32/17: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = septendecimal major seventh, septendecimal diminished octave | |||
| Name = | |||
| Color name = 17u7, su 7th | | Color name = 17u7, su 7th | ||
| Sound = jid_32_17_pluck_adu_dr220.mp3 | | Sound = jid_32_17_pluck_adu_dr220.mp3 | ||
}} | }} | ||
In [[17-limit]] [[just intonation]], '''32/17''' is the '''septendecimal major seventh''' or the '''septendecimal diminished octave''', depending on how one views it. It is also the octave-reduced 17th [[subharmonic]]. Its inversion is [[17/16]], the octave-reduced 17th harmonic. Measuring about 1095{{cent}}, it is the [[mediant]] between [[15/8]] and [[17/9]]. | |||
== Terminology and notation == | |||
There exists a disagreement in different conceptualization systems on whether 32/17 should be a major seventh or a diminished octave. The major seventh view corresponds to [[Functional Just System]], with the formal comma [[4131/4096]] separating it from [[243/128]], the Pythagorean major seventh. The diminished octave view corresponds to [[Helmholtz-Ellis notation]], with the formal comma [[2187/2176]] separating it from [[4096/2187]], the Pythagorean diminished octave. | |||
In practice, the interval category may, arguably, vary by context. One solution for the JI user who uses expanded [[circle-of-fifths notation]] is to prepare a [[Pythagorean comma]] accidental so that the interval can be notated in either category. | |||
== Approximation == | |||
{{Interval edo approximation|32/17}} | |||
== See also == | == See also == | ||
| Line 18: | Line 17: | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
[[Category:Seventh]] | [[Category:Seventh]] | ||
[[Category:Major seventh]] | [[Category:Major seventh]] | ||
[[Category:Octave]] | |||
[[Category:Diminished octave]] | |||
Latest revision as of 13:14, 3 November 2025
| Interval information |
septendecimal diminished octave
reduced subharmonic
[sound info]
In 17-limit just intonation, 32/17 is the septendecimal major seventh or the septendecimal diminished octave, depending on how one views it. It is also the octave-reduced 17th subharmonic. Its inversion is 17/16, the octave-reduced 17th harmonic. Measuring about 1095 ¢, it is the mediant between 15/8 and 17/9.
Terminology and notation
There exists a disagreement in different conceptualization systems on whether 32/17 should be a major seventh or a diminished octave. The major seventh view corresponds to Functional Just System, with the formal comma 4131/4096 separating it from 243/128, the Pythagorean major seventh. The diminished octave view corresponds to Helmholtz-Ellis notation, with the formal comma 2187/2176 separating it from 4096/2187, the Pythagorean diminished octave.
In practice, the interval category may, arguably, vary by context. One solution for the JI user who uses expanded circle-of-fifths notation is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 11 | 10\11 | 1090.91 | -4.14 | -3.79 |
| 12 | 11\12 | 1100.00 | +4.96 | +4.96 |
| 22 | 20\22 | 1090.91 | -4.14 | -7.58 |
| 23 | 21\23 | 1095.65 | +0.61 | +1.16 |
| 24 | 22\24 | 1100.00 | +4.96 | +9.91 |
| 34 | 31\34 | 1094.12 | -0.93 | -2.63 |
| 35 | 32\35 | 1097.14 | +2.10 | +6.12 |
| 45 | 41\45 | 1093.33 | -1.71 | -6.42 |
| 46 | 42\46 | 1095.65 | +0.61 | +2.33 |
| 57 | 52\57 | 1094.74 | -0.31 | -1.46 |
| 58 | 53\58 | 1096.55 | +1.51 | +7.28 |
| 68 | 62\68 | 1094.12 | -0.93 | -5.25 |
| 69 | 63\69 | 1095.65 | +0.61 | +3.49 |
| 79 | 72\79 | 1093.67 | -1.37 | -9.04 |
| 80 | 73\80 | 1095.00 | -0.04 | -0.30 |