Interseptimal interval
In the theory of Margo Schulter, interseptimal is a category of intervals which occupy regions intermediate between two septimal ratios such as 8/7 and 7/6, or 12/7 and 7/4. There are four interseptimal regions given below, with approximate cents ranges from Schulter's article Regions of the Interval Spectrum:
- Maj2-min3 -- intermediate between 8/7 and 7/6 -- 240¢-260¢
- Maj3-4 -- intermediate between 9/7 and 21/16 -- 440¢-468¢
- 5-min6 -- intermediate between 32/21 and 14/9 -- 732¢-760¢
- Maj6-min7 -- intermediate between 12/7 and 7/4 -- 940¢-960¢
Interseptimal intervals are well-represented in 24edo at 250¢, 450¢, 750¢ and 950¢. They also appear in 19edo and 29edo.
As they fall in ambiguous zones between simpler categories, they are inevitably xenharmonic. This also makes them difficult to name: do we classify a 250-cent interval as a second, a third, both, or neither? One option is to give each region a distinct name (analogous to using the word tritone rather than diminished fifth or augmented fourth). Possible names that could be used are:
- 240¢-260¢ -- semifourth -- an interval of this size is around half the size of a perfect fourth. Inthar proposes the name lygic (from Greek lygē, twilight) for this region.
- 440¢-468¢ -- semisixth -- an interval of this size is around half the size of a major sixth. Zhea Erose calls the 440c-464c region naiadic (from naiad, a kind of ancient Greek water spirit) and uses it frequently.
- 732¢-760¢ -- semitenth -- an interval of this size is around half the size of a major tenth (i. e., compound major third). Another possible name is sesquifourth (since this is also about one and a half times the size of a perfect fourth). Inthar proposes the name cocytic (from Dante's Cocytus, the frozen lake in the underworld).
- 940¢-960¢ -- semitwelfth -- an interval of this size is around half the size of a perfect twelfth (i e., a compound perfect fifth, or tritave). All even edts have a semitwelfth of approximately 951 cents, analogous to the 600 cent tritone shared by all even edos. Inthar proposes the name obsidic for this region.
This makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi".
By analogy the tritone could also be called a semioctave, although the term tritone is so well-established that seems is little reason to change it now. A key difference is that the tritone is intermediate between two septimal ratios separated by a jubilisma (50/49), whereas the other interseptimal ranges listed above are between two septimal ratios separated by a slendro diesis (49/48).
Examples
Some interseptimal intervals in all four ranges, both just and tempered, are listed below.
Maj2-min3 - 240-260¢
| Interval | Cents Value | Prime Limit (if applicable) |
|---|---|---|
| 147/128 | 239.607 | 7 |
| 1\5 | 240.000 | - |
| 54/47 | 240.358 | 47 |
| 23/20 | 241.961 | 23 |
| 1152/1001 | 243.238 | 13 |
| 38/33 | 244.240 | 19 |
| 144/125 | 244.969 | 5 |
| 15/13 | 247.741 | 13 |
| 6\29 | 248.276 | - |
| 5\24 | 250.000 | - |
| 52/45 | 250.304 | 13 |
| 37/32 | 251.344 | 37 |
| 81/70 | 252.680 | 7 |
| 4\19 | 252.632 | - |
| 22/19 | 253.805 | 19 |
| 29/25 | 256.950 | 29 |
| 3\14 | 257.143 | - |
| 297/256 | 257.183 | 11 |
| 36/31 | 258.874 | 31 |
| 5\23 | 260.870 | - |
Maj3-4 - 440-468¢
| Interval | Cents Value | Prime Limit (if applicable) |
|---|---|---|
| 5\88cET or 11\30 | 440.000 | - |
| 40/31 | 441.278 | 31 |
| 7\19 | 442.015 | - |
| 31/24 | 443.081 | 31 |
| 10\27 | 444.444 | - |
| 22/17 | 446.363 | 17 |
| 35/27 | 449.275 | 7 |
| 3\8 | 450.000 | - |
| 48/37 | 450.611 | 37 |
| 13/10 | 454.214 | 13 |
| 11\29 | 455.172 | - |
| 125/96 | 456.986 | 5 |
| 8\21 | 457.143 | - |
| 56/43 | 457.308 | 43 |
| 43/33 | 458.245 | 43 |
| 30/23 | 459.994 | 23 |
| 5\13 | 461.538 | - |
| 47/36 | 461.597 | 47 |
| 64/49 | 462.348 | 7 |
| 98/75 | 463.069 | 7 |
| 17/13 | 464.428 | 17 |
| 12\31 | 464.516 | - |
| 7\18 | 466.667 | - |
| 38/29 | 467.936 | 29 |
5-min6 - 732-760¢
| Interval | Cents Value | Prime Limit (if applicable) |
|---|---|---|
| 5\Bohlen-Pierce | 731.521 | - |
| 29/19 | 732.064 | 29 |
| 11\18 | 733.333 | - |
| 19\31 | 735.484 | - |
| 26/17 | 735.572 | 17 |
| 49/75 | 736.931 | 7 |
| 49/32 | 737.652 | 7 |
| 72/47 | 738.403 | 47 |
| 23/15 | 740.006 | 23 |
| 66/43 | 741.755 | 43 |
| 43/28 | 742.692 | 43 |
| 13\21 | 742.857 | - |
| 182/125 | 743.014 | 5 |
| 18\29 | 744.828 | - |
| 20/13 | 745.786 | 13 |
| 37/24 | 749.389 | 37 |
| 5\8 | 750.000 | - |
| 54/35 | 750.725 | 7 |
| 17/11 | 753.637 | 17 |
| 17\27 | 755.556 | - |
| 48/31 | 756.919 | 31 |
| 12\19 | 757.895 | - |
| 31/20 | 758.722 | 31 |
| 19\30 | 760.000 | - |
Maj6-min7 - 940-960¢
| Interval | Cents Value | Prime Limit (if applicable) |
|---|---|---|
| 18\23 | 939.130 | - |
| 31/18 | 941.126 | 31 |
| 512/297 | 942.817 | 11 |
| 11\14 | 942.857 | - |
| 50/29 | 943.050 | 29 |
| 19/11 | 946.195 | 19 |
| 140/81 | 947.320 | 7 |
| 15\19 | 947.368 | - |
| 64/37 | 948.656 | 37 |
| 45/26 | 949.696 | 13 |
| 19\24 | 950.000 | - |
| 23\29 | 951.724 | - |
| 26/15 | 952.259 | 13 |
| 125/72 | 955.031 | 5 |
| 33/19 | 955.760 | 19 |
| 1001/576 | 956.762 | 13 |
| 40/23 | 958.039 | 23 |
| 47/27 | 959.642 | 47 |
| 4\5 | 960.000 | - |
| 256/147 | 960.393 | 7 |