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Thread: What is a diatonic Scale?

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    Senior Member isorhythm's Avatar
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    Quote Originally Posted by millionrainbows View Post
    No, your problem is that you accept axiomatic definitions without really discovering them for yourself. This much is obvious. And it's very boring.
    My problem is that you start these threads and never seem to move beyond the axiomatic definitions. If the questions you're really interested in are WHY the diatonic scale is what it is, and why it's so widely used across cultures, just say so. I agree those are interesting questions.

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    Quote Originally Posted by isorhythm View Post
    My problem is that you start these threads and never seem to move beyond the axiomatic definitions. If the questions you're really interested in are WHY the diatonic scale is what it is, and why it's so widely used across cultures, just say so. I agree those are interesting questions.
    Well, gee, Iso, I'm sorry you have a problem with that. Maybe you can figure out a way, within yourself, to enjoy them.

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    Senior Member isorhythm's Avatar
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    What are your thoughts about the origin and significance of the diatonic scale?

    Just skip the definitions - we all know those already. Move on to the substance of your ideas, if you have any.

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    Quote Originally Posted by isorhythm View Post
    What are your thoughts about the origin and significance of the diatonic scale?

    Just skip the definitions - we all know those already. Move on to the substance of your ideas, if you have any.
    \

    Well, after that little remark, I seem to have lost my motivation.

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    For this post we look at intervals on the piano as if they were pure ratios, sort of suspending disbelief, as it were--think close enough for jazz. This is not about micro-tuning (until the very end); it requires the opposite, that we're cool with the close-enough-to-pure nature of the intervals of ET, so that we can explore the intriguing symmetry of the 12-tone system and figure out how the Diatonic scale arose from it.

    Okay, so here is the scenario and these are my observations: We're not concerned with where the Diatonic scale came from historically right now; we're in the present tense, at the piano, looking at it as an eternal thing, this set of 7 diatonic and 5 pentatonic modes (that is, the pattern of white keys and black keys). Pianos will have the exact same pattern a thousand years from now. Why does this pattern seem eternal? Because if you try changing the pattern of blacks and whites, it can't be done within a 12 tone system. There is only one way lay them out, and that's with 5 blacks and 7 whites, in the pattern that the are in. Any other pattern (like 4 blacks and 8 whites, 6 and 6, etc.) fails the test of giving each individual note a distinct identity within the sequence. For instance, in the 6 plus 6 version you get two whole tone sequences, but every white key looks the same, and every black key looks the same. Try it for yourself with pencil and paper--there is only one system, the one we have. So my theory is that Mother Nature requires this system which divides 12 into 7 and 5, regardless of how it came to be in human history.

    Of course there are all manner of ancient scales with different sequences of whole and half steps, and some with leaps, but they would all involve a mixture of these two opposing symmetry forces, 7 and 5, or Diatonic and Pentatonic. For some reason the Greek church modes all wound up being Diatonic or "pure" in this mathematical way, partaking of only the white keys. The gypsy modes and so on were more of a hybrid. That's cool and they sound great and everything, but they are just mathematically less "meant to be" by the shape of nature itself. Like a trapazoid is less "pure" than a square. Maybe it was because the Greeks themselves were so in to mathematical purity? In any case, there's no denying that the modes are all on the white keys, and there is only one possible set of those, the ones we have.

    There is another way to witness this mathematical truth, that 7 and 12 are inextricably linked in sound and music, not just by cultural bias, but by Nature itself. If you look at a P5th on the piano, say A to E, that would be 7 half steps, or 7/12ths of the way up to the octave C. But in Hz, it would be 440 to 660, which is only half the way to A 880 (the octave of A 440). Interesting, huh? To go half way by one measure you've travelled 7/12ths of the way by the other measure. Of course we all know it's a logarithmic scale, so we wouldn't expect the answer to be 6 half steps equals 660, but it's just interesting that the answer is, again, 7 out of 12. The headline is that the P5th and P4th divide the octave into 7 and 5, if we are to approve of the resulting sounds as "consonance", just as the physical keyboard divides the octave into 7 and 5, if we are to make any sense of it by looking at it. It just is, like a cube or an icosahedron, two of the Platonic solids. And, as an aside, there are only 5 Platonic solids (objects with perfect 3d symmetry), and two of them--the Icosahedron and the Dodecahedron--exhibit the inherent truth of the shape of Nature by having 12 axes of symmetry, just like our system of music. (Somebody stop me before I lapse into talking about the golden ration...)

