21/20: Difference between revisions

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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]].

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>[https://~~marsbat~~.~~space~~/pdfs/crystal-growth.pdf ~~Marc Sabat (2008)~~ Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space]</ref>.

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 21/20'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.

~~<references~~/>

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}}

== See also ==

* [[40/21]] – its [[octave complement]]

* [[10/7]] – its [[fifth complement]]

* [[1ed21/20]] - its equal multiplication

* [[List of superparticular intervals]]

* [[Gallery of just intervals]]

* [[Septisemi temperaments]], where it is tempered out

== References ==

[[Category:Second]]

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[[Category:Chroma]]

[[Category:Septisemi]]

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

@@ 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]].
+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 ==
-/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>[https://marsbat.space/pdfs/crystal-growth.pdf Marc Sabat (2008) Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space]</ref>.
+/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''').
+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.
-<references/>
+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}}
 == See also ==
 * [[40/21]] – its [[octave complement]]
 * [[10/7]] – its [[fifth complement]]
-* [[1ed21/20]] - its equal multiplication
 * [[List of superparticular intervals]]
 * [[Gallery of just intervals]]
 * [[Septisemi temperaments]], where it is tempered out
+== References ==
+<references/>
 [[Category:Second]]
@@ Line 26: / Line 53: @@
 [[Category:Chroma]]
 [[Category:Septisemi]]
-{{todo|improve synopsis}}
+[[Category:Commas named after their interval size]]
+{{Todo|improve synopsis}}