Schismic: Difference between revisions

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| Odd limit 2 = (5-limit) 125 | Mistuning 2 = 0.837 | Complexity 2 = 29

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'''Schismic''', '''schismatic''', or '''helmholtz''' ~~is a~~ [[5-limit]] [[regular temperament|temperament]] which takes ~~a roughly justly tuned~~ [[3/2|perfect fifth]] and stacks it ~~eight times~~ to reach [[8/5]], ~~thus finding the 5th harmonic at~~ the diminished fourth (e.g. C–F♭) and [[tempering out]] the [[schisma]], 32805/32768. ~~[[5~~/4]] can be respelled as a major third flattened by one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5~~/4. It is one of~~ the ~~most basic examples of a~~ [[~~microtemperament~~]], as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between [[8192/6561]] and 5/4, the schisma being tempered out, is approximately 2 cents, which is [[just-noticeable difference|unnoticeable]] to most people). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings on both sides of the just interval work fine.

'''Schismic''', '''schismatic''', or '''helmholtz''' (specifically in the [[5-limit]]) is a[[regular temperament|temperament]] which takes an almost just [[3/2|perfect fifth]] and stacks it eightfold to reach [[8/5]], mapping 5/4 to the diminished fourth (e.g. C–F♭) and [[tempering out]] the [[schisma]], 32805/32768. Put alternatively: 8/5 maps to the [[tetratone]].

Extensions ~~to schismic~~ include [[~~garibaldi~~]], which equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224~~]] and [[5120/5103~~]]) to provide an efficient framework for [[7-limit]] harmony, ~~and unlike~~ 5-limit ~~schismic performs best with a fifth tuned~~ slightly sharp ~~of just;~~ [[~~pontiac~~]], which tempers out [[4375/4374]] to induce very little damage on schismic harmonies, at the cost of 7 being quite complex; and the 2.3.5.19 [[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]). This page, however, focuses on the basic 5-limit temperament.

[[5/4]] maps equivalently to a major third minus one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into one tempered comma, splitting octaves into two major thirds and one ~5/4.

It is one of the simplest [[microtemperament|microtemperaments]], as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the schisma is practically [[Unnoticeable comma|unnoticeable]]). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings with sharper fifths (and worse 5-limit, like in 41 or 94edo) still work fine.

Extensions include:

* [[Garibaldi]], which equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224]]) to provide an efficient framework for [[7-limit]] harmony, though with worse 5-limit intonation since the tuning uses slightly sharp fifths.

* [[Pontiac]], which tempers out [[4375/4374]] to induce very little damage on schismic harmonies, at the cost of 7 being quite complex.

* [[Sesquiquartififths]], which tempers out [[2401/2400]] and splits the fifth in fourths, inducing very little damage with a less complex mapping of 7 at the cost of quadrupling the complexity of 3 and 5.

* The 2.3.5.13 [[subgroup]] extension [[Schismatic family#Maqamschismic (2.3.5.13)|maqamschismic]], (tempering out the [[325/324|marveltwin comma]]) and finds [[13/8]] at the dupminor sixth (^^Ab from C). See [[2.3.5.13 subgroup]] for more details.

* The 2.3.5.19 [[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]).

This page, however, focuses on the basic 5-limit temperament.

See [[Schismatic family #Schismic, schismatic, a.k.a. helmholtz]] for technical data.

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<nowiki/>* In 5-limit CWE tuning

== Notation ==

Using schismic can be a challenge because it defies the tradition of {{w|tertian harmony}} in [[chain-of-fifths notation]]. The just major ~~triad on C~~ is C–F♭–G~~, for example~~. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.

Using schismic can be a challenge because it defies the tradition of diatonic {{w|tertian harmony}} in [[chain-of-fifths notation]]; The just major third is not C–E–G like in [[meantone]], but rather, C–F♭–G. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.

== Scales ==

=== MOS scales ===

* [[5L 7s]] (p-chromatic) [17edo]

* [[12L 5s]] (p-enharmonic) [29edo]

* [[12L 17s]] (pythagotonic) [41edo]

* [[12L 29s]] (pythamystonic) [53edo]

* [[12L 41s]] (antipythomerc) [65edo]

* [[53L 12s]] (m-chro antipythomerc) [118edo]

EDO in brackets represents basic step ratio.

