21/20: Difference between revisions

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{{Infobox Interval

| Name = septimal ~~minor~~ semitone, ~~septimal chromatic~~ semitone, ~~large septimal chroma~~

| Name = septimal chromatic semitone, aberschismic diatonic semitone, minor diatonic semitone

| Color name = zg2, zogu 2nd

| Sound = jid_21_20_pluck_adu_dr220.mp3

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'''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]].

'''21/20''' is a small semitone in [[7-limit]] [[just intonation]] of about 84.5 [[cents]], traditionally called the '''septimal chromatic semitone''' for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it functions as a ''[[diatonic semitone]]'', as is supported by [[Sagittal notation]], [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the [[Pythagorean limma]] altered by an [[aberschisma]]. This gives rise to the more precise name '''aberschismic diatonic semitone''', or as [[Marc Sabat]] has taken to call it, the '''minor diatonic 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>. It may be found as, for example, the difference between [[4/3]] and [[7/5]], between [[8/7]] and [[6/5]], or between [[5/3]] and [[7/4]].

~~== Terminology ==~~

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.

~~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 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]].

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 upwards is ~4¢ from [[11/10]]

* A stack of two 21/20's upwards is ~4{{c}} from [[11/10]];

* A stack of seven 21/20 upwards is ~9¢ from [[7/5]]

* A stack of seven 21/20's upwards is ~9{{c}} from [[7/5]];

* A stack of ten 21/20 upwards is ~4¢ from [[13/8]]

* A stack of ten 21/20's upwards is ~4{{c}} from [[13/8]];

* A stack of twelve 21/20 upwards is ~4¢ from [[9/5]]

* A stack of twelve 21/20's upwards is ~4{{c}} from [[9/5]];

and

* A stack of six 21/20 downwards is ~~~10¢~~ from [[3/2]]

* A stack of six 21/20's downwards is ~10{{c}} from [[3/2]];

* A stack of nine 21/20 downwards is ~5¢ from [[9/7]]

* A stack of nine 21/20's downwards is ~5{{c}} from [[9/7]];

* A stack of eleven 21/20 downwards is ~4¢ from [[7/6]]

* 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.~~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 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|collapsed=1|start=12|columns=12}}

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[[Category:Septisemi]]

[[Category:Commas named after their interval size]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = septimal minor semitone, septimal chromatic semitone, large septimal chroma
+| Name = septimal chromatic semitone, aberschismic diatonic semitone, minor diatonic semitone
 | Color name = zg2, zogu 2nd
 | Sound = jid_21_20_pluck_adu_dr220.mp3
@@ Line 7: / Line 7: @@
 {{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 [[4/3]] and [[7/5]], [[8/7]] and [[6/5]], or [[5/3]] and [[7/4]].
+'''21/20''' is a small semitone in [[7-limit]] [[just intonation]] of about 84.5 [[cents]], traditionally called the '''septimal chromatic semitone''' for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it functions as a ''[[diatonic semitone]]'', as is supported by [[Sagittal notation]], [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the [[Pythagorean limma]] altered by an [[aberschisma]]. This gives rise to the more precise name '''aberschismic diatonic semitone''', or as [[Marc Sabat]] has taken to call it, the '''minor diatonic 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>. It may be found as, for example, the difference between [[4/3]] and [[7/5]], between [[8/7]] and [[6/5]], or between [[5/3]] and [[7/4]].
-== Terminology ==
+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.
-/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 21/20s 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]].
+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 upwards is ~4¢ from [[11/10]]
+* A stack of two 21/20's upwards is ~4{{c}} from [[11/10]];
-* A stack of seven 21/20 upwards is ~9¢ from [[7/5]]
+* A stack of seven 21/20's upwards is ~9{{c}} from [[7/5]];
-* A stack of ten 21/20 upwards is ~4¢ from [[13/8]]
+* A stack of ten 21/20's upwards is ~4{{c}} from [[13/8]];
-* A stack of twelve 21/20 upwards is ~4¢ from [[9/5]]
+* A stack of twelve 21/20's upwards is ~4{{c}} from [[9/5]];
 and
-* A stack of six 21/20 downwards is ~10¢ from [[3/2]]
+* A stack of six 21/20's downwards is ~10{{c}} from [[3/2]];
-* A stack of nine 21/20 downwards is ~5¢ from [[9/7]]
+* A stack of nine 21/20's downwards is ~5{{c}} from [[9/7]];
-* A stack of eleven 21/20 downwards is ~4¢ from [[7/6]]
+* 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''').
-ed21/20 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.
+ed21/20 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.
 ed21/20 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}}
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 [[Category:Septisemi]]
 [[Category:Commas named after their interval size]]
-{{todo|improve synopsis}}
+{{Todo|improve synopsis}}