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Thread: Is the chromatic scale a result of the diatonic scale?

  1. #1
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    Is the chromatic scale a result of the diatonic scale?

    Music theory beginner looking for guidance, here.

    For the longest time I had been battling myself in my own head, trying to understand exactly how our system of music evolved the way it did, especially in regard to the note nomenclature. It struck me as arbitrary, the way that the notes are named according to the diatonic gamut, and then the chromatic notes are seen as "in between" them, even though all twelve notes are equally spaced apart, in terms of cents (at least in 12TET).

    Then somebody suggested to me that, rather than diatonicism being a result of chromaticism, it was the other way around. That the diatonic notes came FIRST, and then a tuning system was built around that. Can anybody verify this? Even better, can somebody point me to a book or a website that explains this, in thorough detail?

    I want to know WHY the diatonic gamut sounds to us the way it does. I want to know WHY there are two semitones, two and three whole tones apart respectively, and WHY it's so damn universal.

  2. #2
    Registered User xyzzy's Avatar
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    I'm sure some others will take the time to give you some exposition on this very interesting topic. Meanwhile here's a document that will keep you occupied :-) ... http://www.midicode.com/tunings/Tuning10102004.pdf

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    Registered User Malcolm's Avatar
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    WOW! Good question. You must be an engineer.

    Going to give you a music dictionary and let you see what you can come up with.

    http://www.music.vt.edu/musicdictionary/

    Diatonic
    (DIE-uh-toh-nik)
    --------------------------------------------------------------------------------
    Proceeding in the order of the octave based on five tones and two semitones. The major and natural minor scales and the modes are all diatonic In the major scale, the semitones fall between the third and fourth tones and the seventh and eighth tones. In the minor scale, the semitones fall between the second and third tones and the fifth and sixth tones.
    Chromatic
    (kroe-MA-tik)
    --------------------------------------------------------------------------------
    Any music or chord that contains notes not belonging to the diatonic scale.
    Music which proceeds in half steps.
    I chalk things like this up to the old guys decided that a long time ago. I accept it as old guy stuff and move on. If I had to come up with which was first I'd say chromatic which would be just half tone movement was first then the diatonic scales with whole tone and half tone came from that. Another reason is when we start out our journey on how scales are made we go to the chromatic to start learning about WWHWWWH. So, my vote is with chromatic first.

    Joh is the resident history buff and I'm sure he will jump on this.
    Last edited by Malcolm; 06-20-2011 at 09:35 PM.

  4. #4
    bitter old fool Jed's Avatar
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    Jon can indeed fill you in on the info you are asking about. But before he does and before you get completely lost in the details - Understand that the answer to why things are this way is very complicated and convoluted. If your interest lies in that direction OK, but know this - understanding how all of this came about will not be particularly helpful with your goal of learning to understand music theory.

    Music theory only makes sense from a musical perspective. That is to say it will make no sense until you've memorized a bunch of music theory fundamentals. By that time, you won't care why and will only care about how to manipulate the rules to make music. So my advise (if you enjoy being sane) is to forget about the why and start learning about the what.

    cheers,

  5. #5
    Registered User xyzzy's Avatar
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    Well, sure, Western Civ equals a bunch of dead white guys .

    Here's my oversimplified take: For a long time, people realized shorter vibrating strings sounded in higher pitches than longer ones at a comparable tension. Some old Greek science types created a "monochord" single-strand instrument to get some more precise measurements. They notice that a subdivision of 1/2 creates a tone with very similar qualities. They further notice that a subdivision of 2/3 or 1/3 on the strings also created a tone which sounds pleasant alongside the tone from a full-length one. That second tone is a perfect fifth, which is the inverse of the perfect fourth.

    Now you can create four tones: two of them an octave apart, and the other two tones in between, and they will comprise a perfect fourth and a perfect fifth. You now have a scale of R-4-5-R ... And these show up in non-Western music as well -- how could they not?

    Now, how to subdivide the two larger "sections" ... How about 1/3 of the 1/3 (or, rather, the length of the full string less this amount) of the length of the original string? That will yield a length that fits between the large gaps about 2 and a half times ... and which is also "about" the same length as the separation between those two intermediate "perfect" lengths between the octaves that yield those nice tones . ... and there are lots of ways to try to do this with ever-more-precise fractions. This is more where the arbitrariness begins, but there are a lot of interesting fractional games you can play with those those non-perfect tones.