=== Scala files ===

* [[Clipper32805]] – in a 1–3–5 equal-beating tuning

@@ Line 15: / Line 15: @@
 | Odd limit 2 = (5-limit) 125 | Mistuning 2 = 0.837 | Complexity 2 = 29
 }}
-'''Schismic''', '''schismatic''', or '''helmholtz''' is a [[5-limit]] [[regular temperament|temperament]] which takes a roughly justly tuned [[3/2|perfect fifth]] and stacks it eight times to reach [[8/5]], thus finding the 5th harmonic at the diminished fourth (e.g. C–F♭) and [[tempering out]] the [[schisma]], 32805/32768. [[5/4]] can be respelled as a major third flattened by one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5/4. It is one of the most basic examples of a [[microtemperament]], as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between [[8192/6561]] and 5/4, the schisma being tempered out, is approximately 2 cents, which is [[just-noticeable difference|unnoticeable]] to most people). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings on both sides of the just interval work fine.
+'''Schismic''', '''schismatic''', or '''helmholtz''' (specifically in the [[5-limit]]) is a[[regular temperament|temperament]] which takes an almost just [[3/2|perfect fifth]] and stacks it eightfold to reach [[8/5]], mapping 5/4 to the diminished fourth (e.g. C–F♭) and [[tempering out]] the [[schisma]], 32805/32768. Put alternatively: 8/5 maps to the [[tetratone]].
-Extensions to schismic include [[garibaldi]], which equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224]] and [[5120/5103]]) to provide an efficient framework for [[7-limit]] harmony, and unlike 5-limit schismic performs best with a fifth tuned slightly sharp of just; [[pontiac]], which tempers out [[4375/4374]] to induce very little damage on schismic harmonies, at the cost of 7 being quite complex; and the 2.3.5.19 [[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]). This page, however, focuses on the basic 5-limit temperament.
+[[5/4]] maps equivalently to a major third minus one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into one tempered comma, splitting octaves into two major thirds and one ~5/4.
+It is one of the simplest [[microtemperament|microtemperaments]], as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the schisma is practically [[Unnoticeable comma|unnoticeable]]). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings with sharper fifths (and worse 5-limit, like in 41 or 94edo) still work fine.
+Extensions include:
+* [[Garibaldi]], which equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224]]) to provide an efficient framework for [[7-limit]] harmony, though with worse 5-limit intonation since the tuning uses slightly sharp fifths.
+* [[Pontiac]], which tempers out [[4375/4374]] to induce very little damage on schismic harmonies, at the cost of 7 being quite complex.
+* [[Sesquiquartififths]], which tempers out [[2401/2400]] and splits the fifth in fourths, inducing very little damage with a less complex mapping of 7 at the cost of quadrupling the complexity of 3 and 5.
+* The 2.3.5.13 [[subgroup]] extension [[Schismatic family#Maqamschismic (2.3.5.13)|maqamschismic]], (tempering out the [[325/324|marveltwin comma]]) and finds [[13/8]] at the dupminor sixth (^^Ab from C). See [[2.3.5.13 subgroup]] for more details.
+* The 2.3.5.19 [[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]).
+This page, however, focuses on the basic 5-limit temperament.
 See [[Schismatic family #Schismic, schismatic, a.k.a. helmholtz]] for technical data.
@@ Line 81: / Line 93: @@
 | 20.77
 | 81/80
+|-
+|13
+|722.49
+|243/160
+|-
+|14
+|224.22
+|256/225
 |}
 <nowiki/>* In 5-limit CWE tuning
 == Notation ==
-Using schismic can be a challenge because it defies the tradition of {{w|tertian harmony}} in [[chain-of-fifths notation]]. The just major triad on C is C–F♭–G, for example. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.
+Using schismic can be a challenge because it defies the tradition of diatonic {{w|tertian harmony}} in [[chain-of-fifths notation]]; The just major third is not C–E–G like in [[meantone]], but rather, C–F♭–G. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.
 == Scales ==
+=== MOS scales ===
+* [[5L 7s]] (p-chromatic) [17edo]
+* [[12L 5s]] (p-enharmonic) [29edo]
+* [[12L 17s]] (pythagotonic) [41edo]
+* [[12L 29s]] (pythamystonic) [53edo]
+* [[12L 41s]] (antipythomerc) [65edo]
+* [[53L 12s]] (m-chro antipythomerc) [118edo]
+EDO in brackets represents basic step ratio.
 === Scala files ===
 * [[Clipper32805]] – in a 1–3–5 equal-beating tuning