9-odd-limit

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 Odd-limits:

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• 5-odd-limit
• 7-odd-limit
• 9-odd-limit
• 11-odd-limit
• 13-odd-limit
• 15-odd-limit
• 17-odd-limit
• 19-odd-limit
• 21-odd-limit
• 23-odd-limit
• 25-odd-limit
• 27-odd-limit

The 9-odd-limit consists of all intervals where the largest allowable odd factor in the numerator and denominator is 9. Reduced to an octave, these are:

Table of 9-odd-limit intervals

Reduced to an octave, the intervals of the 9-odd-limit are:

Interval	Cents	Name
1/1	0.0	Unison
10/9	182.4	Minor whole tone, Ptolemaic major 2nd
9/8	203.9	Major whole tone, Pythagorean major 2nd
8/7	231.2	Septimal major 2nd
7/6	266.9	Septimal minor 3rd
6/5	315.6	Classical minor 3rd
5/4	386.4	Classical major 3rd
9/7	435.1	Septimal major 3rd
4/3	498.0	Perfect 4th
7/5	582.5	Lesser septimal tritone
10/7	617.5	Greater septimal tritone
3/2	702.0	Perfect 5th
14/9	764.9	Septimal minor 6th
8/5	813.6	Classical minor 6th
5/3	884.4	Classical major 6th
12/7	933.1	Septimal major 6th
7/4	968.8	Septimal minor 7th
16/9	996.1	Pythagoran minor 7th
9/5	1017.6	Classical minor 7th
2/1	1200.0	Octave

Approximation by edos

The first edo to be consistent to the 9-odd-limit is 5edo, giving a rough outline for harmony with its relatively accurate 3/2 and 7/4 of 720 and 960 cents respectively, while very sharply mapping 5/4 to 480 cents. The first edo to be distinctly consistent in this limit is 41edo.

Intervals of the 9-odd-limit

Here are the intervals in the 9-odd-limit which are not part of any lower odd-limit. Note that the intervals of the 9-odd-limit are all contained within the 7-prime-limit, as 9 factors as 3 × 3.

10/9

10/9, often called the minor whole tone or ptolemaic whole tone, is an interval of 182.4 cents. It is often considered a counterpart to 9/8, the major whole tone, as 9/8 and 10/9 add up to 5/4, the classical major third. Meantone temperament eliminates the distinction between 10/9 and 9/8 by tempering out 81/80, the syntonic comma. Due to its smaller size and more complex ratio, it is generally considered to be somewhat more dissonant than 9/8, and has a darker quality than 9/8. It is notably the octave complement of 9/5, the classical minor seventh.

In terms of chthonic harmony, it can be considered a subminor latus, as the fourth complement of 6/5.

Porcupine temperament uses a very flat (~163c) 10/9 as a generator, where it is equated with 11/10 and 12/11.

9/8

See also: Pythagorean tuning § Major second

9/8 can be called the Pythagorean whole tone or major whole tone, and sometimes simply the whole tone. It is reached by stacking up 2 perfect fifths and down an octave. It is a very important melodic interval in the diatonic scale, being a major second. In temperaments generated by the fifth, it is often equated with nearby intervals; for example, Meantone equates it with 10/9, while Archy equates it with 8/7. An octave above 9/8 is the major ninth 9/4, which often appears in chords such as 1–5/4–3/2–9/4 (4:5:6:9).