9-odd-limit: Difference between revisions
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==Approximation by edos== | == 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 == | |||
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-limit|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 fifth]]s 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). | |||
=== 9/7 === | |||
=== 14/9 === | |||
=== 16/9 === | |||
=== 9/5 === | |||
Revision as of 18:10, 21 June 2026
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
| 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
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
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).
