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.
- 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 minor tenth (i. e., an octave plus a minor third). Another possible name is sesquifourth (since this is also about one and a half times the size of a perfect fourth).
- 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.
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 |