    And either those are a couple of tetrachords lined up into a few variations of 7-note sets to create the Major Scale, or I'm a big fat lyre.
    Last edited by xyzzy; 06-20-2011 at 09:19 PM.

  6. #6
    Registered User JonR's Avatar
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    Quote Originally Posted by inf4nticide View Post
    Music theory beginner looking for guidance, here.

    For the longest time I had been battling myself in my own head, trying to understand exactly how our system of music evolved the way it did, especially in regard to the note nomenclature. It struck me as arbitrary, the way that the notes are named according to the diatonic gamut, and then the chromatic notes are seen as "in between" them, even though all twelve notes are equally spaced apart, in terms of cents (at least in 12TET).

    Then somebody suggested to me that, rather than diatonicism being a result of chromaticism, it was the other way around. That the diatonic notes came FIRST, and then a tuning system was built around that. Can anybody verify this? Even better, can somebody point me to a book or a website that explains this, in thorough detail?
    The midicode site xyzzy linked to is the best I know.
    In particular check out the so-called "Greater Perfect System", which shows the Pythagorean proportions across two octaves:
    http://www.midicode.com/tunings/greek.shtml#2.6
    Quote Originally Posted by inf4nticide View Post
    I want to know WHY the diatonic gamut sounds to us the way it does.
    Frequency ratio, in short. If the frequencies of two pitches are in a simple ratio with one another, they will sound related - they will blend, or sound "consonant". The common consonant intervals have the following ratios:
    Octave = 2:1
    Perfect 5th = 3:2
    Perfect 4th = 4:3
    Major 3rd = 5:4
    Major 6th = 5:3
    Minor 3rd = 6:5

    Notice all of these use only factors of 2, 3 or 5. This is known as "5-limit" tuning, and produces a scale very close to our diatonic scale of tones and semitones. Other intervals in the scale have the following ratios:

    Minor 6th = 8:5 = inverted major 3rd
    Major 2nd = 9:8 (or 10:9)
    Minor 7th = 16:9 (or 9:5) = inverted major 2rd
    Minor 2nd =

    You can see this system ends up with a "tone" of two sizes (9:8 is a little larger than 10:9), and a semitone i not quite half a tone.



    I want to know WHY there are two semitones, two and three whole tones apart respectively, and WHY it's so damn universal.[/QUOTE]

  7. #7
    Registered User JonR's Avatar
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    Quote Originally Posted by inf4nticide View Post
    Music theory beginner looking for guidance, here.

    For the longest time I had been battling myself in my own head, trying to understand exactly how our system of music evolved the way it did, especially in regard to the note nomenclature. It struck me as arbitrary, the way that the notes are named according to the diatonic gamut, and then the chromatic notes are seen as "in between" them, even though all twelve notes are equally spaced apart, in terms of cents (at least in 12TET).

    Then somebody suggested to me that, rather than diatonicism being a result of chromaticism, it was the other way around. That the diatonic notes came FIRST, and then a tuning system was built around that. Can anybody verify this? Even better, can somebody point me to a book or a website that explains this, in thorough detail?
    The midicode site xyzzy linked to is the best I know.
    In particular check out the so-called "Greater Perfect System", which shows the Pythagorean proportions across two octaves:
    http://www.midicode.com/tunings/greek.shtml#2.6
    Quote Originally Posted by inf4nticide View Post
    I want to know WHY the diatonic gamut sounds to us the way it does.
    Frequency ratio, in short.
    If the frequencies of two pitches are in a simple ratio with one another, they will sound related - they will blend, or sound "consonant".
    This, in turn, is because of the harmonic series. Every musical note is a spectrum of partial vibrations, most of them mujtiples of the "fundamental" pitch that we hear (by which we identify the note). Notes whose fundamentals are in a simple ratio will share some harmonics, which is how we perceive the affinity between them. We can't, of course, hear the vibrations (as vibrations), but we can hear interference, when frequencies clash.

    The common consonant intervals have the following ratios:

    Octave = 2:1
    Perfect 5th = 3:2
    Perfect 4th = 4:3
    Major 3rd = 5:4
    Major 6th = 5:3
    Minor 3rd = 6:5

    Notice all of these use only factors of 2, 3 or 5. This is known as "5-limit" tuning, and produces a scale very close to our diatonic scale of tones and semitones.

