256/255: Difference between revisions

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{{Infobox Interval

| Name = septendecimal kleisma, charisma, char comma, 255th subharmonic

| Name = septendecimal kleisma, charisma, char comma, octave-reduced 255th subharmonic

| Color name = 17ug1, sugu 1sn, Sugu comma

| Comma = yes

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'''256/255''', the '''septendecimal kleisma''', '''charisma''', '''char comma''' or '''255th subharmonic''' is a [[small comma|small]] [[17-limit]] [[superparticular]] comma about 6.8 [[cent]]s in size. It is the difference between [[16/15]] (the classical diatonic semitone) and [[17/16]] (the large septendecimal semitone) and forms the amount by which a stack consisting of [[15/8]] and 17/16 falls short of an [[octave]].

'''256/255''', the '''septendecimal kleisma''', '''charisma''', '''char comma''' or '''octave-reduced 255th subharmonic''' is a [[small comma|small]] [[17-limit]] [[superparticular]] comma about 6.8 [[cent]]s in size. It is the difference between [[16/15]] (the classical diatonic semitone) and [[17/16]] (the large septendecimal semitone) and forms the amount by which a stack consisting of [[15/8]] and 17/16 falls short of an [[octave]].

== Temperaments ==

By tempering it out is defined the ~~'''charismic temperament''' (~~full 17-limit rank-6~~) or~~ '''~~charic~~ temperament''' (2.3.5.17 subgroup rank-3), which enables the [[charismic chords]].

By tempering it out is defined the full 17-limit rank-6 '''charismic temperament''' or 2.3.5.17 subgroup rank-3 '''charic temperament''', which enables the [[charismic chords]].

=== Charic ===

[[Subgroup]]: 2.3.5.17

~~Patent EDO tunings with [[relative error]] < 1/3 for all~~ generators: ~~{{Optimal ET sequence|~~3, ~~9, 10, 12, 15, 21, 22, 24, 31, 32, 34, 43, 44, 46, 55, 56, 58, 65, 68, 77, 80, 111, 114, 145 }}~~

: sval mapping generators: ~2, ~3, ~5

[[CTE]] ~~generators~~: (2/1,) 3/2 = 702.~~647~~, 5/4 = 387.~~798~~

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, 3/2 = 702.6467, 5/4 = 387.7981

{{Optimal ET sequence|legend=1| 5, 7, 9, 10, 12, 22, 31, 34, 65, 87, 99, 343cgg, 442cgg, 541bcggg, 640bcgggg }}

=== Charismic ===

[[Subgroup]]: ~~full [[~~17~~-limit]]~~

[[Subgroup]]: 2.3.5.7.11.13.17

[[Mapping]]: ~~same as charic plus trivial pure extra primes~~

[[Mapping]]:

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 8 || ],

|-

| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || -1 || ],

|-

| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || -1 || ],

|-

| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]

|}

: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

~~Patent EDO tunings with~~ [[~~relative error~~]] ~~<~~ 1/~~3 for all generators: {{Optimal ET sequence|9~~, 31, 43, 46, ~~56 }}~~

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, 3/2 = 702.6467, 5/4 = 387.7981, ~7/4, ~11/8, ~13/8

~~[[CTE]] generators: same as charic plus trivial pure extra primes~~

{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 41, 46, 58, 77, 80, 87, 99ef, 111, 121, 152fg, 167, 198g, 256cfg, 319fgg, 377cdefgg, 507cdeefggg, 705bccdeeffggg }}

=== [[Srutal archagall]] ===

=== Srutal archagall ===

By also tempering the semitonisma ([[289/288]]), an efficient temperament known as [[Srutal archagall]] is achieved, which is equivalently described as [[charic]] [[semitonic]].

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = septendecimal kleisma, <br>charisma, char comma, <br>255th subharmonic
+| Name = septendecimal kleisma, <br>charisma, char comma, <br>octave-reduced 255th subharmonic
 | Color name = 17ug1, sugu 1sn, <br>Sugu comma
 | Comma = yes
@@ Line 6: / Line 6: @@
 {{Redirect-distinguish|Charisma|Horcrux}}
-'''256/255''', the '''septendecimal kleisma''', '''charisma''', '''char comma''' or '''255th subharmonic''' is a [[small comma|small]] [[17-limit]] [[superparticular]] comma about 6.8 [[cent]]s in size. It is the difference between [[16/15]] (the classical diatonic semitone) and [[17/16]] (the large septendecimal semitone) and forms the amount by which a stack consisting of [[15/8]] and 17/16 falls short of an [[octave]].
+'''256/255''', the '''septendecimal kleisma''', '''charisma''', '''char comma''' or '''octave-reduced 255th subharmonic''' is a [[small comma|small]] [[17-limit]] [[superparticular]] comma about 6.8 [[cent]]s in size. It is the difference between [[16/15]] (the classical diatonic semitone) and [[17/16]] (the large septendecimal semitone) and forms the amount by which a stack consisting of [[15/8]] and 17/16 falls short of an [[octave]].
 == Temperaments ==
-By tempering it out is defined the '''charismic temperament''' (full 17-limit rank-6) or '''charic temperament''' (2.3.5.17 subgroup rank-3), which enables the [[charismic chords]].
+By tempering it out is defined the full 17-limit rank-6 '''charismic temperament''' or 2.3.5.17 subgroup rank-3 '''charic temperament''', which enables the [[charismic chords]].
 === Charic ===
 [[Subgroup]]: 2.3.5.17
-{{Mapping|legend=1| 1 1 2 5 | 0 1 0 -1 | 0 0 1 -1 }}
+{{Mapping|legend=2| 1 0 0 8 | 0 1 0 -1 | 0 0 1 -1 }}
-Patent EDO tunings with [[relative error]] &lt; 1/3 for all generators: {{Optimal ET sequence|3, 9, 10, 12, 15, 21, 22, 24, 31, 32, 34, 43, 44, 46, 55, 56, 58, 65, 68, 77, 80, 111, 114, 145 }}
+: sval mapping generators: ~2, ~3, ~5
-[[CTE]] generators: (2/1,) 3/2 = 702.647, 5/4 = 387.798
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, 3/2 = 702.6467, 5/4 = 387.7981
+{{Optimal ET sequence|legend=1| 5, 7, 9, 10, 12, 22, 31, 34, 65, 87, 99, 343cgg, 442cgg, 541bcggg, 640bcgggg }}
 === Charismic ===
-[[Subgroup]]: full [[17-limit]]
+[[Subgroup]]: 2.3.5.7.11.13.17
-[[Mapping]]: same as charic plus trivial pure extra primes
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 8 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
-Patent EDO tunings with [[relative error]] &lt; 1/3 for all generators: {{Optimal ET sequence|9, 31, 43, 46, 56 }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, 3/2 = 702.6467, 5/4 = 387.7981, ~7/4, ~11/8, ~13/8
-[[CTE]] generators: same as charic plus trivial pure extra primes
+{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 41, 46, 58, 77, 80, 87, 99ef, 111, 121, 152fg, 167, 198g, 256cfg, 319fgg, 377cdefgg, 507cdeefggg, 705bccdeeffggg }}
-=== [[Srutal archagall]] ===
+=== Srutal archagall ===
 By also tempering the semitonisma ([[289/288]]), an efficient temperament known as [[Srutal archagall]] is achieved, which is equivalently described as [[charic]] [[semitonic]].