Schismic: Difference between revisions

(8 intermediate revisions by 3 users not shown)

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]]

| ~~Generator~~ = 3/2

| 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

~~| Edo join 1 = 12 | Edo join 2 = 41~~

~~| Optimization method = CWE~~

~~| Generator tuning = 701.731~~

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

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

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

| 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''' 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]].

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 diatonic major thirds and one downmajor third representing 5/4.

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 85:

Line 93:

== 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 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

Line 119:

Line 166:

|-

| 38\65 || || 701.5385 ||

|-

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

|-

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

Line 124:

Line 173:

| 69\118 || || 701.6949 ||

|-

| || 5/4 || 701.7108 || 1/8-comma

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

|-

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

Line 135:

Line 184:

|-

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

|-

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

|-

| 31\53 || || 701.8868 ||

|-

| || 3/2 || 701.9550 || Pythagorean tuning

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

|-

| 24\41 || || 702.4390 ||

Line 151:

Line 202:

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

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

[[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]]
-| Generator = 3/2
+| 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
-| Edo join 1 = 12 | Edo join 2 = 41
-| Optimization method = CWE
-| Generator tuning = 701.731
-| MOS scales = [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[12L&nbsp;5s]]
 | Odd limit 1 = 5 | Mistuning 1 = 0.217 | Complexity 1 = 12
-| Odd limit 2 = (5-limit) 125 | Mistuning 2 = 0.837 | Complexity 2 = 29
+| 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''' 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]].
-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 diatonic major thirds and one downmajor third representing 5/4.
+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 85: / Line 93: @@
 == 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 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 119: / Line 166: @@
 |-
 | 38\65 ||  || 701.5385 ||
+|-
+|  || 45/32 || 701.6294 || 1/6-comma
 |-
 |  || 15/8 || 701.6759 || 1/7-comma
@@ Line 124: / Line 173: @@
 | 69\118 ||  || 701.6949 ||
 |-
-|  || 5/4 || 701.7108 || 1/8-comma
+|  || 5/4 || 701.7108 || 1/8-comma, lower bound of 5-odd-limit diamond tradeoff
 |-
 |  || 25/24 || 701.7252 || 2/17-comma
@@ Line 135: / Line 184: @@
 |-
 |  || 9/5 || 701.7596 || 1/10-comma
+|-
+|  || 81/80 || 701.7922 || 1/12-comma
 |-
 | 31\53 ||  || 701.8868 ||
 |-
-|  || 3/2 || 701.9550 || Pythagorean tuning
+|  || 3/2 || 701.9550 || Pythagorean tuning, upper bound of 5-odd-limit diamond tradeoff
 |-
 | 24\41 ||  || 702.4390 ||
@@ Line 151: / Line 202: @@
 * [https://x31eq.com/schismic.htm ''Schismic Temperaments''] by [[Graham Breed]]
-[[Category:Schismatic| ]] <!-- main article -->
+[[Category:Schismic| ]] <!-- main article -->
 [[Category:Rank-2 temperaments]]
+[[Category:Microtemperaments]]
 [[Category:Schismatic family]]