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A while ago (a long while ago, I guess — hey, I have a job, so I’m busy), I posted about how 19-tone equal temperament is a diatonic tuning.  But is it the only one?  Other than 12-TET, that is?

Obviously not.  Let’s take a look at how it works!

Under our harmonic system, we base everything on the fifth.  A perfect fifth is a frequency ratio of exactly 1.5, which translates to about 701.955 cents.  Our 12-TET scale only uses multiples of 100 cents, so it approximates the perfect fifth quite well at 700 cents.  This is far from the only possible value, however!  There’s a small range where we can arbitrarily choose our fifth, and that’s what we’re about to explore.

Let’s also define what we mean by a “diatonic tuning”.  For our purposes, a diatonic tuning is one where we can use our usual note names in a way consistent with the Circle of Fifths, aside from enharmonics.  If we count fifths up from C, we get C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, etc., and if we count down from C, we get C, F, Bb, Eb, Ab, Db, Gb, Cb, Fb, Bbb, Ebb, Abb, Dbb, etc.  In a diatonic tuning, these relationships are preserved.  Eb and Bb are always one fifth apart, for instance, and Bb and D# are eleven fifths apart.  In 12-TET, Eb and D# are enharmonic equivalents — the same note — and this makes sense because eleven fifths one way is one fifth the other way.  Other diatonic tunings will not have this specific equivalence.

We’ll talk about interval sizes as fractions of the octave rather than frequency ratios.  The 12-TET fifth, in this case, is 7/12.  Let the size of the fifth be f and work out a few relations.  From C4 to D4, we have two fifths minus an octave, so if a whole step is t, t = 2f – 1.  All whole steps have the same size of two fifths minus an octave.  From C4 to B4, we have five fifths minus two octaves, and this is also an octave minus a half step d, so 5f – 2 = 1 – d, or d = 3 – 5f.  From C4 to C#4, we have seven fifths minus four octaves, and this is a half step d, so d = 7f – 4.  Wait, how can the half step be both 3 – 5f and 7f – 4?  This can only happen for one value of f, the 12-TET value of 7/12.  Since we want to explore other tunings, we’ll consider two different sizes of half step, the diatonic half step d = 3 – 5f between different letter names like B and C or E and F, and the chromatic half step c = 7f – 4 between accidentals in the same letter name, like Bb and B or F and F#.  Notice that a diatonic half step plus a chromatic half step means C to Db to D, so it’s a whole step t.  Mathematically, t = 2f – 1 = d + c = 3 – 5f + 7f – 4 (you can check the math yourself).  An octave, therefore, is five whole steps and two diatonic half steps (by counting up the major scale, for instance), or 5(2f – 1) + 2(3 – 5f) = (10f – 5) + (6 – 10f) = 1.  The f cancels out — not very useful, huh?  That’s just a function of how we defined the intervals in terms of the fifth and octave.  The octave can’t be written in terms of the fifth because we’re defining the octave as 1.  We can, however, write it in terms of the two kinds of half steps: 7d + 5c = 1.

Notice that we have one equation in two variables.  Picking one value gives us the other, as well as the size of the fifth.  For example, if we pick d, we have c = (1 – 7d)/5 and f = (3 – d)/5.  If c and d are nonnegative, that means 0 ≤ d ≤ 1/7, and therefore 0 ≤ c ≤ 1/5 and 4/7 ≤ f ≤ 3/5.  f is therefore restricted to be between about 685.714 and 720 cents.  Since c and d are rational functions of f, any rational f, written p/q in lowest terms, will be the fifth in a diatonic equal-tempered tuning with q equal divisions of the octave.  To find diatonic tunings, it’s enough to find rational values of f between 4/7 and 3/5.  Turns out, though, that it’s easier to solve 7d + 5c = 1, and we can calculate f = 4d + 3c.

