21/20: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = septimal minor semitone, septimal chromatic semitone, large septimal chroma | |||
| Name = minor semitone, | |||
| Color name = zg2, zogu 2nd | | Color name = zg2, zogu 2nd | ||
| Sound = jid_21_20_pluck_adu_dr220.mp3 | | Sound = jid_21_20_pluck_adu_dr220.mp3 | ||
| Comma = yes | |||
}} | }} | ||
{{Wikipedia|Septimal chromatic semitone}} | |||
'''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 [[8/7]] and [[6/5]], or between [[5/3]] | '''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]]. | ||
In what is known as an authentic cadence, there is a resolution from the V chord to the I chord. If the V is a [[4:5:6:7|harmonic seventh chord]], its harmonic seventh ([[21/16]] above the tonic) resolves down to the major third of the I chord ([[5/4]]) by a step of 21/20. | |||
== 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<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>. | |||
== Approximation == | |||
{{Interval edo approximation|21/20}} | |||
== Interval chain == | |||
An [[interval chain]] of 21/20's 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's upwards is ~4{{c}} from [[11/10]]; | |||
* A stack of seven 21/20's upwards is ~9{{c}} from [[7/5]]; | |||
* A stack of ten 21/20's upwards is ~4{{c}} from [[13/8]]; | |||
* A stack of twelve 21/20's upwards is ~4{{c}} from [[9/5]]; | |||
and | |||
* A stack of six 21/20's downwards is ~10{{c}} from [[3/2]]; | |||
* A stack of nine 21/20's downwards is ~5{{c}} from [[9/7]]; | |||
* A stack of eleven 21/20's downwards is ~4{{c}} 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 14.2067edo, 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]], or alternatively of the 2.9.5.7.11.17 [[subgroup]]. | |||
{{Harmonics in equal|1|21|20|intervals=integer|columns=11}} | |||
{{Harmonics in equal|1|21|20|intervals=integer|collapsed=1|start=12|columns=12}} | |||
== See also == | == See also == | ||
| Line 17: | Line 44: | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[ | * [[Septisemi temperaments]], where it is tempered out | ||
== References == | |||
<references/> | |||
[[Category:Second]] | [[Category:Second]] | ||
[[Category:Semitone]] | [[Category:Semitone]] | ||
[[Category:Chroma]] | |||
[[Category: | [[Category:Septisemi]] | ||
[[Category:Commas named after their interval size]] | |||
{{Todo|improve synopsis}} | |||