21/20: Difference between revisions

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{{Infobox Interval

| Name = septimal minor semitone, large septimal chroma

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

| Color name = zg2, zogu 2nd

| Sound = jid_21_20_pluck_adu_dr220.mp3

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== 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>.

~~<references~~/>

== 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]].

== See also ==

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* [[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 1: / Line 1: @@
 {{Infobox Interval
-| Name = septimal minor semitone, large septimal chroma
+| Name = septimal minor semitone, septimal chromatic semitone, large septimal chroma
 | Color name = zg2, zogu 2nd
 | Sound = jid_21_20_pluck_adu_dr220.mp3
@@ Line 10: / Line 10: @@
 == 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>.
-<references/>
+== 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''').
+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 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 ==
@@ Line 20: / Line 38: @@
 * [[Gallery of just intervals]]
 * [[Septisemi temperaments]], where it is tempered out
+== References ==
+<references/>
 [[Category:Second]]
@@ Line 25: / Line 46: @@
 [[Category:Chroma]]
 [[Category:Septisemi]]
+[[Category:Commas named after their interval size]]
 {{todo|improve synopsis}}