Interordinal

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 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:

1. semifourth (between major 2nd and minor 3rd)
2. semisixth (between major 3rd and perfect 4th)
3. semitenth (between perfect 5th and minor 6th)
4. 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, 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. 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:

Interordinal nomenclatures

24edo interval	"semi" names	"inter" names	"ultra"/"infra"	Greek-derived names	Latal names
50c, 1\24	semisecond	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	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".

• 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

Interval regions

Unison (JND region)

Comma

Quartertone

Semitone

Neutral second

Major second

Inframinor third

Minor third

Neutral third

Major third

Ultramajor third

Fourth

Semiaugmented fourth

Tritone

Semidiminished fifth

Fifth

Inframinor sixth

Minor sixth

Neutral sixth

Major sixth

Ultramajor sixth

Minor seventh

Neutral seventh

Major seventh

Ultramajor seventh

Octave