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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
{{Infobox regtemp
| Title = Schismic
| Title = Schismic
| Subgroups = 2.3.5
| Subgroups = 2.3.5
| Comma basis = 32805/32768
| Comma basis = [[32805/32768]]
| Generator = 3/2
| Edo join 1 = 12 | Edo join 2 = 53
| Mapping = 1; 1 -8
| Mapping = 1; 1 -8
| Pergen = (P8, P5)
| Generators = 3/2
| Color name = Layo
| Generators tuning = 701.731
| Edo join 1 = 12 | Edo join 2 = 41
| Optimization method = CWE
| Optimization method = CWE
| Generator tuning = 701.731
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[12L 5s]]
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[12L 5s]]
| Odd limit 1 = 5 | Mistuning 1 = 0.217 | Complexity 1 = 29
| Pergen = (P8, P5)
| Odd limit 2 = (5-limit) 75 | Mistuning 2 = 0.651 | Complexity 2 = 41
| 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 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]].  
 
[[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 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.  
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.  
See [[Schismatic family #Schismic, schismatic, a.k.a. helmholtz]] for technical data.  
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|}
|}
<nowiki/>* In 5-limit CWE tuning
<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 ==
== 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 ===
=== 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"
{| 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
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Minimax
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| {{Monzo| 0 -10 17 }}
| {{Monzo| 0 -10 17 }}
|}
|}
=== 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 ==
== External links ==
* [https://x31eq.com/schismic.htm ''Schismic Temperaments''] by [[Graham Breed]]
* [https://x31eq.com/schismic.htm ''Schismic Temperaments''] by [[Graham Breed]]


[[Category:Schismatic| ]] <!-- main article -->
[[Category:Schismic| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Microtemperaments]]
[[Category:Schismatic family]]
[[Category:Schismatic family]]

Latest revision as of 19:19, 29 April 2026

This page is about the regular temperament sometimes known as "helmholtz". For the music theorist, see Hermann von Helmholtz.
Schismic
Subgroups 2.3.5
Comma basis 32805/32768
Reduced mapping ⟨1; 1 -8]
ET join 12 & 53
Generators (CWE) ~3/2 = 701.731 ¢
MOS scales 2L 3s, 5L 2s, 5L 7s, 12L 5s
Ploidacot monocot
Pergen (P8, P5)
Color name Layoti
Minimax error 5-odd-limit: 0.217 ¢;
5-limit 125-odd-limit: 0.837 ¢
Target scale size 5-odd-limit: 12 notes;
5-limit 125-odd-limit: 29 notes

Schismic, schismatic, or helmholtz is a 5-limit temperament which takes an almost just 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 commas 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 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). 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 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.

Interval chain

In the following table, odd harmonics 1–9 and their inverses are in bold.

# Cents* Approximate ratios
0 0.00 1/1
1 701.73 3/2
2 203.46 9/8
3 905.19 27/16
4 406.92 81/64
5 1108.65 243/128, 256/135
6 610.38 64/45
7 112.12 16/15
8 813.85 8/5
9 315.58 6/5
10 1017.31 9/5
11 519.04 27/20
12 20.77 81/80

* In 5-limit CWE tuning

Notation

Using schismic can be a challenge because it defies the tradition of diatonic 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

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

Terms: The later mos names are proposals that can be found on the page TAMNAMS Extension.

Scala files

Tunings

Norm-based tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 701.7187 ¢ CWE: ~3/2 = 701.7308 ¢ POTE: ~3/2 = 701.7359 ¢

Target tunings

Delta-rational tunings
Optimized chord Generator value Polynomial Further notes
3:4:5 (+1 +1) ~3/2 = 701.6910 ¢ g9 - 4g8 + 64 = 0 1–3–5 equal-beating tuning
4:5:6 (+1 +1) ~3/2 = 701.7278 ¢ g9 + g8 - 64 = 0 1–3–5 equal-beating tuning
Odd-limit-based target tunings
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
5-odd-limit ~3/2 = 701.7379 ¢ 5/3 ~3/2 = 701.728 ¢ [0 -10 17⟩

Tuning spectrum

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

* Besides the octave

External links

Retrieved from "https://en.xen.wiki/index.php?title=Schismic&oldid=228983"