Harry Partch classed intervals according to their odd limit. This is the largest odd number used in the frequency ratio reduced to be within an octave. The idea is that an interval is more dissonant the larger its odd limit. Chords (or tonalities) within an odd limit are classed as either otonal or utonal. Removing all factors of 2, the otonal 9-limit chord is 1:3:5:7:9. All its intervals are in the 9-limit. This is also true of the utonal version, which has the intervals reversed: 1/9:1/7:1/5:1/3:1/1.
Whether Partch used the word "limit" to refer to odd or prime numbers is a matter of some debate. However, he did seem to think that the consonance of an interval is governed by its odd limit, and called it an identity or something like that.
Paul Erlich, in a post to the Tuning List showed that there are chords which are neither 9-limit utonalities or otonalities, but all of whose intervals are within the 9-limit.
The two examples are 3:5:9:15 and 3:7:9:21. In octave equivalent matrix representation these are:
(1 1) (2 0)h (0 1) (1 0) | (1 0 1) (2 0 0)h (0 0 1) (1 0 0) |
There is an obvious similarity.
I've decided to call these "anomalous saturated suspensions" as they are suspensions of notes to which none can be added without raising the odd limit, and which are outside Partch's original theory. As the phrase is rather unwieldy, the acronym "ass" will be used from henceforth. A general means of constructing asses soon emerged from the discussion, and will be explained here. If you like, skip that for the examples.
Incidentally, in choosing the acronym "ass" I do not mean to allude to the usual meaning of that word, namely a beast of burden. It is certainly not my intention to imply that Partch was an ass for not discovering them. The fact that they were only uncovered over 20 years after his death is good evidence that this is not the case.
All the numbers we'll be meeting today will be odd, so I won't say so every time. Numbers that are not primes or squares of primes have a special significance, and will be called "rhubarb numbers" throughout this page.
For an n-limit ass, first choose a number m. Either n or m, or both, must be composite. m<n. n and m must not share a common factor (I think this makes them "relatively prime"). Providing neither n nor m are rhubarb, an n-limit ass can be constructed form all the divisors of n*m except n*m and 1. This chord will be symmetrical: it's own utonal analog.
To take the first example above, n=9 and m=5. The divisors of 9*5=45 are 45, 15, 9, 5, 3 and 1. 45 in matrix representationis (2 1)h. The divisors are all vectors whose elements are between (0 0)h and this, which is an Euler genus:
(2 1) (1 1) (2 0)h (0 1) (1 0) (0 0)
Removing 1 and 45 from this chord is so trivial an operation that a detailed explanation of it would only confuse you. Anyway, that's the general method.
When m=i*j is rhubarb (i<j) we have a problem. The intervals n*i/j and n*j/i will be beyond the n-limit. The same problem occurs when n=i*j is rhubarb unless i>m so that j*m<n. (j*m<n => j*m<i*j => m<i) In this case, no asses of the required limit exist, as explained below. Now, though, an example that does work: n=15, m=7. (1 1 0)h+(0 0 1)h =(1 1 1)h. Divisors are:
(0 1 1) (1 0 1) (1 1 0)h (0 0 1) (0 1 0) (1 0 0)
This gives us the chord 35:21:15:7:5:3. The intervals 35/3 and21/5 are >15-limit. Therefore [21 or 5] and [35 or 3] must be removed. If 35 and 21 are removed, we have a 15-limit otonality. Removing 5 and 3 gives its utonal analog. The choice is then removing [21 and 3] or [35 and 5]. The resulting 15-limit asses 35:15:7:5 and 21:15:7:3 are then otonal/utonal analogs.
For the case where n=i*j is rhubarb, and i<m, try the example n=21, m=5.3<5 and 7>5. The non-rhubarb method gives us:
(0 1 1) (1 0 1) (1 1 0)h (0 0 1) (0 1 0) (1 0 0)
Or 3:5:7:15:21:35. (This is the 1-3-5-7 hexany). 35/3 is the only interval that exceeds the 21-limit. Remove the 35, and 3:5:7:15:21 is a subset of the 21-limit otonality, hence it is not saturated. Remove the 3, and you get the corresponding utonality:
(0 1 1) (-1 0 0) (1 0 1) ( 0 -1 0) (1 1 0)h - J(1 1 1)h = ( 0 0 -1)h (0 0 1) (-1 -1 0) (0 1 0) (-1 0 -1)
Or 1/3:1/5:1/7:1/15:1/21.
Otonal/utonal pairs seem to arise whenever the rhubarb problem occurs. I haven't looked at any cases where n and m are both rhubarb. The first of these is n=35, m=33. Doubly rhubarb numbers can safely be ignored until someone starts writing 105-limit music.
There the theory stood until Praveen Venkataramana published six new 15-limit chords (27th June 2016):
I didn't include these in what I though was a complete list. I haven't checked them in detail to see what I missed, but one thing is that I didn't consider primes 11 and 13 in the 15-limit. (7, 11, and 13 should be interchangeable, as long as all instances in a chord are interchanged.) So, it's disappointing that for all the mathematical tuning theory it took so long to list all the chords containing only 15-limit intervals, but at least the theory's moving on.
limit | thing | matrix | ratio |
9 | 5 | (1 1) (2 0)h (0 1) (1 0) | 3:5:9:15 |
9 | 7 | (1 0 1) (2 0 0)h (0 0 1) (1 0 0) | 3:7:9:21 |
11 | 9 | (1 0 0 1) (0 0 0 1)h (2 0 0 0) (1 0 0 0) | 3:9:11:33 |
13 | 9 | (1 0 0 0 1) (0 0 0 0 1)h (2 0 0 0 0) (1 0 0 0 0) | 3:9:13:39 |
15 | 7 | (1 0 1) (1 1 0)h (0 0 1) (1 0 0) | 3:7:15:21 |
15 | 7 | (0 1 1) (1 1 0)h (0 0 1) (0 1 0) | 5:7:15:35 |
17 | 9 | (1 0 0 0 0 1) (0 0 0 0 0 1)h (2 0 0 0 0 0) (1 0 0 0 0 0) | 3:9:17:51 |
17 | 15 | (1 0 0 0 0 1) (0 0 0 0 0 1)h (1 1 0 0 0 0) (1 0 0 0 0 0) | 3:15:17:51 |
17 | 15 | (0 1 0 0 0 1) (0 0 0 0 0 1)h (1 1 0 0 0 0) (0 1 0 0 0 0) | 5:15:17:85 |
19 | 9 | (1 0 0 0 0 0 1) (0 0 0 0 0 0 1)h (2 0 0 0 0 0 0) (1 0 0 0 0 0 0) | 3:9:19:57 |
19 | 15 | (1 0 0 0 0 0 1) (0 0 0 0 0 0 1)h (1 1 0 0 0 0 0) (1 0 0 0 0 0 0) | 3:15:19:57 |
19 | 15 | (0 1 0 0 0 0 1) (0 0 0 0 0 0 1)h (1 1 0 0 0 0 0) (0 1 0 0 0 0 0) | 5:15:19:95 |
25 | 9 | (1 2) (2 1) (0 2) (1 1)h (2 0) (0 1) (1 0) | 3:5:9:15:25:45:75 |