Schismic: Difference between revisions

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~~: ''This page is about a~~ regular temperament sometimes known as "helmholtz"~~. For~~ the music theorist, ~~see~~ [[~~Hermann von Helmholtz~~]].''

{{About|the regular temperament sometimes known as "helmholtz"|the music theorist|Hermann von Helmholtz}}

{{Infobox regtemp

| Title = Schismic

| Subgroups = 2.3.5

| Comma basis = [[32805/32768]]

| Edo join 1 = 12 | Edo join 2 = 53

| Mapping = 1; 1 -8

| Generators = 3/2

| Generators tuning = 701.731

| Optimization method = CWE

| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[12L 5s]]

| Pergen = (P8, P5)

| Color name = Layoti

| Odd limit 1 = 5 | Mistuning 1 = 0.217 | Complexity 1 = 12

| 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 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]].

~~'''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.

[[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.

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.

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 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]]).

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.

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 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 ==

{{Idiosyncratic terms|The later mos names are proposals that can be found on the page [[TAMNAMS Extension]].}}

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

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

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

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

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

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

=== Scala files ===

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

== Tunings ==

=== Norm-based tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Tenney

| CTE: ~3/2 = 701.7187{{c}}

| CWE: ~3/2 = 701.7308{{c}}

| POTE: ~3/2 = 701.7359{{c}}

|}

=== Target tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | Delta-rational tunings

|-

! Optimized chord !! Generator value !! Polynomial !! Further notes

|-

| 3:4:5 (+1 +1) || ~3/2 = 701.6910{{c}} || ''g''9 - 4''g''8 + 64 = 0 || 1–3–5 equal-beating tuning

|-

| 4:5:6 (+1 +1) || ~3/2 = 701.7278{{c}} || ''g''9 + ''g''8 - 64 = 0 || 1–3–5 equal-beating tuning

|}

{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"

|+ style="white-space: nowrap;" | ~~Target~~ tunings

|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings

! rowspan="2" | Target

! colspan="2" | Minimax

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|}

[[Category:~~Schismatic~~| ]]

=== Tuning spectrum ===

{| class="wikitable center-all left-4"

|-

! Edo generator !! Eigenmonzo (Unchanged-interval)* !! Generator (¢) !! Comments

|-

| 7\12 || || 700.0000 || Lower bound of 5-limit 9-odd-limit diamond monotone

|-

| 52\89 || || 701.1236 ||

|-

| 45\77 || || 701.2987 ||

|-

| 38\65 || || 701.5385 ||

|-

| || 45/32 || 701.6294 || 1/6-comma

|-

| || 15/8 || 701.6759 || 1/7-comma

|-

| 69\118 || || 701.6949 ||

|-

| || 5/4 || 701.7108 || 1/8-comma, lower bound of 5-odd-limit diamond tradeoff

|-

| || 25/24 || 701.7252 || 2/17-comma

|-

| 169\289 || || 701.7301 ||

|-

| || 5/3 || 701.7379 || 1/9-comma, 5-odd-limit minimax

|-

| 100\171 || || 701.7544 ||

|-

| || 9/5 || 701.7596 || 1/10-comma

|-

| || 81/80 || 701.7922 || 1/12-comma

|-

| 31\53 || || 701.8868 ||

|-

| || 3/2 || 701.9550 || Pythagorean tuning, upper bound of 5-odd-limit diamond tradeoff

|-

| 24\41 || || 702.4390 ||

|-

| 17\29 || || 703.4483 ||

|-

| 10\17 || || 705.8824 || Upper bound of 5-limit 9-odd-limit diamond monotone

|}

<nowiki/>* Besides the octave

== External links ==

* [https://x31eq.com/schismic.htm ''Schismic Temperaments''] by [[Graham Breed]]

[[Category:Schismic| ]]

[[Category:Rank-2 temperaments]]

[[Category:Microtemperaments]]

[[Category:Schismatic family]]

