I really like the microtonal musical works of Ben Johnston. I'd like to make music that good. I don't have a lot of understanding of how he did his stuff, but I want to learn. In this post, I'll talk about some things he did.
He had a scale that was all made of octave-reduced overtones.
Here's how it looks when sorted by numerator:
P5: 3/2
M3: 5/4
m7: 7/4
M2: 9/8
A4: 11/8
m6: 13/8
M7: 15/8
A1: 17/16
m3: 19/16
P4: 21/16
M6: 27/16
And here it is by frequency ratio size, more compactly:
[1/1, 17/16, 9/8, 19/16, 5/4, 21/16, 11/8, 3/2, 13/8, 27/16, 7/4, 15/8, 2/1]
Start yourself out with a scale like that and see how the sound of it change how you compose. That was one of Ben's tricks. He also used a scale made of the octave-complements of that scale, namely:
[2/1, 32/17, 16/9, 32/19, 8/5, 32/21, 16/11, 4/3, 16/13, 32/27, 8/7, 16/15, 1/1]
Here's another one of his tricks: Take a bunch of super-particular ratios that multiplied together make an octave:
[16/15 * 15/14 * 14/13 * 13/12 * 12/11 * 11/10 * 10/9 * 9/8] = 2/1
Now here's Ben's genius: do a cyclic permutation of the scale in thsi way:
[12/11, 11/10, 10/9, 9/8, 16/15, 15/14, 14/13, 13/12]
And now when we accumulate these ratios multiplicatively, we get this scale:
[1/1, 12/11, 6/5, 4/3, 3/2, 8/5, 12/7, 24/13, 2/1]
which has a justly tuned m3, P4, P5, and m6. It only has one (neutral) second, and one (minor) third, so it doesn't support chromaticism at the low end as well as the previous scale did, but you can still make some cool music out of it. I believe he called this scale "Eu15".
Ben also used utonality in addition to otonality, which is no big surprise since he was a student of Harry Partch. I should include some examples here.
Or, let's just explain the concepts first. You can define a chord like [4:5:6:7] as a short hand for (4/4, 5/4, 6/4, 7/4). Just divide everything through by the first element. Overtones, kind of. Divided by four, in this case. Otonality. Nice.
This chord is of course better known as
(1/1, 5/4, 3/2, 7/4).
a major chord with a 5-limit just third and a harmonic seventh. The relative steps of the chord are (5/4, 6/5, 7/6).
Undertones are not produced by harmonic instruments the way that overtones are, but people still make interesting music by inverting overtone chords. If we write descending integers, [7:6:5:4], that's shorthand for (7/7, 7/6, 7/5, 7/4), with relative intervals (7/6, 6/5, 5/4). We can represent that same chord with ascending integers by inverting all the fractions from the otonal chord
(1/1, 4/5, 2/3, 4/7)
then reversing the order, and then multiplying through by the least common multiple of the denominators, which happens to be 105 for (7, 3, 5, 1):
(4/7, 2/3, 4/5, 1/1) * 105 = [60:70:84:105]
So that's an ascending way of representing the otonal inverse of [4:5:6:7], but it sure hides all of the structure, so [7:6:5:4] is probably the better thing to use.
When I say that Ben Johnston used utonality in addition to otonality, I mostly mean that he would play a chord like
(1/1, 5/4, 3/2, 7/4)
and soon after also play a chord like
(1/1, 7/6, 7/5, 7/4)
not necessarily on the same root. When you add a note to an otonal chord, you get a totally different utonal chord. For example, if we didn't have the harmonic seventh but just a plain 5-limit major chord, [4:5:6], then the utonal inverse would be (6/6, 6/5, 6/4), i.e.
(1/1, 6/5, 3/2)
a just minor triad, which is nothing at all like the utonal chord we just saw with 7s in the numerators. But if you don't know how to use factors of 7 or 11 or whatever, you've got to start somewhere, and there are worse things to do than to make a scale with frequency ratios of the form {11/n}, for different values of {n} and noodle around in there.
Since Ben Johnston had overtone scales and undertone scales, I think it's likely he would just go down the undertone scale when he wanted undertone harmony, and go up overtone scale when he wanted overtone harmony. An easy recipe for cool music without having to think about the frequency ratios too much: you find what sounds good and then you just have to figure out post hoc how to notate it; Up from here, down from here, temporary tonicizations everywhere, all over frequency space, never fixed to a P1 of 440 hz or anything like that. That's my guess. For some of his works. He had different tricks for lots of different compositions.
When moving between two chords, Ben might have used voice leading based on super-particular ratios. Like it's okay to go from a D at 10/9 to a D with a bunch of weird high-prime accidentals at 260/243 for example, because they're related by (27/26). Or a weirdly inflected B at (140/81) to a weirdly inflected A at (400/243) is fine because they're related by 21/20. I'm saying he "might" have done this because super particular show up everywhere when you move by small amounts between just frequency ratios. You don't have to try very hard to find them. And all of his prime accidentals are tuned to super-particular frequency ratios, so anytime you move by a comma, like from like a B to a B-, you're getting free super-particularity. Or, like, even just looking at 5-limit frequency ratios, is it supposed to be impressive that a composer moved melodically by (Ac1, A1, m2, M2, AcM2, m3, M3, P4, or P5), which are all super particular? Good luck avoiding it, even while moving between crazy chords spaces. Between C utonal scale and C otonal scale, if you just move to the same letter name pitch or an adjacent letter name pitch, you have about a 1/2 chance of moving by a super-particular ratio. It's possible that Ben chose super-particular voice leading a lot more than that 1/2 chance. I haven't analyzed his pieces enough to know. It's a possibility and people have claimed it about his work.
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If I wanted to make a 7-limit scale using overtones and undertones, I'd choose something that looked like this:
P5 : 3/2 P4 : 4/3
M3 : 5/4 m6 : 8/5
Sbm7 : 7/4 SpM2 : 8/7
AcM2 : 9/8 Grm7 : 16/9
M7 : 15/8 m2 : 16/15
SbAc4 : 21/16 SpGr5 : 32/21
A5 : 25/16 d4 : 32/25
AcM6 : 27/16 Grm3 : 32/27
AcA4 : 45/32 Grd5 : 64/45
SbAc8 : 63/32 SpGr1 : 64/63
AcA2 : 75/64 Grd7 : 128/75
AcM3 : 81/64 Grm6 : 128/81
A7 : 125/64 d2 : 128/125
It has 28 intervals. I bet you could do some cool things with it. If you think it's too limited, keep extending it down. If you think it's too expansive, crop it midway. Try composing harmony that only uses one side or the other. Or don't. I'm not the boss of you.
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