Assume we’re in n-TET.  Both c and d are therefore fractions over n.  If we change the definitions a bit, just to be confusing, we can say 7d + 5c = n, where now c and d are nonnegative integers.  Finding values for n now is easy!  Just pick some integer values for c and d, and there you go!  If we take the additional postulation that we’re only looking for primitively diatonic n-TET scales — that is, scales where every note is part of the circle of fifths — then c and d can’t have any common factors, either.  For example, d = 1, c = 8 gives a 47-TET scale where the diatonic half step is 1/47, the chromatic half step is 8/47, and the fifth is 28/47.  A different example: d = 6, c = 1 gives another 47-TET scale where the diatonic half step is 6/47 and the chromatic half step is 1/47, so the fifth is 27/47.  47-TET turns out to be the smallest equal temperament with two different primitively diatonic tunings.  I’ll hopefully have some labeled examples at the synth for at least some of these.

So, let’s explore them, shall we?  The simplest examples are ones where c or d is 0.  There’s obviously just one of each, because if c = 0 but d = 3, say, they’ll have a common factor of 3.  So we can do c = 0 and d = 1 or d = 0 and c = 1.  In the latter case, we have 5-TET, which comes with some interesting enharmonics.  The diatonic step is 0, so the distance between E and F is 0.  Our 5-note scale is therefore B/C, D, E/F, G, A, (B/C).  C# is actually D, and Db is actually C.  If we do c = 0 and d = 1, the chromatic step between C and C# is 0, so our 7-note scale is C, D, E, F, G, A, B, (C), with C# = C, Db = D, etc.  No amount of flats or sharps will actually alter a note in 7-TET.  5-TET and 7-TET are the extremes of the size of the fifth.

The next example is familiar: d = 1, c = 1.  The diatonic and chromatic half steps are the same size.  This leads to C# = Db and E# = F, and 12-TET.  There’s not much more to be said about it at this time.

After this, we have d = 1, c = 2, which gives n = 17.  The fifth is 10/17, a slightly high 705.88 cents.  Since the perfect fifth is about 701.96 cents, this fifth is about 3.9 cents sharp, only twice as much as 12-TET is flat.  It’s really a rather small amount.  However, the major third is four fifths, so it’s about 23.5 cents higher than 12-TET, which is already about 14 cents above a perfect 5/4 ratio.  This leads to a major third that’s about 40 cents above just intonation, and you can really tell.  Since the minor third is defined to be a major third below the fifth, it’s rather low as well.  The nice part is that there’s a neutral third between them.  A major third is 6/17 and a minor third is 4/17; the neutral third is 5/17, almost exactly halfway between the equal-tempered minor and major thirds.  But the problem with this is, in my opinion, the major limitation of this diatonic tuning system: spelling things correctly doesn’t mean they sound good!

We can assign note names to the 17 notes of the 17-TET scale, based on the circle of fifths.  Simply put:

C, Db, C#, D, Eb, D#, E, F, Gb, F#, G, Ab, G#, A, Bb, A#, B, (C)

Since the chromatic step is 2 divisions and the diatonic step is 1, C and Db are closer than C and C#.  The Cx and Eb are enharmonic equivalents.  It’s an odd way to think about things, I suppose.  Perhaps if our society had this 17-TET scale early on, our musical orthography would be quite different — the neutral third is D#!  On the other hand, we could adapt half-sharps and half-flats to add another dimension to our 17-tone system: Ct = Db, and Dd = C# (t is half-sharp, d is half-flat).  C – E – G and C – Eb – G sound out of tune as triads, but C – Ed – G (or, equivalently, C – D# – G) sounds different but fine.

This brings us to another limitation of our system of diatonic tunings: the third is completely ignored.  Just Intonation is based on perfect 5/4 thirds and perfect 3/2 fifths (at least 5-limit JI), but as we vary the size of the fifth, the third suffers.  This is not technically a problem, but it is if you expect music written in a diatonic tuning to sound like 12-TET.  The diatonic system we’re used to is simply not perfectly compatible with non-12-TET systems.

