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We've talked about the conditions for diatonic and chromatic scales to be well ordered in rank-2 interval space. Briefly: assuming pure octaves, the perfect fifth has to be tuned between its 5-EDO and 7-EDO values.
Things get a lot more interesting in rank-3 interval space. Here below are the rank-3 intervals of a chromatic scale.
P1 : (0, 0, 0) :: (0, 0, 0) # 1/1
m2 : (4, -1, -1) :: (0, 1, 1) # 16/15
M2 : (1, -2, 1) :: (0, 2, 1) # 10/9
m3 : (1, 1, -1) :: (1, 3, 2) # 6/5
M3 : (-2, 0, 1) :: (1, 4, 2) # 5/4
P4 : (2, -1, 0) :: (1, 5, 3) # 4/3
d5 : (2, 2, -2) :: (2, 6, 4) # 36/25
P5 : (-1, 1, 0) :: (2, 7, 4) # 3/2
m6 : (3, 0, -1) :: (2, 8, 5) # 8/5
M6 : (0, -1, 1) :: (2, 9, 5) # 5/3
m7 : (0, 2, -1) :: (3, 10, 6) # 9/5
M7 : (-3, 1, 1) :: (3, 11, 6) # 15/8
P8 : (1, 0, 0) :: (3, 12, 7) # 2/1
They're first expressed in the rank-3 prime harmonic coordinate basis (P8, P12, M17) and then they're expressed in the Lilley comma basis (Ac1, A1, d2). I've also got their just tunings written in.
If we keep the octave pure but mistune P12 and M17 a little, how much can we mistune them while still keeping normal chromatic intervals in their usual order that we find above, the usual order that we know from 12-TET?
Let's figure it algebraically. What tuned values of P12 and M17 would make two chromatic intervals equal? Those values of P12 and M17 will also be a limit of their mistuning, subject to maintaining the chromatic order. For example, surely we don't have the normal order in which (M2 > m2) if (m2 = M2). So any tuning of (P8, P12, M17) which tempers out the augmented unison, i.e. the difference between M2 and m2, will be a chromatic limit of mistuning. Mistune P12 and M17 that far or any farther out and you won't have a normal chromatic scale anymore.
First we need to know the differences between successive chromatic intervals. There happen to be only three distinct difference intervals:
A1 = (-3, -1, 2) :: (0, 1, 0) # 25/24
m2 = (4, -1, -1) :: (0, 1, 1) # 16/15
Acm2 = (0, 3, -2) :: (1, 1, 1) # 27/25
Next, we'll write out algebraic relationships that describe the tempering of each of these intervals. The way to tune A1 that is implied by its coordinates in the (P8, P12, M17) basis is
t(A1) = t(P8)^(-3) * t(P12)^(-1) * t(M17)^(2)
We'll justly tune P8 to 2 and call t(P12) = x and t(M17) = y. Finally, tempering A1 means the whole thing will be equal to 1/1. Thus we have
2^(-3) * x^(-1) * y^2 = 1
y^2 / x = 8
as an equation that describes which tuned values of P12 and M17 will temper out A1. We'll call that equation 1.
Next we'll find an equation that described the tempering of the minor second. The prime harmonic coordinates for m2 are:
(4, -1, -1)
If we want to temper out m2 and keep octaves pure, then we have this relation:
2^(4) * x^(-1) * y^(-1) = 1
which simplifies to
x * y = 16
That's equation 2. Finally we have prime harmonic coordinate for the acute minor second, Acm2:
(0, 3, -2)
and tempering that is like
x^3 / y^2 = 1
Let's plot the equations:
There's a little region near x = 3 and y = 5 that supports chromaticism . Let's zoom in on it.
It looks a lot like a triangle, but of course we know the equations for its sides and none of them are straight lines. I'm going to call it a tricorn instead of a triangle. This is the prime harmonic tricorn of chromatic mistuning limits. I've also added lines at x = 3 and y = 5.
Let's figure out the vertices of the chromatic tricorn by by solving pairs of equations that describe its sides.
Eqns 1 and 2): (x = 2 * 2^(2/3)), (y = 4 * 2^(1/3)) // Bottom right vertex
Eqns 1 and 3): (x = 2 * 2^(1/2)), (y = 4 * 2^(1/4)) // Bottom left vertex
Eqns 2 and 3): (x = 2 * 2^(3/5)), (y = 4 * 2^(2/5)) // Top vertex
I've given the vertex coordinates in a form where its easy to see that the tuned P12 is a factor of 2 (i.e. a justly tuned octave) over a value for P5, and likewise the tuned M17 is a factor of 4 (i.e. two octaves) over a value for M3, as they should be.
