Interordinal
Interordinals are interval categories halfway between adjacent interval classes of the diatonic scale. For example, 250c is an interordinal because it falls between 200c (the 12edo major second) and 300c (the 12edo minor third). Interordinals may sometimes be called interseptimals;, however the term "interseptimal" also has certain other senses on this wiki.
There are usually considered to be four interordinal regions:
- semifourth (between major 2nd and minor 3rd)
- semisixth (between major 3rd and perfect 4th)
- semitenth (between perfect 5th and minor 6th)
- semitwelfth (between major 6th and minor 7th).
19edo, 24edo, and 29edo are notable edos with a complete set of interordinals; the MOS scales manual (5L1s) and semiquartal (5L4s) in certain tunings have all four interordinal regions as well. Notable JI interordinals include 15/13 (247.7c, a semifourth) and 13/10 (454.2c, a semisixth); thus 10:13:15 is a fairly low-complexity JI triad with a semisixth.
There is no unified nomenclature for interordinal regions. The following table shows various ways to name interordinals:
| 24edo interval | "semi" names | "inter" names | "ultra"/"infra" | Greek-derived names |
|---|---|---|---|---|
| 250c, 5\24 | semifourth | second-inter-third | ultramajor second inframinor third |
chthonic |
| 450c, 9\24 | semisixth | third-inter-fourth | ultramajor third infrafourth |
naiadic |
| 750c, 15\24 | semitenth | fifth-inter-sixth | ultrafifth inframinor sixth |
cocytic |
| 950c, 19\24 | semitwelfth | sixth-inter-seventh | ultramajor sixth inframinor seventh |
ouranic |
Hemipythagorean interordinals
The Pythagorean-based sizes for interordinals are the logarithmic midpoints (mathematically, geometric means) of the corresponding Pythagorean diatonic intervals:
- semifourth: sqrt(9/8 * 32/27) = sqrt(4/3) = 249.0c
- semisixth: sqrt(81/64 * 4/3) = sqrt(27/16) = 452.9c
- semitenth: sqrt(3/2 * 128/81) = sqrt(64/27) = 747.1c
- semitwelfth: sqrt(27/16 * 16/9) = sqrt(3/1) = 951.0c