    Other intervals in the scale have the following ratios:

    Minor 6th = 8:5 = inverted major 3rd
    Major 2nd = 9:8 or 10:9
    Minor 7th = 16:9 or 9:5 = inverted major 2rd
    Major 7th = 15:8
    Minor 2nd = 16:15 (inverted major 7th)

    If you play guitar, you can check some of these against the fretboard (because frequency ratio relates to string fractions - half-string length doubles the frequency, etc):

    12th fret (octave) = half-way between bridge and nut
    7th fret (perfect 5th) = 2/3 of way between bridge and nut
    5th fret (perfect 4th) = 3/4 of way between bridge and nut
    4th fret (major 3rd) = 4/5 of way between bridge and nut
    3rd fret (minor 3rd) = 5/6 of way between bridge and nut
    2nd fret (major 2nd) = 8/9 (or is 10/9?) of way between bridge and nut

    Of course, these are not exact, because the frets are calculated according to 12-TET, but they are pretty close.
    The difference between 2/3 and 3/4 (frets 5-7) is 1/12 - ie 1/12 of the string length. The ratio difference is a whole tone of 8:9 (because we are comparing 8/12 with 9/12.)
    The difference between 3/4 and 4/5 (frets 4-5) is 1/20 (ratio difference of 15:16).
    So there is your "tone" and "semitone" - you can see that the latter is a little more than half a tone (it's 1/20, and not 1/24).
    IOW, the frets don't quite coincide with the "pure" ratio positions. 4th fret, in particular, is a little higher than its pure 4/5 position would be.

    Anyway, this system ends up with a "tone" of two sizes (9:8 is a little larger than 10:9 - the scale contains both kinds), and a semitone is not exactly half a tone.
    So while this scale sounds good, it's not very practical for designing or tuning instruments. If you tune an instrument to these frequencies, you can only play in one key (guitar frets would need to be spaced differently on different strings - and differently again if the guitar was tuned differently). You can't take another note in the scale, make that a key note, and include any of the other notes, because some of them will be out of tune. (Of course you will need chromatic alterations of some notes, but then how do you divide a tone, if a semitone is not exactly half a tone? and which of the two tones do you pick?)

    In fact, you can cobble together other approximate major scales (without retuning), but they will sound different: the ratios will not be quite the same. (This is the source of the idea that different keys have different qualities, eg F major being "pastoral", or whatever. Before 12-TET, this idea had some foundation, at least it was true that different keys had different sounds. The more remote the key from your default one (C major), the more out of tune it would be. And because keys are therefore distinguishable, it's possible to attach different meanings to them.)

    The earlier Pythagorean system was "3-limit" - using factors of 2 and 3 only. This was considered more "pure" (philosophically at least), but resulted in a major 3rd of 81:64 - a complex ratio, and therefore dissonant. (It's a little sharp of both the 5:4 "pure" 3rd, and our ET 3rd.) This was why, in early medieval music, 3rds were not used: harmony was very basic, and used 4ths and 5ths only. This meant the music sounded very pure, but was (obviously) limited. (But then in those days, even 4ths and 5ths were considered adventurous!)

    5-lmit tuning gave better-sounding 3rds, but brought other problems (principally the two sizes of tone). This led to keyboard instruments with 17 keys per octave (because A# and Bb would be two different notes).

    12-TET was a long time coming (centuries), and resistance against it was no doubt due to the fact that it meant every note (apart from octaves) would be out of tune. But in the end, the advantage of being able to play in any of 12 different keys - all of them equivalent - on keyboards with only 12 notes per octave, without retuning, outweighed the slight out-of-tuneness of everything.
    (Luckily our ears have a threshold of tolerance: an interval can be "near enough" to its pure equivalent for it to be acceptable - at least to most ears. There are still sensitive people who find our major 3rd too sharp, being 14 cents away from 5:4.)
    Quote Originally Posted by inf4nticide View Post
    I want to know WHY there are two semitones, two and three whole tones apart respectively, and WHY it's so damn universal.
    Well, it's not exactly universal. The west adopted 12-TET to suit the demands of our developing harmonic system: it was crucial for us that - first of all - our pitches could be combined in various simutaneous stacks ("chords") that sounded consistent and meaningful; and secondly that these chords could be translated through all the possible keys without retuning our instruments.
    The diatonic scale with its semitones placed where they are means 12 different keys are possible. And if we make the 12 semitones exactly equal, then all keys are equivalent, and chords are easy to construct and handle.