Let's also figure out the chromatic limits of M17 when P12 is pure, and the chromatic limits of P12 when M17 is pure.
For a purely tuned M17, we have a low value of P12 at
x^3 / y^2 = 1, y = 5, solve for x
x = 5^(2/3)
which is the intersection of the horizontal line, (y = 5), with the left side of the tricorn.
The highest we can tune P12 while M17 is pure is
y^2 / x = 8, y = 5, solve for x
x = 25/8
this is the intersection of the red line, (y = 5), with the right side of the tricorn.
The lowest we can tune M17 while P12 is pure is
y^2 / x = 8, x = 3, y > 0, solve for y
y = 2 * sqrt(6)
and the highest we can tune M17 while P12 is pure is
x^3 / y^2 = 1, x = 3, y > 0, solve for y
y = 3 * sqrt(3)
Pretty cool right? If you want to know if some weird tuning system keeps the natural rank-3 intervals in their usual chromatic order, check how it tunes P12 and M17 and see if they fall in the tricorn. I gave equations for the sides of the tricorn before, but I suppose to see if intervals fall within it, we should have inequalities. The interior of the chromatic tricorn is defined by
y^2 / x > 8
x * y < 16
x^3 / y^2 > 1
Nice.
: Rank-3 Diatonicity
If we look at the differences between intervals of a major scale, instead of a chromatic scale, and we do the same math, we'll find a diatonic tricorn. Let's derive equations for its sides. All of the differences between adjacent major scale degrees are one of these:
m2 : (4, -1, -1) :: (0, 1, 1) # 16/15
M2 : (1, -2, 1) :: (0, 2, 1) # 10/9
AcM2 : (-3, 2, 0) :: (1, 2, 1) # 9/8
If we keep octaves pure and mistune the P12 and M17, then our prime harmonic tempering curves for each of these intervals are:
x * y = 16
x^2 = 2 * y
x^2 = 8
Lets' graph them.
More than equalities of tempering, we'd like inequalities that tell us which side of each line the point of pure tuning (x, y) = (3, 5) is on, so we can see the whole region of diatonic mistuning bounds. These happen to be
x * y < 16
x^2 < 2 * y
x^2 > 8
The overlap of the regions defined by these inequalities gives us the interior of the diatonic tricorn, which in the figure above is the obvious little tricorn with the(3, 5) point inside. Now now let's super-impose the chromatic tricorn.
The larger cyan region shows rank-3 diatonicity and the smaller dark blue-gray region shows rank-3 chromaticity. The take-away is that diatonicity is a slightly larger space than chromaticity. I think I care about chromaticity more than diatonicity, but they're both neat things to know about, say, a given tuning system.
: Chromaticity and Diatonicity for Given EDOs
Which EDOs are well diatonic or chromatic and which aren't? Let's use canonical prime harmonic definitions for EDOs, and we'll work in rank-3 interval space.
An EDO of a given number of divisions {N} will mistune P12, which is purely tuned to the third harmonic, to
round(N * log2(3))
steps, which means a frequency ratio of
t(P12) = 2 ** (round(N * log_2(3)) / N)
Likewise a given EDO will mistune M17, which is purely tuned to the fifth harmonic, to a frequency ratio of
t(M17) = 2 ** (round(N * log_2(5) / divisions)
If we call those two {x} and {y} respectively, then the tests for chromaticity and diatonicity in python look like:
is_chromatic = (y ** 2 / x > 8) and (x * y < 16) and (x ** 3 / y ** 2 > 1)
is_diatonic = (x * y < 16) and (x ** 2 < 2 * y) and (x ** 2 > 8)
Almost every EDO lands within both regions. In particular, [3, 4, 5, 6, and 8]-EDO are the only non-diatonic EDOs, and we could have guessed that everything under 7 divisions per octave would have been non-diatonic, since you need 7 distinct pitches for the diatonic scale. Also [3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, and 20]-EDO are the only non-chromatic EDOs, and we could have guessed that everything under 12 divisions per octave would have been non-chromatic since you need 12 distinct pitches for a chromatic scale. It's interesting to see that 17-EDO is poorly behaved with regard to rank-3 chromaticism since 17-EDO is usually considered well behaved when considered as a temperament over rank-2 interval space.
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