256/255: Difference between revisions

(42 intermediate revisions by 12 users not shown)

Line 1:

{{Infobox Interval

~~| Ratio = 256/255~~

| Name = charisma, charic comma, septendecimal kleisma

~~| Monzo = 8 -1 -1 0 0 0 -1~~

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

~~| Cents = 6.77588~~

| Comma = yes

| Name = septendecimal kleisma~~, <br/> 255th subharmonic~~

| Color name =

| ~~FJS name =~~

~~| Sound~~ =

}}

{{Redirect|Charisma|the temperament that used to go by this name|Horcrux}}

'''256/255''', the '''~~septendecimal kleisma~~''', or '''~~255th subharmonic~~''', is a [[small comma|small]] [[17-limit]] [[superparticular]] comma about 6.8 [[cent]]s in size. It ~~forms~~ the amount by which a stack consisting of [[15/8]] and [[17/16]] falls short of an [[octave]]. It ~~differs from~~ [[~~352~~/~~351~~]] (the ~~minthma~~) by [[~~936~~/~~935~~]]- an [[~~unnoticeable~~ comma]] ~~measuring about 1~~.85 ~~cents.~~

'''256/255''', the '''charisma''', '''charic comma''' or '''septendecimal kleisma''' 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), the difference between [[128/85]] (the archagall fifth) and [[3/2]] (the just perfect fifth), and the amount by which a stack consisting of [[15/8]] and 17/16 falls short of an [[octave]].

It is the octave-reduced 255th subharmonic. By virtue of {{nowrap| 255 {{=}} 2<sup>8</sup> - 1 }}, it is a [[Mersenne comma]].

== Temperaments ==

[[Tempering out]] this comma defines the full 17-limit rank-6 '''charismic''' temperament or 2.3.5.17-subgroup rank-3 '''charic''' temperament. In either case, it enables the [[charismic chords]].

By also tempering out the semitonisma, [[289/288]] ({{S|17}}), charic can be tempered to an efficient but lower-accuracy rank-2 temperament known as [[srutal archagall]], which has [[17/15]]~[[9/8]] and therefore makes the minor third reached by the circle of fifths equal to ~[[20/17]]. Alternatively, by tempering out the marvel comma, [[225/224]] ({{S|15}}), charic can be extended to the 2.3.5.7.17 subgroup, known as [[char]], which is also of lower accuracy compared to either charic or [[marvel]] as it introduces the equivalences [[17/15]]~[[8/7]] and [[20/17]]~[[7/6]].

=== Charic ===

[[Subgroup]]: 2.3.5.17

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.3878{{c}}, 3/2 = 702.7586{{c}}, 5/4 = 387.9493{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, 3/2 = 703.0148{{c}}, 5/4 = 388.0713{{c}}

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

[[Badness]] (Sintel): 0.157

=== Charismic ===

[[Subgroup]]: 2.3.5.7.11.13.17

[[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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.3878{{c}}, 3/2 = 702.7586{{c}}, 5/4 = 387.9493{{c}}, ~7/4 = 970.0490{{c}}, ~11/8 = 553.1530{{c}}, ~13/8 = 842.3626{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, 3/2 = 703.0148{{c}}, 5/4 = 388.0713{{c}}, ~7/4 = 970.1701{{c}}, ~11/8 = 552.9744{{c}}, ~13/8 = 842.2995{{c}}

{{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 }}

[[Badness]] (Sintel): 2.75

== Etymology and history ==

This interval was described as ''septendecimal kleisma'' as early as 2020.

The "char" in the name ''charisma'' (or ''charic comma'') refers to the char data type in C-derived programming languages, where the char represents a byte of at least or exactly 8 bits. Thereby, the maximum unsigned value for an 8-bit char is 255 and the number of values an 8-bit char can take is 256, hence 256/255.

The former name ''diasemisma'' was proposed by [[User:Xenllium|Xenllium]] in May 2023. It is a contraction of ''diatonic semitone'' into a single word. In some contexts, both [[16/15]] and [[17/16]] are considered minor seconds (i.e. [[diatonic semitone]]), namely classical diatonic semitone and minor diatonic semitone respectively. However, a rename to ''charisma'' was proposed as part of an effort to make naming more standardised and for a number of reasons including potential confusion with [[diasem]] and the former nonconforming naming of [[horcrux]] in the 11- and 13-limit (which were formerly named "charisma" and "charismic", creating a potential false impression that the former was a comma, not a temperament, and that the latter was the temperament defined by tempering out that comma in the corresponding prime limit).

The rename took place as [[Starshine]] (from the [[XA Discord]] server) suggested (half-jokingly) that a chance had been missed to name it the ''charisma'' in December 2023, a name which [[Godtone]] took favor to and championed which then caused awareness of nonconforming names of two horcrux temeraments. A revision to ''charsma'' (no-''i'' spelling) was proposed by Xenllium in January 2024 for disambiguation but this would cause inconsistency with the -ic/-ismic/-isma rule which is a reason that those same temperaments were already being proposed to be renamed.