    Other cultures don't use chords, so they are freer to (a) use pure intonation in their scales, and (b) use a greater variety of scales, and even different ways of dividing the octave. This means the music of other cultures tends to be more sophisticated melodically than western music - and often rhythmically and timbrally too. Our harmonic system necessitates a simple fixed scale (so that our chords can work reliably), and also fairly simple rhythms, so that we can hear the chords working in succession.
    (Some cultures do have a sense of harmony - of the way notes interact - such as Indonesian gamelan, but they tune to make notes deliberately clash, for the sonorities that produces; they don't have a chord system like ours.)
    Last edited by JonR; 06-22-2011 at 12:09 AM.

  8. #8
    Registered User jimc8p's Avatar
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    Quote Originally Posted by inf4nticide View Post
    ...the diatonic notes came FIRST, and then a tuning system was built around that.
    Basically yes. The diatonic scale is based on a ladder of perfect fifths. If you give a perfect fifth its own fifth, then that fifth its own fifth, and so on, you end up with the Major Pentatonic scale after 3 repetitions, the Lydian scale after 5 repetitions and the chromatic scale after 10 repetitions.

  9. #9
    Registered User JonR's Avatar
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    Quote Originally Posted by jimc8p View Post
    Basically yes. The diatonic scale is based on a ladder of perfect fifths. If you give a perfect fifth its own fifth, then that fifth its own fifth, and so on, you end up with the Major Pentatonic scale after 3 repetitions, the Lydian scale after 5 repetitions and the chromatic scale after 10 repetitions.
    That's a neat way of putting it:

    C G D A E = C major pentatonic, stack of four 5ths (consonant scale, popular in folk around the world)

    C G D A E B F# = C lydian mode, stack of six 5ths (most stable mode)

    C G D A E B F#/Gb Db Ab Eb Bb F = chromatic scale, stack of eleven 5ths (the "circle of 5ths", including a return to C as the 12th step)

    As a note to the OP, if this is done using a pure ratio of 3:2 each time, the C you arrive at (on top of that last F) is out of tune with the C you'd get by raising the octave 7 times. The difference is around 23 cents (the "pure" C on the stacked 5ths is sharp) and is known as the "Pythagorean comma": the built-in error of the Pythagorean tuning system.
    http://en.wikipedia.org/wiki/Pythagorean_comma
    Less than quarter of a semitone might not seem like much (over seven octaves, just about the entire gamut of western music), but it bedevilled European music for centuries - it seemed insoluble, until they said (essentially) "the hell with it, let's just detune all our 5ths by around 2 cents - no one will notice - and then everything will fit."

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    Though the historical evolution of the diatonic scale is long and probably not perfectly understood, the scale itself has some properties that make it easy to derive objectively as a very useful scale.

    There is a property of some scales called "Myhill's Property" (http://en.wikipedia.org/wiki/Myhill's_property). Scales with this property, are perceptually very friendly, they are very easy to follow, think about, and sing. The reason for this is they have only two step sizes for any given "generic interval". A generic interval is one like "second" or "third" that merely describes how many steps are spanned within a scale, not the exact size. The diatonic scale has Myhill's Property because there are two types of every generic interval: there are major and minor seconds, major and minor thirds, etc. (Fourths and fifths have different names like perfect and augmented or diminished, but still two sizes in the diatonic scale.)

    The only way to arrive at scales with Myhill's property is using two intervals, a "generator" and a "period". One takes the generator and stacks it on top of itself, taking each newly arrived note in the stack of generators and putting it in each period.

    For the (let's say, C major) diatonic scale, the period is the octave (an approximate frequency ratio of 2:1) and the generator is a fifth (or an approximate frequency ratios of 3:2.) Stacking the fifth of seven times arrives at these notes:

    F-C-G-D-A-E-B

    But in order for these notes to make sense, we must reduce each one back into the original octave. Doing so arrives at this sequence:

    F-G-A-B-C-D-E (or any order there-of.)