    This has been a lot about the 7 part of the 12 (Diatonic), but it's also worth mentioning that the negative image of Diatonic, which is of course Pentatonic, also seems to arise naturally from all ends of the earth as a sort of cross-cultural norm, or "fact of Nature". I'm running out of mojo right now, so I'll just leave it at that.

    One more thing before I call it a night, and this one goes out to BabyGiraffe:

    We start with the stipulation that the reason 12-tone music works out well for the ear is the mathematical “coincidence” that the 12th root of 2, raised to various whole number powers, happens to create a set of intervals that are reasonably close to the pure, Pythagorean simple ratio intervals (1/2; 2/3; 3/4; 4/5; 5/6; 3/5; 5/8; etc.). For instance the 12th root of 2 to the 7th is approx. 3/2. So the question is: What do the other possible equal tempered (ET) scales sound like?

    I found 5-tone ET tuning by ear, playing a DX-7 with microtuning capability years ago, and also discovered an interesting phenomenon concerning the numbers 7 and 5, which also happen to be the number of white and black keys on the piano keyboard. Seems the next ET tuning (after 5-tone) that sounded musical (in a primitive sense) was the 7-tone ET scale. Then of course there is the 12-tone scale, then the next two that work out reasonably well are the 17-tone and 19-tone scales, then the 29-tone and 31-tone scales, and so on. I began to see a pattern emerging (besides the fact that the last two are Mersenne pairs). The scales that come close to musical simple ratio consonances, 5; 7; 12; 17; 19; 29; 31; etc., can be broken down like this: 5; 7; 5+7; 5+7+5; 7+5+7; 5+7+5+7+5; 7+5+7+5+7; etc.

    So what is a diatonic scale? It's yet another manifestation of what is true and eternal in music and more broadly in reality, like the Platonic Solids, the overtone series, standing waves, and many other musical and acoustical natural phenomena.

    Thanks for reading.

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    Quote Originally Posted by Wes Lachot View Post
    This has been a lot about the 7 part of the 12 (Diatonic), but it's also worth mentioning that the negative image of Diatonic, which is of course Pentatonic, also seems to arise naturally from all ends of the earth as a sort of cross-cultural norm, or "fact of Nature". I'm running out of mojo right now, so I'll just leave it at that.





    I found 5-tone ET tuning by ear, playing a DX-7 with microtuning capability years ago, and also discovered an interesting phenomenon concerning the numbers 7 and 5, which also happen to be the number of white and black keys on the piano keyboard. Seems the next ET tuning (after 5-tone) that sounded musical (in a primitive sense) was the 7-tone ET scale. Then of course there is the 12-tone scale, then the next two that work out reasonably well are the 17-tone and 19-tone scales, then the 29-tone and 31-tone scales, and so on. I began to see a pattern emerging (besides the fact that the last two are Mersenne pairs). The scales that come close to musical simple ratio consonances, 5; 7; 12; 17; 19; 29; 31; etc., can be broken down like this: 5; 7; 5+7; 5+7+5; 7+5+7; 5+7+5+7+5; 7+5+7+5+7; etc.
    7note diatonic is made of 2 pentatonics on F and G, giving a third pentatonic as incidence structure.

    I hope you have heard about syntonic comma - both 5 and 7 equal are meantone temperaments.

    About other scales -
    12 as 5+7, ok, but it also supports the anti-meantone-12 -> schismatic temperament (it can be see called "Helmholtz" on microtonal wiki, because H. von Helmholtz was a fan of it.)

    So, 12 + 5 = 17 has also two mappings - one schismatic and one meantone and the meantone one is bad - major and minor thirds
    are on the same step -> 352.941176 cents.

    17 + 7 = 24 = 1/4 tone scale, which is neither meantone, nor schismatic unless we use that neutral third generator and it will have two mappings again.

    17 + 5 = 22 - This has 3 mappings and meantone is the worst one, the best one is diaschismic (the difference between syntonic comma and schisma is diaschisma).