== See also ==

* [[List of superparticular intervals]]

* [[255/128]] – its [[octave complement]]

* [[85/64]] – its [[fourth complement]]

[[Category:Charismic]]

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

[[Category:Commas referencing a famous use of a number]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Ratio = 256/255
+| Name = charisma, charic comma, septendecimal kleisma
-| Monzo = 8 -1 -1 0 0 0 -1
+| Color name = 17ug1, sugu 1sn, Sugu comma
-| Cents = 6.77588
+| Comma = yes
-| Name = septendecimal kleisma, <br/> 255th subharmonic
-| Color name =
-| FJS name =
-| Sound =
 }}
+{{Redirect|Charisma|the temperament that used to go by this name|Horcrux}}
-'''256/255''', the '''septendecimal kleisma''', or '''255th subharmonic''', is a [[small comma|small]] [[17-limit]] [[superparticular]] comma about 6.8 [[cent]]s in size.  It forms the amount by which a stack consisting of [[15/8]] and [[17/16]] falls short of an [[octave]].  It differs from [[352/351]] (the minthma) by [[936/935]]- an [[unnoticeable comma]] measuring about 1.85 cents.
+'''256/255''', the '''charisma''', '''charic comma''' or '''septendecimal kleisma''' 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), the difference between [[128/85]] (the archagall fifth) and [[3/2]] (the just perfect fifth), and the amount by which a stack consisting of [[15/8]] and 17/16 falls short of an [[octave]].
+It is the octave-reduced 255th subharmonic. By virtue of {{nowrap| 255 {{=}} 2<sup>8</sup> - 1 }}, it is a [[Mersenne comma]].
+== Temperaments ==
+[[Tempering out]] this comma defines the full 17-limit rank-6 '''charismic''' temperament or 2.3.5.17-subgroup rank-3 '''charic''' temperament. In either case, it enables the [[charismic chords]].
+By also tempering out the semitonisma, [[289/288]] ({{S|17}}), charic can be tempered to an efficient but lower-accuracy rank-2 temperament known as [[srutal archagall]], which has [[17/15]]~[[9/8]] and therefore makes the minor third reached by the circle of fifths equal to ~[[20/17]]. Alternatively, by tempering out the marvel comma, [[225/224]] ({{S|15}}), charic can be extended to the 2.3.5.7.17 subgroup, known as [[char]], which is also of lower accuracy compared to either charic or [[marvel]] as it introduces the equivalences [[17/15]]~[[8/7]] and [[20/17]]~[[7/6]].
+=== Charic ===
+[[Subgroup]]: 2.3.5.17
+{{Mapping|legend=2| 1 0 0 8 | 0 1 0 -1 | 0 0 1 -1 }}
+: mapping generators: ~2, ~3, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.3878{{c}}, 3/2 = 702.7586{{c}}, 5/4 = 387.9493{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, 3/2 = 703.0148{{c}}, 5/4 = 388.0713{{c}}
+{{Optimal ET sequence|legend=1| 5, 7, 9, 10, 12, 22, 31, 34, 65, 87, 99, 343cgg, 442cgg, 541bcggg, 640bcgggg }}
+[[Badness]] (Sintel): 0.157
+=== Charismic ===
+[[Subgroup]]: 2.3.5.7.11.13.17
+[[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
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.3878{{c}}, 3/2 = 702.7586{{c}}, 5/4 = 387.9493{{c}}, ~7/4 = 970.0490{{c}}, ~11/8 = 553.1530{{c}}, ~13/8 = 842.3626{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, 3/2 = 703.0148{{c}}, 5/4 = 388.0713{{c}}, ~7/4 = 970.1701{{c}}, ~11/8 = 552.9744{{c}}, ~13/8 = 842.2995{{c}}
+{{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 }}
+[[Badness]] (Sintel): 2.75
+== Etymology and history ==
+This interval was described as ''septendecimal kleisma'' as early as 2020.
+The "char" in the name ''charisma'' (or ''charic comma'') refers to the char data type in C-derived programming languages, where the char represents a byte of at least or exactly 8 bits. Thereby, the maximum unsigned value for an 8-bit char is 255 and the number of values an 8-bit char can take is 256, hence 256/255.
+The former name ''diasemisma'' was proposed by [[User:Xenllium|Xenllium]] in May 2023. It is a contraction of ''diatonic semitone'' into a single word. In some contexts, both [[16/15]] and [[17/16]] are considered minor seconds (i.e. [[diatonic semitone]]), namely classical diatonic semitone and minor diatonic semitone respectively. However, a rename to ''charisma'' was proposed as part of an effort to make naming more standardised and for a number of reasons including potential confusion with [[diasem]] and the former nonconforming naming of [[horcrux]] in the 11- and 13-limit (which were formerly named "charisma" and "charismic", creating a potential false impression that the former was a comma, not a temperament, and that the latter was the temperament defined by tempering out that comma in the corresponding prime limit).
+The rename took place as [[Starshine]] (from the [[XA Discord]] server) suggested (half-jokingly) that a chance had been missed to name it the ''charisma'' in December 2023, a name which [[Godtone]] took favor to and championed which then caused awareness of nonconforming names of two horcrux temeraments. A revision to ''charsma'' (no-''i'' spelling) was proposed by Xenllium in January 2024 for disambiguation but this would cause inconsistency with the -ic/-ismic/-isma rule which is a reason that those same temperaments were already being proposed to be renamed.
+== See also ==
+* [[List of superparticular intervals]]
+* [[255/128]] – its [[octave complement]]
+* [[85/64]] – its [[fourth complement]]
+[[Category:Charismic]]
+[[Category:Commas named after their interval size]]
+[[Category:Commas referencing a famous use of a number]]