    This combination of notes has Myhill's Property as is described above, which makes it perceptually friendly. It is also very simple, that is, it has relatively few notes before it repeats, which is perceptually helpful. We usually need somewhere between 5 and 10 notes in our scales for them to seem functional.

    Finally, it also has very low "error", or difference between the frequency ratios we say its intervals represent. That is, we say from C to E is an approximate 5/4 (which is actually 386 cents*), we say from C to Eb is an approximate 6/5 (which is actually 314 cents), and we say that from C to G is an approximate 3/2 (which is actually 702 cents). Though the diatonic scale never has all of those intervals approximated perfectly, it gets pretty close, and so is low error.
    http://en.wikipedia.org/wiki/Cent_(music)

    (How we decide on what frequency ratios we are going to say each interval represents has to do with what's called the "mapping", but that's a little much to get into now. Better I leave you with this and ask if I haven't explained anything well.)

    The chromatic scale can ALSO be reached by stacking fifths on top of one another, and it ALSO has Myhill's Property. (The chromatic scale actually has TWO sizes of intervals, the minor second AND the augmented unison. The fact that they are the same size has to do with our use of 12-TET or 12-edo) It also is just as *accurate* as the diatonic scale. The one thing it doesn't have that the diatonic scale has is *simplicity*. Once you get to 12 notes in an octave, it is much harder to follow along. So the diatonic scale tends to work as the fundamental scale, while it operates within a (more complicated) chromatic sort of gamut.

    Does any of that make sense?

    John M

  11. #11
    Registered User JonR's Avatar
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    Quote Originally Posted by JlMoriart View Post
    Though the historical evolution of the diatonic scale is long and probably not perfectly understood, the scale itself has some properties that make it easy to derive objectively as a very useful scale.

    There is a property of some scales called "Myhill's Property" (http://en.wikipedia.org/wiki/Myhill's_property). Scales with this property, are perceptually very friendly, they are very easy to follow, think about, and sing. The reason for this is they have only two step sizes for any given "generic interval". A generic interval is one like "second" or "third" that merely describes how many steps are spanned within a scale, not the exact size. The diatonic scale has Myhill's Property because there are two types of every generic interval: there are major and minor seconds, major and minor thirds, etc. (Fourths and fifths have different names like perfect and augmented or diminished, but still two sizes in the diatonic scale.)

    The only way to arrive at scales with Myhill's property is using two intervals, a "generator" and a "period". One takes the generator and stacks it on top of itself, taking each newly arrived note in the stack of generators and putting it in each period.

    For the (let's say, C major) diatonic scale, the period is the octave (an approximate frequency ratio of 2:1) and the generator is a fifth (or an approximate frequency ratios of 3:2.) Stacking the fifth of seven times arrives at these notes:

    F-C-G-D-A-E-B

    But in order for these notes to make sense, we must reduce each one back into the original octave. Doing so arrives at this sequence:

    F-G-A-B-C-D-E (or any order there-of.)

    This combination of notes has Myhill's Property as is described above, which makes it perceptually friendly. It is also very simple, that is, it has relatively few notes before it repeats, which is perceptually helpful. We usually need somewhere between 5 and 10 notes in our scales for them to seem functional.

    Finally, it also has very low "error", or difference between the frequency ratios we say its intervals represent. That is, we say from C to E is an approximate 5/4 (which is actually 386 cents*), we say from C to Eb is an approximate 6/5 (which is actually 314 cents), and we say that from C to G is an approximate 3/2 (which is actually 702 cents). Though the diatonic scale never has all of those intervals approximated perfectly, it gets pretty close, and so is low error.
    http://en.wikipedia.org/wiki/Cent_(music)

    (How we decide on what frequency ratios we are going to say each interval represents has to do with what's called the "mapping", but that's a little much to get into now. Better I leave you with this and ask if I haven't explained anything well.)

    The chromatic scale can ALSO be reached by stacking fifths on top of one another, and it ALSO has Myhill's Property. (The chromatic scale actually has TWO sizes of intervals, the minor second AND the augmented unison. The fact that they are the same size has to do with our use of 12-TET or 12-edo) It also is just as *accurate* as the diatonic scale. The one thing it doesn't have that the diatonic scale has is *simplicity*. Once you get to 12 notes in an octave, it is much harder to follow along. So the diatonic scale tends to work as the fundamental scale, while it operates within a (more complicated) chromatic sort of gamut.

