S-expression
Sk (square-particulars)
A square superparticular, or square-particular for short, is a superparticular interval whose numerator is a square number, which is to say, a superparticular of the form
n2/(n2 - 1) = (n/(n-1)) / ((n+1)/n)
which is square-(super)particular n for a given integer n > 1. A suggested shorthand for this interval is Sn for the nth square superparticular, where the S stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article. Note that this means S2 = 4/3 is the first musically meaningful square-particular, as S1 would be 1/0.
Square-particulars are important structurally because they are the intervals between consecutive superparticular intervals while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (n+1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward.
It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy it can be more beneficial instead to temper differences between square superparticulars, so that consecutive superparticulars become equidistant. If we define a sequence of commas Un = Sn/S(n+1), where "U" stands for "Ultraparticular", in analogy with the "super-", "ultra-" progression and because these would be differences between differences between differences between superparticulars (which means a higher order of "particular", and as we will see, no longer superparticular), we get Ultraparticulars. Ultraparticulars have a secondary consequence: Because (n+2)/(n+1) and n/(n-1) are equidistant from (n+1)/n (because of tempering Sn/S(n+1)), this means that another expression for Un = Sn/S(n+1) is (n+2)/(n-1) / ((n+1)/n)^3, so you can read the n and n+1 from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence.
Sk/S(k+1) (ultraparticulars)
Note that while a lot of these have pages, not all of them do.
| S-expression | cube relation | comma |
|---|---|---|
| S2/S3 = (4/3)/(9/8) | (4/1) / (3/2)3 | 32/27 |
| S3/S4 = (9/8)/(16/15) | (5/2) / (4/3)3 | 135/128 |
| S4/S5 = (16/15)/(25/24) | (2/1) / (5/4)3 | 128/125 |
| S5/S6 = (25/24)/(36/35) | (7/4) / (6/5)3 | 875/864 |
| S6/S7 = (36/35)/(49/48) | (8/5) / (7/6)3 | 1728/1715 |
| S7/S8 = (49/48)/(64/63) | (3/2) / (8/7)3 | 1029/1024 |
| S8/S9 = (64/63)/(81/80) | (10/7) / (9/8)3 | 5120/5103 |
| S9/S10 = (81/80)/(100/99) | (11/8) / (10/9)3 | 8019/8000 |
| S10/S11 = (100/99)/(121/120) | (4/3) / (11/10)3 | 4000/3993 |
| S11/S12 = (121/120)/(144/143) | (13/10) /(12/11)3 | 17303/17280 |
| S12/S13 = (144/143)/(169/168) | (14/11) / (13/12)3 | 24192/24167 |
| S13/S14 = (169/168)/(196/195) | (5/4) / (14/13)3 | 10985/10976 |
| S14/S15 = (196/195)/(225/224) | (16/13) / (15/14)3 | 43904/43875 |
| S15/S16 = (225/224)/(256/255) | (17/14) / (16/15)3 | 57375/57344 |
| S16/S17 = (256/255)/(289/288) | (6/5) / (17/16)3 | 24576/24565 |
Note from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably consistent) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3k+1)/S(3k+2) for a positive integer k, because then the superparticular can be expressed as:
((3k+3)/(3k))/((3k+2)/(3k+1))3 = (k+1)/k / ((3k+2)/(3k+1))3
Also note that if you temper multiple adjacent ultraparticulars, you sometimes aren't required to use those ultraparticulars in the comma list due to an interesting case of Sk/S(k+2), discussed next.
Sk/S(k+2) (semiparticulars)
For differences between square-particulars of the form S(k+1)/S(k+3) the resulting comma is either superparticular or "odd-particular", meaning an interval of the form (2n+1)/(2n-1) for some positive integer n. (This terminology also suggests "throdd-particular" for intervals of the form (3n+2)/(3n) and (3n+1)/(3n-1) and maybe "quodd-particular" (sounding like "quad-particular") for (4n+3)/(4n-1) and (4n+1)/(4n-3).) Furthermore, S(k+1)/S(k+3), when tempered, implies that (k+4)/k is divisible exactly into two halves of (k+3)/(k+1) which is equated with (k+1)/k * (k+4)/(k+3). It is for this reason that the indexing choice of S(k+1)/S(k+3) was chosen, as two (k+3)/(k+1)'s make a (k+4)/k. This form of comma does not yet have an official name, but a proposed name is "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered they all cause an interval to be divided into two equal parts where each part is a tempered version of a superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular.
Specifically, Sk/S(k+2) is superparticular when k is not a multiple of 4, and odd-particular otherwise. See derivation for details on this and other facts stated here.
Derivation
Work in progress.