Our next tuning, though, fixes the issue of the third: d = 2, c = 1.  This is the 19-TET of my previous post.  The fifth is 11/19, the major third is 6/19, the minor third is 5/19.  In cents, these are 694.74, 378.95, and 315.79.  (In 12-TET, they’re 700, 400, 300; in JI, they’re 701.96, 386.31, 315.64.) The fifth is a little flat by about 7 cents.  The major third is a bit flat as well (7 cents), but it’s much closer to JI than in 12-TET, where it’s about 14 cents sharp.  The minor third, though, is about 0.15 cents sharp, which is nearly perfect!  So what’s the big caveat?  It’s the size of the diatonic half step.  Harmonically, 19-TET has very nice, resonant intervals, but melodically, it has terrible leading tones!  Their size is about 126.32 cents, compared to 100 in 12-TET.  They’re close enough to hear as leading tones, but far enough for their resolutions to sound grating.  A V-i progression will have two beautifully in tune chords whose connection is unsatisfying, at least to 12-TET-accustomed ears.  And there’s really no good way of fixing this without a gigantic number of notes per octave; the lower the third, the lower the fifth, and the lower the leading tone.  For the record, here are the notes of 19-TET:

C, C#, Db, D, D#, Eb, E, E#/Fb, F, F#, Gb, G, G#, Ab, A, A#, Bb, B, B#/Cb, (C)

Here, E# = Fb, and Cx = Db.

Harry Partch was famous for using 31-TET and 43-TET.  These would be d = 3, c = 2 and d = 4, c = 3, respectively.  31-TET has a fifth of 18/31 (696.77 cents), a third of 10/31 (387.10 cents), and the diatonic half step is still bigger than the chromatic at 3/31 (116.13 cents), though not as much.  This is probably one of the better compromises between note count and the technical details of chords and leading tones.  43-TET has a fifth of 25/43 (697.67), third of 14/43 (390.70), diatonic half step of 4/43 (111.63).  The fifth is a bit closer, the third a bit farther, the leading tone a bit closer than 31-TET.

You may, by the way, have noticed a lack of symmetry in all of these scales other than 12-TET: they’re all prime numbers.  Of these, only 12-TET will let you have symmetric subsets like the symmetric augmented triad and the diminished seventh.  Why is that?  Honestly, it’s by accident.  These are simply the numbers that can be written as 7d + 5c where d and c have no common factors.  By the way, the next one is 22 (d = 1, c = 3), which at least has a halfway point.  The one after that is 26 (d = 3, c = 1), also with a halfway point.  27 (d = 1, c = 4) has an augmented triad; 29 (d = 2, c = 3) has nothing, 31 (d = 3, c = 2, which we’ve already considered) has nothing, 32 (d = 1, c = 5) has loads of symmetry, 33 (d = 4, c = 1) has an augmented triad, 37 (d = 1, c = 6) has nothing, 39 (d = 2, c = 5) has an augmented triad, 40 (d = 5, c = 1) has loads of symmetry, 41 (d = 3, c = 4) has nothing, 42 (d = 1, c = 7) has a bunch, 43 (d = 4, c = 3) has nothing, 45 (d = 5, c = 2) has some, 46 (d = 3, c = 5) has a little, 47 (d = 6, c = 1 and d = 1, c = 8) has none but has two different diatonic representations, etc.  In general, if we want to get symmetries from a low-count diatonic tuning, we have to go to the equivalent of microtones, like quarter steps or the like.

In all this, there’s one thing we’re not doing at all: considering other harmonic possibilities.  We’re only talking about retuning our usual diatonic 12-TET to have more notes and different qualities, but there’s no radical change in concept.  Notes are spelled the same.  At the same time, just because we can use the diatonic system with a particular tuning doesn’t mean we have to.  If we don’t insist on in-tune triads, instead focusing on major seconds or something entirely different, or ditch the concept altogether, we can make beautiful microtonal and xenharmonic music without being tied to standard 12-TET concepts.

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