    Does any of that make sense?
    Does to me!

    At least, it's a good mathematical description of the kind of scales that are most popular (worldwide).

    However, I don't believe it explains much. Unless one has a good singing voice, or a good ear, one can't tell that a scale is composed only of whole or half steps, or that a whole step is twice a half-step - at least I can't. (We might just be able to tell that some steps are larger than others.) In any case, the harmonic minor and byzantine scales are pretty popular (at least in other cultures), and they contain one or two augmented 2nds, as well as major and minor ones. And Indian music recognises 22 octave divisions (although it does generally use 7-note scales in practice).
    We don't really care how many notes are in a scale, as long as it's not too many, and that no two notes are too close to tell the difference between them. (But I think we'd accept the notion of bent notes: that one note can swoop smoothly to another.)

    IMO, it comes down to frequency ratios, as I said. Of course, that's still not somethng we perceive in absolute terms - "hey, that pitch is vibrating a 261 times a second!" - but we perceive the effects of frequency, in terms of how notes either blend or clash (due to interference between overtones). We hear beats between notes that are out of tune. (We don't like that effect, but they do in Indonesia.)
    This is not just a harmonic effect (notes played simultaneously), but would apply to melodies too. Eg, we can hear that the two notes of a melodic 4th, 5th or octave have an affinity, just as we can those in a harmonic 4th, 5th or octave (maybe not quite as clearly).

    If we believe the Pythagoras myth (at least as the origin of the European system), it began from a perception of consonance; that led to the discovery of simple ratio, and the fact that a good scale could be built using factors of 2 and 3 only - between string lengths or weights of metal bars - (allowing that some of the later notes needed to be tweaked into tune a little). It so happened - a result of using those simple factors - that we ended up with approximate whole and half-steps (once we reached 7 notes).
    Of course, the pentatonic scale we get to first is made of major 2nds and minor 3rds (augmented 2nds?), and is a much more friendly and popular scale than the lydian mode we get from the first 7 notes.

    Naturally, it would be possible to arrive at a good scale using one's ear only - but I doubt there would be such a neat arrangement as the WWHWWWH major scale. Different people might tune a scale differently. There would probably be between 5 and 7 different notes in it, and intervals between notes would differ. And I suspect there'd be at least a perfect 5th in the middle. But I think we might find quite a lot of variability in how the other notes are placed. (But then such an experiment would be inevitably coloured by the subjects' musical experience, both of listening and playing. That would be enough to epxlain the probable emergence of something close to the major scale, if the subjects were western.)

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    Quote Originally Posted by JonR View Post
    Does to me!

    At least, it's a good mathematical description of the kind of scales that are most popular (worldwide).

    However, I don't believe it explains much.
    Yep, that's a job for the more traditional sense of music theory ;-) Chord resolutions and form are generally not touched by any of this type of theory (yet!). It does, however, predict very well what scales we're likely to be able to hear in a musical way (or "functionally", if you will.)

    Quote Originally Posted by JonR View Post
    Unless one has a good singing voice, or a good ear, one can't tell that a scale is composed only of whole or half steps, or that a whole step is twice a half-step - at least I can't. (We might just be able to tell that some steps are larger than others.)
    You hit that one on the nose there. All that seems to matter is that the scale fits the L-L-S-L-L-L-S pattern (L=Large, S=Small) in order for it to sound "diatonic", and we *can* hear that certain intervals are bigger than others, as you mention. The closer the L and S get in size the more ambiguous the scale (like you said about two steps being too close to tell the difference), the farther apart they get the more stark the difference between major and minor. It all still sounds diatonic though.

    For example:
    In 19 equal divisions of the octave the steps for the diatonic scale are (out of 19) 3-3-2-3-3-3-2. There the ratio in size between the large and small steps is 3:2, the difference between major and minor being more ambiguous than in 12-edo (where the ratio is 2:1). In 17-edo the steps are 3-3-1-3-3-3-1. There the ratio L:S is 3:1, the difference between major and minor being far more stark.
    The ratio between the large and small steps varies depending on the size of your generator, the range of possible diatonic scales stopping at 5-edo (where the small step becomes 0 cents) and 7-edo (when the small steps equal the large steps):
    http://upload.wikimedia.org/wikipedi..._Continuum.jpg