    19 is meantone = 12+7

    29 has both meantone and schismatic, but again schismatic is better. The interesting part about 29 is that it's the opposite of 12 equal - the error of all intervals is almost the same, but they are tuned in the opposite directions - flat major thirds, sharp minor thirds etc (assuming schismatic temperament). Also, if we use the best approximations of intervals relative to just intonation, it is neither schismatic or meantone, but some kind of inconsistent system.
    22 and 29 are anti-duals in the 17-tone scales that come after 12. 22's temperament is actually "Super- or Hyper-pythagorean", because the fifth is sharper than in Schismatic.

    19+5 = 26, again meantone, but is almost out of tune.

    22+5 = 27 = Superpythagorean

    26+5 =31 or 19+12 = meantone - 1/4 comma

    31+19 = 50 or (43 + 7) meantone - 2/7 comma
    31+12 = 43 meantone -1/5 comma
    43+12 = 55 meantone - 1/6 comma

    31, 43, 50 and 55 are the most famous historical meantone systems.

    About fibonacci or golden ratio = in meantone it is 5+7 or maybe 2+5 sequence, if we assume 2 equal (tritone tuning ) supports meantone; let's see - 2,5, 7, 12, 19, 31, 50,81 etc.




    Still, diatonic scale doesn't have to be tempered in meantone. Imo, it's a rank-3 tuning with 3 generators - major third, perfect fifth and octave - the most obvious is to construct it as chain of three triads. The problem is that we can get various permutations of this, differing by syntonic comma, so this rank3 JI- diatonic scale has 10 "modes" as permutations (and neapolitan major and melodic minor also have 10 permutations each).

    Another cool scale is Double harmonic. The difference between augmented second in 5-limit and septimal minor third ( or very sharp second) 7/6 is 225/224. So, all these scales like harmonic major or minor/double harmonic/neapolitan minor etc that temper this can be thought as higher rank scales. Out of these double harmonic looks like it has the most symmetry.

    Another way to look into double harmonic etc - if we start from 7 equal and project it into 10 equal, we get the most even even heptatonic in 10. We then project it into 12. Various modes will correspond to double harmonic or harmonic minor etc.
    Diatonic and pentatonic can be thought as projections of 7 or 5 equal into 12.

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    The negative image of meantone heptatonic is actually in 16 equal (in semi-decent tuning). There, if you stack 3 fifths (C-G-D-A), you get a major (A-C is major third), not minor third as difference to octave. So, it's super flat and sounds somewhat out of tune or like when some untrained singers try to perform. It was actually found in some African country as xylophone tuning.
    But you can transfer music from meantone-7 to it - like we can transfer music from meantone-12 to schismatic-12.

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    Yeah, BG, I've heard of this stuff, read Partch etc. I knew you'd get off on this. Thanks for your extremely illuminating post.
    Last edited by Wes Lachot; Aug-14-2020 at 19:13.

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    Quote Originally Posted by BabyGiraffe View Post
    The negative image of meantone heptatonic is actually in 16 equal (in semi-decent tuning).
    By "negative image" I mean just what you see on the piano--the blacks are a "negative image" of the whites, in terms of their spacings, in ET, of course. Sorry if I did not make this clear enough.

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    Quote Originally Posted by Wes Lachot View Post
    By "negative image" I mean just what you see on the piano--the blacks are a "negative image" of the whites, in terms of their spacings, in ET, of course. Sorry if I did not make this clear enough.
    https://en.wikipedia.org/wiki/Complement_(set_theory)

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    Quote Originally Posted by Wes Lachot View Post
    So what is a diatonic scale? It's yet another manifestation of what is true and eternal in music and more broadly in reality, like the Platonic Solids, the overtone series, standing waves, and many other musical and acoustical natural phenomena.
    So what is a diatonic scale? Half-steps and whole steps only.

    An octave can be divided 2+2+1+2+2+2+1.

    This has been a lot about the 7 part of the 12 (Diatonic), but it's also worth mentioning that the negative image of Diatonic, which is of course Pentatonic, also seems to arise naturally from all ends of the earth as a sort of cross-cultural norm, or "fact of Nature". I'm running out of mojo right now, so I'll just leave it at that.

    It comes from the same place 12 came from: stacking fifths. C-G-D-A-E. But stopping early.
    Last edited by millionrainbows; Aug-15-2020 at 13:11.

