Interordinal: Difference between revisions
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[[File:Ytwy.png|thumb|502x502px|The interval regions surrounding neutral (blue) and mosdiatonic (red/yellow) intervals. Interordinal regions are the "gaps" in this scheme.]] | [[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" | '''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''', as interordinals are midpoints of septimal intervals (such as 8/7 and 7/6) separated by 49/48; however, the term "interseptimal" on this wiki may refer to 49/48 itself. | ||
There are usually considered to be four interordinal regions: | There are usually considered to be four interordinal regions: | ||
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Sometimes the '''interizer'''/'''semisecond''', and its octave-complement, the '''antiinterizer'''/'''semifourteenth''', are included. The ''interizer'' is defined as the interval that separates interordinals from adjacent diatonic ordinals; it is half of the diatonic small step. | Sometimes the '''interizer'''/'''semisecond''', and its octave-complement, the '''antiinterizer'''/'''semifourteenth''', 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 [[ | [[19edo]], [[24edo]], [[29edo]], and [[53edo]] 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. See [[Chthonic harmony]] for a compositional theory using interordinals. | ||
== Naming == | == Naming == | ||
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!|50c, 1\24 | !|50c, 1\24 | ||
||semisecond | ||semisecond | ||
||unison-inter-second | ||interizer, unison-inter-second | ||
||ultraunison<br/>inframinor second | ||ultraunison<br/>inframinor second | ||
||''-'' | ||''-'' | ||
| Line 59: | Line 59: | ||
!|1150c, 23\24 | !|1150c, 23\24 | ||
||semifourteenth | ||semifourteenth | ||
||seventh-inter-octave | ||antiinterizer, seventh-inter-octave | ||
||ultramajor seventh<br/>infraoctave | ||ultramajor seventh<br/>infraoctave | ||
||''-'' | ||''-'' | ||
| Line 66: | Line 66: | ||
== Pythagorean-based interordinals == | == 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. These are also the mathematically exact "interseptimals". | 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. These are also the mathematically exact "interseptimals", separated from the surrounding septimal intervals by sqrt(49/48). | ||
* interizer: sqrt(256/243) = 45.1c = midpoint of 64/63 and 28/27 | * interizer: sqrt(256/243) = 45.1c = midpoint of 64/63 and 28/27 | ||
* semifourth: sqrt(9/8 * 32/27) = sqrt(4/3) = 249.0c = midpoint of 8/7 and 7/6 | * semifourth: sqrt(9/8 * 32/27) = sqrt(4/3) = 249.0c = midpoint of 8/7 and 7/6 | ||
Latest revision as of 10:44, 15 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, as interordinals are midpoints of septimal intervals (such as 8/7 and 7/6) separated by 49/48; however, the term "interseptimal" on this wiki may refer to 49/48 itself.
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/semisecond, and its octave-complement, the antiinterizer/semifourteenth, 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, 29edo, and 53edo 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. See Chthonic harmony for a compositional theory using interordinals.
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 | Latal names |
|---|---|---|---|---|---|
| 50c, 1\24 | semisecond | interizer, unison-inter-second | ultraunison inframinor second |
- | latal semichroma |
| 250c, 5\24 | semifourth | second-inter-third | ultramajor second inframinor third |
chthonic | neutral (uni)latus |
| 450c, 9\24 | semisixth | third-inter-fourth | ultramajor third infrafourth |
naiadic | neutral bilatus |
| 750c, 15\24 | semitenth | fifth-inter-sixth | ultrafifth inframinor sixth |
cocytic | neutral trilatus |
| 950c, 19\24 | semitwelfth | sixth-inter-seventh | ultramajor sixth inframinor seventh |
ouranic | neutral antilatus |
| 1150c, 23\24 | semifourteenth | antiinterizer, seventh-inter-octave | ultramajor seventh infraoctave |
- | latal antisemichroma |
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. These are also the mathematically exact "interseptimals", separated from the surrounding septimal intervals by sqrt(49/48).
- interizer: sqrt(256/243) = 45.1c = midpoint of 64/63 and 28/27
- semifourth: sqrt(9/8 * 32/27) = sqrt(4/3) = 249.0c = midpoint of 8/7 and 7/6
- semisixth: sqrt(81/64 * 4/3) = sqrt(27/16) = 452.9c = midpoint of 9/7 and 21/16
- semitenth: sqrt(3/2 * 128/81) = sqrt(64/27) = 747.1c = midpoint of 32/21 and 14/9
- semitwelfth: sqrt(27/16 * 16/9) = sqrt(3/1) = 951.0c = midpoint of 12/7 and 7/4
- semifourteenth: sqrt(243/128 * 2/1) = sqrt(243/64) = 1154.9c = midpoint of 27/14 and 63/32
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
