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#REDIRECT [[Schismic and garibaldi]]
{{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]].
 
[[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.
 
== Interval chain ==
In the following table, odd harmonics 1–9 and their inverses are in '''bold'''.
 
{| class="wikitable center-1 right-2"
|-
! #
! 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
|}
<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="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~3/2 = 701.7379{{c}}
| 5/3
| ~3/2 = 701.728{{c}}
| {{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 ==
* [https://x31eq.com/schismic.htm ''Schismic Temperaments''] by [[Graham Breed]]
 
[[Category:Schismic| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Microtemperaments]]
[[Category:Schismatic family]]
Retrieved from "https://en.xen.wiki/w/Schismic"