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    Quote Originally Posted by millionrainbows View Post
    So what is a diatonic scale? Half-steps and whole steps only.

    An octave can be divided 2+2+1+2+2+2+1.



    It comes from the same place 12 came from: stacking fifths. C-G-D-A-E. But stopping early.
    MR, honestly, are you trolling or what? Did you read anything I wrote before in the last few year?
    Or ancient people had time machines, so they could visit us in more modern times and use logarithms to and tuners to "invent" your "canonical" theory?

    CGDAE has no proper tuning without false relations outside of meantone tuning (the same can be said about heptatonic). Stacking pure fifths doesn't even give meantone system, it gives the opposite - schismic (until 665 equal) -> the octave will consist of 7 limmas (90.225 cents ) and 5 apotomes ( 113.685 cents). So, medieval music was in the opposite tuning of Renaissance and Baroque, and most of Classical era.

    In meantone the octave is 7 diatonic semitones and 5 chromatic semitones (and everyone knows that diatonic semitone is bigger than chromatic semitone). The difference between these two type of steps is called "Minor diesis".
    The difference between apotome and limma is called Pythagorean comma.

    Even the idol of "Talkclassical" Bach didn't like or use 12 equal (which should have been possible to be tuned even by ear back then by listening to beats). I don't know what is your obsession with trying to ahistorically to come with wrong theories.

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    Quote Originally Posted by BabyGiraffe View Post
    CGDAE has no proper tuning without false relations outside of meantone tuning (the same can be said about heptatonic). Stacking pure fifths doesn't even give meantone system, it gives the opposite - schismic (until 665 equal) -> the octave will consist of 7 limmas (90.225 cents ) and 5 apotomes ( 113.685 cents). So, medieval music was in the opposite tuning of Renaissance and Baroque, and most of Classical era.
    None of that matters, because we're in ET. The reason the pentatonic is more consonant than a major scale is because of what it doesn't have: the tritone.

    In meantone the octave is 7 diatonic semitones and 5 chromatic semitones (and everyone knows that diatonic semitone is bigger than chromatic semitone). The difference between these two type of steps is called "Minor diesis".
    The difference between apotome and limma is called Pythagorean comma.
    What does it matter if there were originally two different sized semitones? All that matters is the "12" cycle and how fifths are a 12-cycle. ET has taken care of this, but it's always been essentially true.

    Even the idol of "Talkclassical" Bach didn't like or use 12 equal (which should have been possible to be tuned even by ear back then by listening to beats). I don't know what is your obsession with trying to ahistorically to come with wrong theories.
    But Bach was primarily interested in diatonic music, so he came up with his own "well" tempering system; but the fact that it makes all 12 keys usable points towards "12-ness."

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    Quote Originally Posted by millionrainbows View Post
    None of that matters, because we're in ET. The reason the pentatonic is more consonant than a major scale is because of what it doesn't have: the tritone.



    What does it matter if there were originally two different sized semitones? All that matters is the "12" cycle and how fifths are a 12-cycle. ET has taken care of this, but it's always been essentially true.



    But Bach was primarily interested in diatonic music, so he came up with his own "well" tempering system; but the fact that it makes all 12 keys usable points towards "12-ness."


    Nah, intervals around semitones in any tuning are more dissonant than any interval around tritone (in any tuning).


    You are now free to play in any theoretical tuning or temperament. Cultural police won't come to arrest you for ruining the status quo.

    Btw, acoustic instruments are not really in any ET or even any precise "theoretical" temperament... unless you autotune the recording (and even guitars and pianos are not perfectly in-tune to 12 ET, because of inharmonicity). (You can play in up to 1200 ET on saxophone, if you are that good, right?)

    The fact that Bach used all keys means that he played organ or harpsichords with split keys or had higher tolerance to mistuning than all of his contemporaries (well temperaments were never mainstream - it was all about meantone for 3 centuries at least), because well temperaments have more alien intervals than any regular tuning - in one key the fifth may be perfect in another up to 10-15 cents sharper or flatter; the same is valid for major or minor thirds.
    Last edited by BabyGiraffe; Aug-15-2020 at 15:17.

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    The current thinking is that diet tonic is as bad for us as the regular kind. It can definitely put weight on you, so it's wise to keep a diet tonic scale in the bathroom.

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