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]].

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 ==

@@ 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]].
 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>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 ==