Interordinal: Difference between revisions
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'''Interordinals''' | [[File:Ytwy.png|thumb|502x502px|The interval regions surrounding neutral (blue) and mosdiatonic (red/yellow) intervals. Interordinal regions are the "gaps" in this scheme.]] | ||
'''Interordinals''' are interval categories halfway between adjacent [[ordinal]]s, or 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). | |||
Sometimes the '''interizer''' and its octave-complement are included. The ''interizer'' is defined as the interval that separates interordinals from adjacent diatonic ordinals; it is half of the diatonic small step. | |||
The following table shows various ways to name interordinals: | [[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. | ||
== Naming == | |||
There is no unified nomenclature for interordinal regions. The following table shows various ways to name interordinals: | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ Interordinal | |+ Interordinal nomenclatures | ||
!|24edo interval | !|24edo interval | ||
!|"semi" names | !|"semi" names | ||
| Line 36: | Line 45: | ||
||ouranic | ||ouranic | ||
|} | |} | ||
== Pythagorean-based interordinals == | |||
The Pythagorean-based sizes for interordinals are the logarithmic midpoints (mathematically, geometric means) of the corresponding Pythagorean diatonic intervals, hence being the canonical sizes for interordinals in some sense: | |||
* 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 | |||
The Pythagorean-based interizer is sqrt(256/243) = 45.1c, and its octave complement is 1154.9c. | |||
== Some JI interordinals == | |||
* semifourth: 15/13; 22/19; 37/32 | |||
* semisixth: 13/10; 22/17; 31/24; 35/27 | |||
* semitenth: 17/11; 20/13; 37/24; 99/64 | |||
* semitwelfth: 19/11; 26/15; 45/26 | |||
{{Interval regions}} | |||
Revision as of 04:45, 13 February 2026
Interordinals are interval categories halfway between adjacent ordinals, or 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).
Sometimes the interizer and its octave-complement are included. The interizer is defined as the interval that separates interordinals from adjacent diatonic ordinals; it is half of the diatonic small step.
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.
Naming
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 |
Pythagorean-based interordinals
The Pythagorean-based sizes for interordinals are the logarithmic midpoints (mathematically, geometric means) of the corresponding Pythagorean diatonic intervals, hence being the canonical sizes for interordinals in some sense:
- 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
The Pythagorean-based interizer is sqrt(256/243) = 45.1c, and its octave complement is 1154.9c.
Some JI interordinals
- semifourth: 15/13; 22/19; 37/32
- semisixth: 13/10; 22/17; 31/24; 35/27
- semitenth: 17/11; 20/13; 37/24; 99/64
- semitwelfth: 19/11; 26/15; 45/26