    Quote Originally Posted by JonR View Post
    IMO, it comes down to frequency ratios, as I said... This is not just a harmonic effect (notes played simultaneously), but would apply to melodies too. Eg, we can hear that the two notes of a melodic 4th, 5th or octave have an affinity, just as we can those in a harmonic 4th, 5th or octave (maybe not quite as clearly).
    Though harmonic consonance does affect melodic blending a little bit, I do think it is not NEAR as much of a factor. I've played some very high error scales that ended up being my favorites and, conversely, I've heard some very high accuracy scales that were really awkward melodically. If you ask me, it comes down more to the error and complexity of what's called the "mapping" (which does rely on ratios, but in a more complex way). I'll explain a little bit below if you're interested:

    I've talked a lot about generators and periods, and how certain perceptually friendly scales are derived from them, like those with Myhill's Property. For example, stacks of 3:2s brought down by octaves give you the diatonic scale (5L2S). Stacking 5:4s and 6:5s gives completely different scale structures, one with a 3L4S scale and the other with a 4L3S scale. Stacking 10:9s gives you a 1L7S scale.

    Doing the math with stacking the 3:2s, you'll realize that no matter how many 3:2s you stack and bring down into the original octave, you'll never actually get a ratio with a 5 in it, like 5:4. (This is because 3:2 contains only the primes 3 and 2, and so no combination thereof will ever yield a multiple of 5.) You can get intervals pretty close to 5:4 though. For instance, up four 3:2s and dropping two 2:1s gives you 81:64, when 5/4 equals 80:64. The two are very close.

    As you well know, however, we do USE this interval (gotten to by going up four fifths and down two octaves) as an approximation of 5/4, and in meantone tunings we actually flat the fifth slightly to reach an even better approximation of 5:4. So what are we saying mathematically when we say that up four fifths and down two octaves is equal to 5:4? We're saying that 81/64 is equal to 80/64:
    81/64=80/64
    Rearranging with some algebra gives us
    81/80=1/1
    So what we're really saying is that 81:80 is treated as 1/1, or a unison. This process is called "tempering out 81/80". (As you may know, small intervals like 81/80 are called commas, and I'll refer to them as so from now on.) Mathematically the statement 81/80=1/1 is obviously never true, but musically we're basically just saying how to represent intervals with fives in them in terms of intervals with just 3s and 2s.

    *That*, right there, is the mapping. Saying how to represent intervals (containing primes like 5) using other intervals (that lack that prime like 3:2 and 2:1.)

    81/80 is a pretty small, so the difference between what we're calling a 5/4 (an approximation using 3:2s and 2:1s) and an actual 5/4 is pretty small. That means that the mapping is pretty accurate, or low error. And no matter how you actually tune the diatonic scale (like the different meantone tunings including 19-, 31-, 12-, and 17-edo), the *mapping* you're using will always be the same, and the same error. That is, you are *always* saying that up four 3:2s and down two 2:1s gives you a 5:4, regardless of the exact tuning.

    If instead you use the 4L3S scale structure you get by stacking 5:4s, you end up never hitting an interval with three in it like 3:2 (because this time 5:4 has only the primes 5 and 2). You do get pretty close to 3:2 though by stacking five 5:4s and dropping down an octave. That gives you 3125:2048, where 3:2 equals 3000:2000. If you use that intervals AS as 3:2 (like we used an 81:64 AS an 80/64 aka 5:4) then mathematically you are now tempering out 3125/3072.

    This is one of the first commas (the "magic comma") that has a comparable accuracy to 81/80 (81/80 is about 20 cents small, 3125/3072 is about 30 cents small), and as you can see it is far more complex than 81/80. There are many other scale structures with other generators and other commas to temper, but not one of them comes close to the accuracy and simplicity of 81/80. Others are extremely accurate (aka are very small) but are far more complex than 81/80 (like 32805:32768), while some are extremely simple but very inaccurate (aka are very large, like 128:125.)

    Now, does this mean we should only use diatonic scale structures and forget about all these other tunings and scales because diatonic scale structures kick everyone else's butt? No, not for me at least, but it does show why the extremely vast majority of the world's musics tended towards using pentatonic, diatonic, and chromatic scale structures: it's the accuracy and simplicity of the mapping you get when you stack 3:2s and temper out 81/80.

    John M

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