@@ Line 1: / Line 1: @@
-: ''This page is about a regular temperament sometimes known as "helmholtz". For the music theorist, see [[Hermann von Helmholtz]].''
+{{About|the regular temperament sometimes known as "helmholtz"|the music theorist|Hermann von Helmholtz}}
+{{Infobox regtemp
+| Title = Schismic
+| Subgroups = 2.3.5
+| Comma basis = [[32805/32768]]
+| Edo join 1 = 12 | Edo join 2 = 53
+| Mapping = 1; 1 -8
+| Generators = 3/2
+| Generators tuning = 701.731
+| Optimization method = CWE
+| MOS scales = [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[12L&nbsp;5s]]
+| Pergen = (P8, P5)
+| Color name = Layoti
+| Odd limit 1 = 5 | Mistuning 1 = 0.217 | Complexity 1 = 12
+| 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 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]].
-'''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.
+[[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.
-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.
+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 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]]).
+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.
+This page, however, focuses on the basic 5-limit temperament.
 See [[Schismatic family #Schismic, schismatic, a.k.a. helmholtz]] for technical data.
@@ Line 69: / Line 91: @@
 |}
 <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 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 ==
+{{Idiosyncratic terms|The later mos names are proposals that can be found on the page [[TAMNAMS Extension]].}}
+* [[5L 7s]] (p-chromatic)
+* [[12L 5s]] (p-enharmonic)
+* [[12L 17s]] (pythagotonic)
+* [[12L 29s]] (pythamystonic)
+* [[12L 41s]] (antipythomerc)
+* [[53L 12s]] (m-chro antipythomerc)
+=== Scala files ===
+* [[Clipper32805]] – in a 1–3–5 equal-beating tuning
 == Tunings ==
+=== Norm-based tunings ===
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Tenney
+| CTE: ~3/2 = 701.7187{{c}}
+| CWE: ~3/2 = 701.7308{{c}}
+| POTE: ~3/2 = 701.7359{{c}}
+|}
 === Target tunings ===
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | Delta-rational tunings
+|-
+! Optimized chord !! Generator value !! Polynomial !! Further notes
+|-
+| 3:4:5 (+1 +1) || ~3/2 = 701.6910{{c}} || ''g''<sup>9</sup> - 4''g''<sup>8</sup> + 64 = 0 || 1–3–5 equal-beating tuning
+|-
+| 4:5:6 (+1 +1) || ~3/2 = 701.7278{{c}} || ''g''<sup>9</sup> + ''g''<sup>8</sup> - 64 = 0 || 1–3–5 equal-beating tuning
+|}
 {| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
-|+ style="white-space: nowrap;" | Target tunings
+|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
 ! rowspan="2" | Target
 ! colspan="2" | Minimax
@@ Line 90: / Line 154: @@
 |}
-[[Category:Schismatic| ]] <!-- main article -->
+=== Tuning spectrum ===
+{| class="wikitable center-all left-4"
+|-
+! Edo<br>generator !! Eigenmonzo<br>(Unchanged-interval)* !! Generator (¢) !! Comments
+|-
+| 7\12 ||  || 700.0000 || Lower bound of 5-limit 9-odd-limit diamond monotone
+|-
+| 52\89 ||  || 701.1236 ||
+|-
+| 45\77 ||  || 701.2987 ||
+|-
+| 38\65 ||  || 701.5385 ||
+|-
+|  || 45/32 || 701.6294 || 1/6-comma
+|-
+|  || 15/8 || 701.6759 || 1/7-comma
+|-
+| 69\118 ||  || 701.6949 ||
+|-
+|  || 5/4 || 701.7108 || 1/8-comma, lower bound of 5-odd-limit diamond tradeoff
+|-
+|  || 25/24 || 701.7252 || 2/17-comma
+|-
+| 169\289 ||  || 701.7301 ||
+|-
+|  || 5/3 || 701.7379 || 1/9-comma, 5-odd-limit minimax
+|-
+| 100\171 ||  || 701.7544 ||
+|-
+|  || 9/5 || 701.7596 || 1/10-comma
+|-
+|  || 81/80 || 701.7922 || 1/12-comma
+|-
+| 31\53 ||  || 701.8868 ||
+|-
+|  || 3/2 || 701.9550 || Pythagorean tuning, upper bound of 5-odd-limit diamond tradeoff
+|-
+| 24\41 ||  || 702.4390 ||
+|-
+| 17\29 ||  || 703.4483 ||
+|-
+| 10\17 ||  || 705.8824 || Upper bound of 5-limit 9-odd-limit diamond monotone
+|}
+<nowiki/>* Besides the octave
+== External links ==
+* [https://x31eq.com/schismic.htm ''Schismic Temperaments''] by [[Graham Breed]]
+[[Category:Schismic| ]] <!-- main article -->
 [[Category:Rank-2 temperaments]]
+[[Category:Microtemperaments]]
 [[Category:Schismatic family]]