Octave

This page is about the interval region. For the octave as a just ratio, see 2/1.

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Octave

A perfect octave (P8) or octave (8ve) is an interval that is approximately 1200 cents in size. While a rough tuning range for octaves is sharper than 1170 cents according to Margo Schulter's theory of interval regions, the term octave tends to imply a function within music that only works with intervals that corresponding to a just ratio of 2/1 or a close approximation thereof, usually preferred to be sharp-tempered if tempered. Other intervals are also classified as perfect octaves, sometimes called wolf octaves or imperfect octaves, if they are reasonably mapped to 7\7 and 24\24 (precisely seven steps of the diatonic scale and twelve steps of the chromatic scale). The use of 24edo's 24\24 as the mapping criteria here rather than 12edo's 12\12 better captures the characteristics of many intervals in the 11- and 13-limit.

The aforementioned function is the interval of equivalence, or equave, because tones separated by an octave are perceived to have the same or similar pitch class to the average human listener. The reason for this phenomenon is probably due to the strong concordance of the octave or the strong amplitude of the second harmonic in most harmonic instruments. As such, it is common practice to octave-reduce intervals so that they lie within the octave.

Because of that, this page only covers intervals of 1200 cents and flatter, as sharper intervals octave-reduce to commas and dieses.

For the sake of simplicity, this page also covers interseptimal seventh-octaves, which are approximately 1150 cents in size. Thus, the interval region considered as "octave" for the purpose of this page is 1140-1200 cents.

Todo: review Mention concordance before harmonic entropy, since harmonic entropy is a single model of concordance

In just intonation

By prime limit

The only "perfect" octave is the interval 2/1, which can be stacked to produce all other 2-limit intervals. It is 1200 cents in size, by definition. However, various "out-of-tune" octaves exist, usually flat or sharp of an octave by a small interval such as a comma.

Several notable ones are:

Todo: complete list

In tempered scales

As the just octave of 2/1 is the interval being equally divided in EDOs, it is represented perfectly in all of them. It is also represented perfectly in all octave-period MOSes. Note both of these statements assume the octave is untempered. The following table lists other octave-sized intervals (> 1140 cents) that exist in various significant EDOs.

EDO	Suboctaves
22	1145 ¢
24	1150 ¢
25	1152 ¢
26	1154 ¢
27	1156 ¢
29	1159 ¢
31	1161 ¢
34	1165 ¢
41	1142 ¢, 1171 ¢
53	1155 ¢, 1177 ¢

2/1 is also represented perfectly in most temperaments, or the most common tunings thereof, and is mainly involved in octave-reducing intervals (such as saying that, in meantone, four 3/2s (octave-reduced) stack to 5/4).

Todo: complete table