12)
    C ^octave = 2.0 = 2^(12/12)
    Equal tempered 12-t.
    These 12 pitches are special because not only do they 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 music (apart from when a pitch 'slides', but this is usually done inbetween beats, so it's more of an 'effect' than tonal deliberation).
    Just Intonation (ratios)
    Just ratio Closest to ET...
    1 / 1 = 1 (is) C
    2 / 1 = 2 (is) upper C
    3 / 2 = 1.5 G, very slightly off
    4 / 3 = 1.333 F, slightly off
    5 / 3 = 1.666 A, sounds a bit off
    5 / 4 = 1.25 E, but arguably worse
    6 / 5 = 1.2 D# / Eb, slightly off
    7 / 5 = 1.4 F# / Gb, slightly off
    8 / 5 = 1.6 G# / Ab, slightly off
    9 / 5 = 1.8 A# / Bb, a bit off
    9 / 8 = 1.125 D, very slightly off
    17 / 9 = 1.888 B, very slightly off
    21 / 20 = 1.05 C# / Bb, a bit off
    'Just' pitches: 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 appears that the more dissonant ratios (17/9 & 21/20 etc.) seem arbitrarily chosen.
    Other questionable ratios include 7:4 and 7:6. These tones clash quite a bit and I also think are ultimately spurious, but there are some people who do use them in music, so I could be wrong.

    The big debate

    What is assumed (and for now which I'm arguing against), is that the equal tempered 12 note scale has always been an admittedly good but imperfect 'compromise' when put up against the original 'Just interval', 'Mean-tone' or 'Pythagorean' pitch ratios.
    But I'm wondering if there's any truth in the possibility that the 12-note system contains the perfect pitches after all (the 12 'golden' notes), and that they are therefore /not/ a compromise.
    This would mean that the two notes... (for example) D# and Eb - are, for all intents and purposes the same (or should be the same) note. And.....
    .....Ratio defining pitches such as 3/2, 4/3 and 5/4 look mathematically neat, but I think are also ultimately spurious for melodic (and even harmonic) purposes.

    So why then do most knowledgable music theorists think that the equal tempered 12 note scale is a compromise?

    As I have said previously, it's quite easy for the mind to adjust and develop a certain degree of tolerance to certain pitches initally. But once someone has heard a new pitch a sufficient amount of times (whether it be in casual listening to certain just/mean tempered music, or training to hear beatless chords), the mind will not only adapt to the (possibly) wrong pitches, but it might also discard the original (and possibly correct) ones*.
    * I've since questioned this 'tolerance theory' somewhat with my further research - detailed near the end of the page.

    The general consensus is that pitches defined by simple ratios (we'll overlook the pythagoras pitches for now) make for the purest chords in music. Here are some good reasons that would appear to give evidence to Just temperament:
    a: Pitches represented by numbers like 1.5 and 1.3333 'look' better than irrational numbers like 1.059463094359 or 1.259921049895 (but that is no real proof at all - since many important numbers in mathematics (such as PI) are irrational)

    b: Beatless chords (no very low frequency overtones). This is related to the above. (once again, this provides little evidence that these pitches are actually sweeter than the 12-eT pitches (which contain supposedly imperfect low-frequency harmonics) )

    c: Some people simply prefer the Mean/Just temperaments in music But as said above, if someone gets used to a new set of pitches (whether they are right or wrong), the brain will 'adapt'... (but 12-eT was possibly perfect in the first place!)

    d: Maybe some instruments such as the piano cause the subtle overtones (timbre) to clash with the fundamental tones of 12-eT. If this really is the case, then the problem isn't specifically with 12-eT, but rather with the overtones that a particular instrument produces - meaning that 12-eT possibly isn't a compromise after all.

    So, my theory of the equal tempered 12 note scale being perfect (and not a compromise in any way) 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 prefer the tone of 1.25 (5/4) over 1.259 (2^(4/12)) against Root C. I've even tried comparing the differences by trying out both in a melody. In my opinion, a Major third tuned at the logarithmic pitch of ~1.2599 (12-eT) pitch was 'sweeter' and better.

    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 (2^(3/12)) or 1.1892 is the proper interval in my opinion. Why not judge for yourself at the end of the page... there's a unique     poll waiting to be answered.

    Are more than 12 notes in music valid?

    Now is as good as time as any to explain why some people use more than 12 notes in a 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 scale with pitches close to the Just Intervals (more notes in the scale will be more likely to do this). In essence, I suppose one could call these scales 12-note, because only really 12 notes are actually used (the ones closest to the supposedly 'ideal' Just Intervals). All of the the rest are mostly scrapped - with a few exceptions*. It turns out that some of the best equal tempered scales to accomplish this are 19, 31 or 72 notes per octave. However as I have already explained, it is my opinion that these ratio pitches (or 'Just intervals') are ultimately flawed, so I think it's a false objective to try and get close to them... =P

    Anyway, there are quite a few people who seem to think otherwise. Take a look at this (off-site) URL for all the scales with more than 12 notes:     List of unorthodox equal tempered scales
    * These exceptions include notes not available in 12-eT such as the ratios: 7/6, 7/4, 9/7, or 11/9. Once again though, I believe that such intervals aren't needed (or even appropriate) in the best music.

    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' 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 microtonal 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 microtonal music would argue that it's only because of western 'cultural conditioning' that music with more than 12 (equally spaced) notes sounds so strange.
    On the contrary, I think personally that microtonal music is essentially 'atonal' music - where the emphasis is on the 'textural' sound and ryhthm - rather than any actual harmony. If the intervals derived from these scales actually sound any good, I think it's because that those particular notes are close to the 12 'golden' pitches anyway. 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 link and see what you think: Microtonal music

    ...But don't take my word for it.

    Why not decide for yourself what interval you prefer? I have compiled a poll for you to select your favourite interval. In each temperament, you'll hear a pair of chords - one minor and one major. All you need to do is see which pair you prefer; it's as simple as that. Try and listen a few times to each one before making up your mind.

    12 equal versions

    Download the 'Ramp' wave version
    Download the 'Sine' wave version

    Just intoned versions

    Download the 'Ramp' wave version
    Download the 'Sine' wave version


    Survey: After listening to both versions, which one sounds better in your opinion?
    I prefer the 12 equal version
    I prefer the 12 equal version by a whisker
    They're both about as good
    I prefer the Just Intoned version by a whisker
    I prefer the Just Intoned version
    I hear no difference between the two
    I think both 12 Equal and Just are about as good in different ways, so both are valid intervals
    What exactly will you be hearing?
    Each version is in WAV format (if anyone wants me to upload in AIFF, email me), and each contains one minor chord followed by one major chord.
    The timbre of the waveform is a pure ramp or sine wave. The only thing I've done to all of them is add a slight vibrato. This is mainly because without this tiny (sinusoidal) vibrato, the Just intervals would sound too 'plain' (this is because the sound timbre doesn't 'evolve' with pure ratios). Basically, it takes the emphasis off the timbre, and concentrates on the raw pitch.
    For the curious, the degree (amplitude) of vibrato (for all 3 pitches in the major/minor triad) is a difference of 1.003 (5.2 cents) either way. But the wavelength (frequency) of vibrato is slightly different for each pitch in the triad, so as to limit the amount of constructive interference.









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