13/8: Difference between revisions
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Wikispaces>Andrew_Heathwaite **Imported revision 254165610 - Original comment: ** |
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{{Infobox Interval | |||
| Name = (lesser) tridecimal neutral sixth | |||
| Color name = 3o6, tho 6th | |||
| Sound = jid_13_8_pluck_adu_dr220.mp3 | |||
}} | |||
'''13/8''' is the '''(lesser) tridecimal neutral sixth''', which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In [[13-limit]] [[just intonation]], 13/8, as the octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of [[3/2]] and [[7/4]], separated from each by the [[superparticular]] ratios [[13/12]] and [[14/13]], respectively. The word "lesser" is added when necessary to differentiate it from [[64/39]], another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of [[5/4]] and [[13/10]]. | |||
13/8 differs from the Pythagorean minor sixth [[128/81]] by [[1053/1024]], about 48¢, from the classic minor sixth [[8/5]] by [[65/64]], about 27¢, from the undecimal neutral sixth [[18/11]] by [[144/143]], about 12¢, and from the rastmic neutral sixth [[44/27]] by [[352/351]], about 4.9¢. | |||
See | == Approximation == | ||
13/8 is a fraction of a cent away from the neutral sixth found in the [[10edo|10''n''-edo]] family (7\10). | |||
This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the [[golden ratio]]. In this case, 13/8 is ~7.4 [[cent|¢]] sharp of the golden ratio. | |||
{{Interval edo approximation}} | |||
== See also == | |||
* [[16/13]] – its [[octave complement]] | |||
* [[64/39]] – the greater tridecimal neutral sixth | |||
* [[Gallery of just intervals]] | |||
[[Category:Sixth]] | |||
[[Category:Neutral sixth]] | |||
[[Category:Golden ratio approximations]] | |||
Latest revision as of 17:13, 30 May 2026
| Interval information |
reduced harmonic
[sound info]
13/8 is the (lesser) tridecimal neutral sixth, which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In 13-limit just intonation, 13/8, as the octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of 3/2 and 7/4, separated from each by the superparticular ratios 13/12 and 14/13, respectively. The word "lesser" is added when necessary to differentiate it from 64/39, another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of 5/4 and 13/10.
13/8 differs from the Pythagorean minor sixth 128/81 by 1053/1024, about 48¢, from the classic minor sixth 8/5 by 65/64, about 27¢, from the undecimal neutral sixth 18/11 by 144/143, about 12¢, and from the rastmic neutral sixth 44/27 by 352/351, about 4.9¢.
Approximation
13/8 is a fraction of a cent away from the neutral sixth found in the 10n-edo family (7\10).
This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the golden ratio. In this case, 13/8 is ~7.4 ¢ sharp of the golden ratio.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 7 | 5\7 | 857.14 | +16.62 | +9.69 |
| 10 | 7\10 | 840.00 | -0.53 | -0.44 |
| 17 | 12\17 | 847.06 | +6.53 | +9.25 |
| 20 | 14\20 | 840.00 | -0.53 | -0.88 |
| 27 | 19\27 | 844.44 | +3.92 | +8.81 |
| 30 | 21\30 | 840.00 | -0.53 | -1.32 |
| 37 | 26\37 | 843.24 | +2.72 | +8.37 |
| 40 | 28\40 | 840.00 | -0.53 | -1.76 |
| 47 | 33\47 | 842.55 | +2.03 | +7.93 |
| 50 | 35\50 | 840.00 | -0.53 | -2.20 |
| 57 | 40\57 | 842.11 | +1.58 | +7.49 |
| 60 | 42\60 | 840.00 | -0.53 | -2.64 |
| 67 | 47\67 | 841.79 | +1.26 | +7.05 |
| 70 | 49\70 | 840.00 | -0.53 | -3.08 |
| 77 | 54\77 | 841.56 | +1.03 | +6.61 |
| 80 | 56\80 | 840.00 | -0.53 | -3.52 |
See also
- 16/13 – its octave complement
- 64/39 – the greater tridecimal neutral sixth
- Gallery of just intervals