Interseptimal interval
In the theory of Margo Schulter, an interseptimal interval is an interval that belongs in one of four interval regions which are 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 essay 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¢
Additionaly, there are also these 2 interseptimal regions near the unison and octave:
- 1–min2 – intermediate between 64/63 and 28/27 – 40¢–60¢
- Maj7-8 – intermediate between 27/14 and 63/32 – 1140¢–1160¢
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 both diatonic and chromatic categories, they are inevitably xenharmonic.
A JI-agnostic synonym is interordinal; here, ordinal refers to the interval classes of the diatonic scale the interordinal intervals lie between, conventionally denoted with ordinal numbers.
See Neutral and interordinal k-mossteps for a partial generalization of interseptimal categories to other mosses.
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.[1]
- 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, by analogy with naiadic.
- 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.
One might want to use a mixture of above terms. Flora Canou criticizes semisixth and semitenth as they fail to make clear whether the interval to be split is major or minor, and prefers naiadic and cocytic. However, semifourth and semitwelfth are clear enough, so the Greek terms seems practically redundant.
The terminology 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
"Inter" names
Both the "semi-nth" names and the Greek-derived names above are less intuitive than they could be and require some amount of memorization. For this reason, Inthar has proposed the following terms that explicitly name the diatonic interval categories that the interseptimals fall between:
- semifourth/chthonic = second-inter-third (2×3)
- semisixth/naiadic = third-inter-fourth (3×4)
- semitenth/cocytic = fifth-inter-sixth (5×6)
- semitwelfth/ouranic = sixth-inter-seventh (6×7)
Decimal ordinal names
CompactStar has proposed names using decimal ordinals to indicate how these fall between diatonic categories:
- semifourth/chthonic = 2.5th
- semisixth/naiadic = 3.5th
- semitenth/cocytic = 5.5th
- semitwelfth/ouranic = 6.5th
Within a pentatonic framework
A pentatonic framework, as elucidated in Kite Giedraitis's Alternative Tuning guide, is far more amenable to interseptimal intervals than the traditional Western heptatonic framework.
| names | quality | boundaries | heptatonic equivalent | ||
|---|---|---|---|---|---|
| 1sn | unison | perfect | 1/1 to 64/63 | perfect | 1sn |
| half-augmented | (interseptimal) | ||||
| augmented | 28/27 to 16/15 | minor | 2nd | ||
| (interpental) | neutral | ||||
| penta-2nd | subthird | minor | 10/9 to 8/7 | major | |
| neutral | (interseptimal) | ||||
| major | 7/6 to 6/5 | minor | 3rd | ||
| (interpental) | neutral | ||||
| penta-3rd | fourthoid | diminished | 5/4 to 9/7 | major | |
| half-diminished | (interseptimal) | ||||
| perfect | 21/16 to 27/20 | perfect | 4th | ||
| half-augmented | (interpental) | half-augmented | |||
| augmented | 7/5 to 10/7 | augmented | |||
| penta-4th | fifthoid | diminished | diminished | 5th | |
| half-diminished | (interpental) | half-diminished | |||
| perfect | 40/27 to 32/21 | perfect | |||
| half-augmented | (interseptimal) | ||||
| augmented | 14/9 to 8/5 | minor | 6th | ||
| (interpental) | neutral | ||||
| penta-5th | subseventh | minor | 5/3 to 12/7 | major | |
| neutral | (interseptimal) | ||||
| major | 7/4 to 9/5 | minor | 7th | ||
| (interpental) | neutral | ||||
| hexave | octoid | diminished | 15/8 to 27/14 | major | |
| half-diminished | (interseptimal) | ||||
| perfect | 63/32 to 2/1 | perfect | 8ve | ||
Note the two additional interseptimal regions. The boundary ratios are mostly either 81/80 or 64/63 away from a 3-limit interval. The exceptions are 7/5 and 10/7, which are only a Saruyo comma away from the 3-limit diminished 5th and augmented 4th respectively.
Interseptimal intervals are now easily named. However there are now hard-to-name "interpental" intervals which would be neutral intervals in the heptatonic framework, containing such ratios as 12/11, 11/9, etc. This is because interseptimal intervals are the neutral intervals with respect to the parent mos 2L 3s of the diatonic mos 5L 2s.
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 (semifourth/chthonic)
| 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 (semisixth/naiadic)
| 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 (semitenth/cocytic)
| 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 (semitwelfth/ouranic)
| 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
Notes
- ↑ As per Primodal Archive.