Interseptimal

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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.
  • 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).
  • 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\5edo 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\29edo 248.276 -
5\24edo 250.000 -
52/45 250.304 13
37/32 251.344 37
81/70 252.680 7
4\19edo 252.632 -
22/19 253.805 19
29/25 256.950 29
3\14edo 257.143 -
297/256 257.183 11
36/31 258.874 31
5\23edo 260.870 -

Maj3-4 - 440-468

Interval Cents Value Prime Limit (if applicable)
5\88cET or 11\30edo 440.000 -
40/31 441.278 31
7\19edo 442.015 -
31/24 443.081 31
10\27edo 444.444 -
22/17 446.363 17
35/27 449.275 7
3\8edo 450.000 -
48/37 450.611 37
13/10 454.214 13
11\29edo 455.172 -
125/96 456.986 5
8\21edo 457.143 -
56/43 457.308 43
43/33 458.245 43
30/23 459.994 23
5\13edo 461.538 -
47/36 461.597 47
64/49 462.348 7
98/75 463.069 7
17/13 464.428 17
12\31edo 464.516 -
7\18edo 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\18edo 733.333 -
19\31edo 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\21edo 742.857 -
182/125 743.014 5
18\29edo 744.828 -
20/13 745.786 13
37/24 749.389 37
5\8edo 750.000 -
54/35 750.725 7
17/11 753.637 17
17\27edo 755.556 -
48/31 756.919 31
12\19edo 757.895 -
31/20 758.722 31
19\30edo 760.000 -

Maj6-min7 - 940-960

Interval Cents Value Prime Limit (if applicable)
18\23edo 939.130 -
31/18 941.126 31
512/297 942.817 11
11\14edo 942.857 -
50/29 943.050 29
19/11 946.195 19
140/81 947.320 7
15\19edo 947.368 -
64/37 948.656 37
45/26 949.696 13
19\24edo 950.000 -
23\29edo 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\5edo 960.000 -
256/147 960.393 7

See also