I've never found good principles for making tuned chords that use high-prime-limit frequency ratios, but I recently heard of a cool trick that seems to work.
A tuned chord can be written "otonally" as a sequence of integers, like [4, 5, 6]. Divide all the list elements through by the first list entry to get the frequency ratios, e.g. [4, 5, 6] -> (1/1, 5/4, 3/2). That's a major chord and it sounds great.
You can generate random sequences of integers all day to make chords, and some will be okay, some will be bad, some will be good. But it's not easy to listen to 500 random chords, pick your favorites, try to remember how they sounds based on a name that is a list of integers, and then try to use them to compose. It would be nicer if we had some principles for finding tuned chords that sounded good almost surely.
I heard about such a method from Kite Giedraitis; you just use sequences of odd integers. I find that things sound better if you also keep all your integers within an octave, i.e. the largest integer in your list is smaller than twice the first integer in your list.
These chords almost all sound good to my ear. Or maybe I was just in a very-open-to-weirdness headspace when I played with them last night. But I think it's the former. And that's very exciting, because you can get very high prime limit chords.
Here's a list of frequency ratios between 1/1 and 2/2 that show up in such chords, with numerators up to 31: [1/1, 5/3, 7/5, 9/5, 9/7, 11/7, 11/9, 13/7, 13/9, 13/11, 15/11, 15/13, 17/9, 17/11, 17/13, 17/15, 19/11, 19/13, 19/15, 19/17, 21/11, 21/13, 21/17, 21/19, 23/13, 23/15, 23/17, 23/19, 23/21, 25/13, 25/17, 25/19, 25/21, 25/23, 27/17, 27/19, 27/23, 27/25, 29/15, 29/17, 29/19, 29/21, 29/23, 29/25, 29/27, 31/17, 31/19, 31/21, 31/23, 31/25, 31/27, 31/29].
Here they are sorted by increasing size: [1/1, 31/29, 29/27, 27/25, 25/23, 23/21, 21/19, 19/17, 17/15, 31/27, 15/13, 29/25, 27/23, 13/11, 25/21, 23/19, 11/9, 21/17, 31/25, 29/23, 19/15, 9/7, 9/7, 17/13, 25/19, 31/23, 23/17, 15/11, 29/21, 7/5, 27/19, 13/9, 19/13, 25/17, 31/21, 29/19, 23/15, 17/11, 11/7, 27/17, 21/13, 31/19, 5/3, 5/3, 5/3, 29/17, 19/11, 23/13, 9/5, 9/5, 31/17, 13/7, 17/9, 21/11, 25/13, 29/15].
If you sounds those in order against 1/1, you'll notice some elements are very close to each other. Like right at the start,
(29/27) / (31/29) = 841/837 @ 8 cents
is a fairly small comma, and
(25/23) / (27/25) = 625/621 @ 11 cents
is similarly small.
They're not unnoticeable commas, but if you want very-high-prime-limit temperaments, then I've heard of worse methods to generate very-high-prime-limit tuned commas.
The Bohlen-Pierce scale also doesn't use any even harmonics, but it limits things to 7-limit frequency ratios. I think it's cool that this extends Bohlen-Pierce.
I have no idea why this works. But let's figure it out.
I've forgotten almost everything I've written about otonality and utonality. I haven't written much. Time to redo it, and better.
Chords made of frequency ratios are called tuned chords. They differ from chords made of intervals. Those are intervallic chords, like [P1, M3, P5]. We'll be looking at tuned chords in this post.
Here are the just frequency ratios for a 5-limit major chord: [1/1, 5/4, 3/2]. We can multiply through by the least common multiple of the denominators to get a sequence of integers: [4, 5, 6]. This is the otonal representation. To get back to the ratios, divide everything through by the first element of the otonal list.
We can find "inversions" of tuned chords, by which I mean cyclic permutations modulo the octave. Pop the first element off of the list, multiply by 2, and stick it at the end of the list. Divide all the list elements by the new first element. Multiply through by the least common multiple of the denominators if you want that chordal inversion to be represented otonally.
The otonal major chord [4, 5, 6] has cyclic inversions of [5, 6, 8] and [3, 4, 5].
We can also find the elementwise inverse of chords. For each element {e} in a list, take {1/e} for the new element in that position. Sort by size and multiply through by the least common multiple of denominators. We'll call this the utonal inverse of the chord. Optionally you can divide through by the first element if you want frequency ratios. Cyclic inversions versus utonal inverses.
Here are the three inversions of the otonal major chord and the utonal inverse for each after a double colon:
[4, 5, 6] :: [10, 12, 15]
[5, 6, 8] :: [15, 20, 24]
[3, 4, 5] :: [12, 15, 20]
The utonal inverses are also inversions of each other, i.e. the set is closed under cyclic permutation modulo the octave. They happen to also be justly-tuned minor chords.
What if we space things out more, beyond the octave? If we remove factors of two, i.e. raise and lower things by octaves until the factors of two are gone from the utonal integer list, then the major chord has only one all-odd voicing: [1, 3, 5] and its utonal inverse has only one all-odd voicing, [3, 5, 15].
And.... all-odd voicings usually sound good, at least within an octave, not that these necessarily are within an octave. So can we take a bad sounding chord and make it sound better by spacing it out in an all-odd voicing? Are there chords that sound better when compressed into an octave than when expanded to the all-odd voicing?
I think a normally voiced harmonic seventh chord, [4, 5, 6, 7], sounds way better than its all-odd voicing, [1, 3, 5, 7]. So there's one.
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