Interseptimal

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

• Gallery of Just Intervals