Schismic: Difference between revisions

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| 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
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}}
'''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''' (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]].  


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 major thirds and one ~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.
 
Extensions include: 
 
* [[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.
* [[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.


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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| 20.77
| 20.77
| 81/80
| 81/80
|-
|13
|722.49
|243/160
|-
|14
|224.22
|256/225
|}
|}
<nowiki/>* In 5-limit CWE tuning
<nowiki/>* In 5-limit CWE tuning


== Notation ==
== 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 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.  


== Scales ==
== Scales ==
=== MOS scales ===
* [[5L 7s]] (p-chromatic) [17edo]
* [[12L 5s]] (p-enharmonic) [29edo]
* [[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 ===
=== Scala files ===
* [[Clipper32805]] – in a 1–3–5 equal-beating tuning
* [[Clipper32805]] – in a 1–3–5 equal-beating tuning

Revision as of 12:26, 22 January 2026

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

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). Schismic, schismatic, or helmholtz (specifically in the 5-limit) is atemperament 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. Put alternatively: 8/5 maps to the tetratone.

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 major thirds and one ~5/4.

It 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 include:

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
13 722.49 243/160
14 224.22 256/225

* 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 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.

Scales

MOS scales

  • 5L 7s (p-chromatic) [17edo]

EDO in brackets represents basic step ratio.

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
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
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

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