Essential tempering commas

Suppose S is a set of JI intervals i including 1 and 2 with 1 <= i <= 2 such that if i is in S, so is 2/i. S is intended to represent a set of pitch classes defining "consonance". A JI interval c is an essential tempering comma for S if:

1. c is greater than 1 but less than the smallest interval between any two members of S

2. There are three intervals i, j, and k in S such that c = i*j/k

For various odd limit diamonds, we get the following essential tempering commas:

5: 128/125

7: 126/125, 64/63

9: 225/224, 126/125, 245/243

11: 540/539, 441/440, 385/384, 243/242, 225/224, 896/891, 176/175, 126/125, 245/243

13: 1001/1000, 2200/2197, 729/728, 540/539, 441/440, 847/845, 385/384, 364/363, 352/351, 351/350, 325/324, 1573/1568, 243/242, 1188/1183, 225/224, 640/637, 196/195, 1287/1280, 896/891, 176/175

15: 1001/1000, 1575/1573, 2200/2197, 729/728, 676/675, 540/539, 441/440, 847/845, 385/384, 364/363, 352/351, 351/350, 325/324, 1573/1568, 3388/3375, 243/242, 1188/1183

17: 2601/2600, 2431/2430, 1275/1274, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 936/935, 833/832, 1575/1573, 2200/2197, 729/728, 715/714, 676/675, 595/594, 561/560, 540/539, 442/441, 441/440, 847/845, 2880/2873, 2028/2023, 385/384, 375/374, 364/363, 352/351, 351/350, 4928/4913, 2295/2288, 325/324, 1573/1568

19: 4200/4199, 3136/3135, 2926/2925, 2601/2600, 2432/2431, 2431/2430, 5491/5488, 1729/1728, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330, 1275/1274, 1216/1215, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 969/968, 936/935, 2720/2717, 6144/6137, 833/832, 1575/1573, 5415/5408, 3762/3757, 2200/2197, 729/728, 715/714, 676/675, 1862/1859, 595/594, 2912/2907, 2299/2295, 3978/3971, 561/560, 540/539, 513/512, 495/494, 476/475, 2304/2299, 456/455, 442/441, 441/440, 4704/4693, 847/845, 1235/1232, 2880/2873, 2057/2052, 2028/2023, 400/399, 385/384, 375/374, 364/363

21: 5985/5984, 4914/4913, 4200/4199, 4096/4095, 3136/3135, 2926/2925, 2601/2600, 2432/2431, 2431/2430, 2080/2079, 2058/2057, 3971/3969, 5491/5488, 1729/1728, 1701/1700, 3213/3211, 1540/1539, 1521/1520, 1445/1444, 6864/6859, 1331/1330, 1275/1274, 1216/1215, 1156/1155, 1089/1088, 2025/2023, 1001/1000, 969/968, 936/935, 2720/2717, 3553/3549, 4394/4389, 6144/6137, 833/832, 1617/1615, 1575/1573, 5415/5408, 3762/3757, 2200/2197, 729/728, 715/714, 9261/9248, 676/675, 1862/1859, 595/594, 2912/2907, 2299/2295, 3978/3971, 561/560, 6080/6069, 540/539, 513/512, 495/494, 476/475, 2304/2299, 456/455

We don't need to use the full q-limit diamond; from Diamond([1,3,5,7,9,11,15]) we get: 540/539, 441/440, 385/384, 3388/3375, 243/242

On Wed, Dec 14, 2011 at 11:05 AM, genewardsmith
<genewardsmith@...> wrote:
>
> Suppose S is a set of JI intervals i including 1 and 2 with 1 <= i <= 2 such that if i is in S, so is 2/i. S is intended to represent a set of pitch classes defining "consonance". A JI interval c is an essential tempering comma for S if:
>
> 1. c is greater than 1 but less than the smallest interval between any two members of S
>
> 2. There are three intervals i, j, and k in S such that c = i*j/k

I think that this is a fantastic idea. It might also be useful to look
at the case where c = (i/j)/(j/k) = i/j * k/j, which then creates a
single chroma turning k into j and j into i. A good example of this is
875/864, which has the property that 7/6 -> 6/5 -> 5/4 -> 9/7 are all
separated by the same chroma. Another good thing to look at is the
case where c = (i/j) / (k/l), which simply tempers out the difference
between the chromata of two disparate sets of intervals, regardless of
whether or not they form any sort of "chain."

-Mike