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.

Categorical and Notational Approaches

While interseptimals are interesting for falling right in between the typical western interval categories, this also makes them difficult to name and notate: do we classify a 250-cent interval as a second, a third, both, or neither?

Singular, unique names

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.
  • The term chthonic (from khthon, an ancient Greek word referring to spirits of the underworld) refers to the 240-260¢ region by Zhea Erose.
• 440¢–468¢ – semisixth – an interval of this size is around half the size of a major sixth.
  • The term naiadic (from naiad, a kind of ancient Greek water spirit) refers to the 440–464¢ region by Zhea Erose, who 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).
  • The term cocytic was proposed by Inthar, and is carried on with by Flora Canou.
• 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.
  • The term ouranic (by analogy with chthonic, and to match with the other terms) is proposed by Kaiveran.

This makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi". By analogy with the "semi" names, the tritone could also be called a semioctave, although the term tritone is so well-established (and so well represented by an unsplit 3-limit) that there seems 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).

Dual "semichromatic" names

Since interseptimal intervals are typically well represented by any EDO or pergen that divides its approximate 3/1 into 2n parts, another option is to repurpose quartertone accidentals to represent them, which is easy as long as we define our "half-sharps" or "half-flats" to be precisely half of a chromatic semitone. With this in mind, we get the following twinned identities for our interseptimals, with the simplest ones (assuming a half-fifth genchain) listed first:

• semifourth/chthonic = semi-augmented second (+11/2), semi-diminished third (-13/2)
• semisixth/naiadic = semi-diminished fourth (-9/2), semi-augmented third (+15/2)
• semitenth/cocytic = semi-augmented fifth (+9/2), semi-diminished sixth (-15/2)
• semitwelfth/ouranic = semi-diminished seventh (-11/2), semi-augmented sixth (+13/2)

While this does not give the interseptimals a single distinct notational name, it does reflect their ambiguity and flexibility with regards to the surrounding interval categories that many are so fond of. Furthermore, as both identities are exactly 12 notational fifths apart (i.e a direct analogue of the Pythagorean comma), composers can use a mechanism similar to the "po and qu" of Color Notation, or the plus and minus accidentals (+/-) proposed in Rational Comma Notation, to freely switch between the two identities.

Alternatively, one can use the ultra- prefix for sharpening by ~50¢ and infra- for flattening by ~50¢, analogous to super- and sub- for modifications by 64/63 (in a 12edo-related context such as 36edo, 33¢).

• semifourth/chthonic = ultramajor second, inframinor third
• semisixth/naiadic = ultramajor third, infrafourth
• semitenth/cocytic = ultrafifth, inframinor sixth
• semitwelfth/ouranic = ultramajor sixth, inframinor seventh

Ultra- and infra- also work for intervals that are very close to 11/8 or 16/11:

• ~11/8 or ~550¢ = ultrafourth, infratritone, infrasemioctave
• ~16/11 or ~650¢ = infrafifth, ultratritone, ultrasemioctave

Within a pentatonic framework

A pentatonic framework, as elucidated in Kite Giedraitis's Alternative Tuning guide, is far more amenable to interseptimal intervals that the traditional Western heptatonic framework. In a pentatonic framework, there are 5 categories of intervals in the octave rather than 7, into which interseptimals fit much more cleanly:

• the subthird or penta-2nd (imperfect)
  • <- semifourth/chthonic region
• the fourthoid or penta-3rd (perfect)
  • <- semisixth/naiadic region
• the fifthoid or penta-4th (perfect)
  • <- semitenth/cocytic region
• the subseventh or penta-5th (imperfect)
  • <- semitwelfth/ouranic region
• the octoid or hexave (perfect, reducing to a unison)

Of especial interest are the chthonic and ouranic regions, being very close to intervals of 5EDO, and in central positions within the two imperfect interval categories. In this framework, they can easily function as "neutral/middle" intervals to which other intervals in their pentatonic category are compared. So composing in a pentatonic framework may allow interseptimal intervals to play much more pivotal roles than usual.

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

See also

• Gentle region
• Gallery of just intervals