21/20: Difference between revisions
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== Terminology == | == Terminology == | ||
21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[ | 21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>. | ||
== Interval chain == | |||
An [[interval chain]] of 21/20s stacked on top of one another comes close to approximating some important [[JI]] intervals. The error between the approximation and the target JI interval may be tempered out in some [[regular temperaments]]. | |||
Some examples include: | |||
* A stack of two 21/20 upwards is ~4¢ from [[11/10]] | |||
* A stack of seven 21/20 upwards is ~9¢ from [[7/5]] | |||
* A stack of ten 21/20 upwards is ~4¢ from [[13/8]] | |||
* A stack of twelve 21/20 upwards is ~4¢ from [[9/5]] | |||
and | |||
* A stack of six 21/20 downwards is ~10¢ from [[3/2]] | |||
* A stack of nine 21/20 downwards is ~5¢ from [[9/7]] | |||
* A stack of eleven 21/20 downwards is ~4¢ from [[7/6]] | |||
When treated as a scale, this interval chain can be called the '''[[ambitonal sequence]] of 21/20''' ('''AS21/20''' or '''1ed21/20'''). | |||
1ed21/20 is equal to approximately 14.2067 EDO, and as a result of tethering between compressed 14 and heavily stretched 15 it is quite xenharmonic in its sound. It is related to the [[nautilus]], [[sextilifourths]] and [[floral]] temperaments. | |||
1ed21/20 offers a possible approximation of the no-3s [[11-limit]]. It might alternatively be used as a [[dual-fifth tuning]]. | |||
{{Harmonics in equal|1|21|20|intervals=prime}} | |||
== See also == | == See also == | ||
* [[40/21]] – its [[octave complement]] | * [[40/21]] – its [[octave complement]] | ||
* [[10/7]] – its [[fifth complement]] | * [[10/7]] – its [[fifth complement]] | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[Septisemi temperaments]], where it is tempered out | * [[Septisemi temperaments]], where it is tempered out | ||
== References == | |||
<references/> | |||
[[Category:Second]] | [[Category:Second]] | ||
| Line 26: | Line 46: | ||
[[Category:Chroma]] | [[Category:Chroma]] | ||
[[Category:Septisemi]] | [[Category:Septisemi]] | ||
[[Category:Commas named after their interval size]] | |||
{{todo|improve synopsis}} | {{todo|improve synopsis}} | ||
Latest revision as of 23:07, 1 August 2025
| Interval information |
septimal chromatic semitone,
large septimal chroma
reduced
S7 × S8 × S9
[sound info]
21/20 is a small semitone of about 85 cents. It may be found in 7-limit just intonation as, for example, the difference between 4/3 and 7/5, 8/7 and 6/5, or 5/3 and 7/4.
Terminology
21/20 is traditionally called a chroma, perhaps for its proximity (and conflation in systems like septimal meantone) with the major chroma 135/128. However, it is a diatonic semitone in both Helmholtz–Ellis notation and Functional Just System, viewed as the Pythagorean minor second 256/243 altered by 5120/5103. Marc Sabat has taken to call it the minor diatonic semitone in the same material where 15/14 is also named as the major chromatic semitone[1].
Interval chain
An interval chain of 21/20s stacked on top of one another comes close to approximating some important JI intervals. The error between the approximation and the target JI interval may be tempered out in some regular temperaments.
Some examples include:
- A stack of two 21/20 upwards is ~4¢ from 11/10
- A stack of seven 21/20 upwards is ~9¢ from 7/5
- A stack of ten 21/20 upwards is ~4¢ from 13/8
- A stack of twelve 21/20 upwards is ~4¢ from 9/5
and
- A stack of six 21/20 downwards is ~10¢ from 3/2
- A stack of nine 21/20 downwards is ~5¢ from 9/7
- A stack of eleven 21/20 downwards is ~4¢ from 7/6
When treated as a scale, this interval chain can be called the ambitonal sequence of 21/20 (AS21/20 or 1ed21/20).
1ed21/20 is equal to approximately 14.2067 EDO, and as a result of tethering between compressed 14 and heavily stretched 15 it is quite xenharmonic in its sound. It is related to the nautilus, sextilifourths and floral temperaments.
1ed21/20 offers a possible approximation of the no-3s 11-limit. It might alternatively be used as a dual-fifth tuning.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -17.5 | +40.8 | +1.1 | +9.9 | -12.4 | +36.2 | -5.9 | -29.5 | -22.4 | -1.3 | -32.3 |
| Relative (%) | -20.7 | +48.3 | +1.3 | +11.7 | -14.7 | +42.9 | -6.9 | -34.9 | -26.5 | -1.6 | -38.3 | |
| Step | 14 | 23 | 33 | 40 | 49 | 53 | 58 | 60 | 64 | 69 | 70 | |
See also
- 40/21 – its octave complement
- 10/7 – its fifth complement
- List of superparticular intervals
- Gallery of just intervals
- Septisemi temperaments, where it is tempered out
References
- ↑ Marc Sabat. Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space. Plainsound Music Edition, 2008.