Schismic: Difference between revisions

Line 15:

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

}}

'''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~~]].

'''Schismic''', '''schismatic''', or '''helmholtz''' is a [[5-limit]] [[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]].

[[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.

[[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 diatonic major thirds and one downmajor third representing 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.

Schismic is one of the simplest [[microtemperament]]s, 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 [[41edo|41-]] or [[94edo]]) still work fine.

Extensions include:

Extensions of schismic include [[garibaldi]] and [[pontiac]]. Garibaldi 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, though with worse 5-limit intonation since the tuning favors 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. Besides these, there is 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]]).

* [[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.

A notable example of a [[weak extension]] is [[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.

* [[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.

Line 93:

Line 89:

| 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 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.

Using schismic can be a challenge because it defies the tradition of diatonic {{w|tertian harmony}} in [[chain-of-fifths notation]]; The just major triad on C 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 ==

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

~~=== MOS scales ===~~

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

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

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

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

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

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

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

* [[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 ===

@@ Line 15: / Line 15: @@
 | Odd limit 2 = (5-limit) 125 | Mistuning 2 = 0.837 | Complexity 2 = 29
 }}
-'''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]].
+'''Schismic''', '''schismatic''', or '''helmholtz''' is a [[5-limit]] [[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]].
-[[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.
+[[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 diatonic major thirds and one downmajor third representing 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.
+Schismic is one of the simplest [[microtemperament]]s, 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 [[41edo|41-]] or [[94edo]]) still work fine.
-Extensions include:
+Extensions of schismic include [[garibaldi]] and [[pontiac]]. Garibaldi 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, though with worse 5-limit intonation since the tuning favors 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. Besides these, there is 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]]).
-* [[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.
+A notable example of a [[weak extension]] is [[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.
-* [[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.
@@ Line 93: / Line 89: @@
 | 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 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.
+Using schismic can be a challenge because it defies the tradition of diatonic {{w|tertian harmony}} in [[chain-of-fifths notation]]; The just major triad on C 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 ==
+* [[5L 7s]] (p-chromatic)
-=== MOS scales ===
+* [[12L 5s]] (p-enharmonic)
+* [[12L 17s]] (pythagotonic)
-* [[5L 7s]] (p-chromatic) [17edo]
+* [[12L 29s]] (pythamystonic)
+* [[12L 41s]] (antipythomerc)
-* [[12L 5s]] (p-enharmonic) [29edo]
+* [[53L 12s]] (m-chro antipythomerc)
-* [[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 ===