Schismic: Difference between revisions
Added and reworded to make more succint |
Restore some deletions. Distinguish strong and weak extensions. Don't add "maqamschismic" until it's proven notable. - mysterious extra steps in the interval chain |
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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''' | '''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 major thirds and one | [[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 include | 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. | This page, however, focuses on the basic 5-limit temperament. | ||
| Line 93: | Line 89: | ||
| 20.77 | | 20.77 | ||
| 81/80 | | 81/80 | ||
|} | |} | ||
<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 diatonic {{w|tertian harmony}} in [[chain-of-fifths notation]]; The just major | 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 == | == Scales == | ||
* [[5L 7s]] (p-chromatic) | |||
* [[12L 5s]] (p-enharmonic) | |||
* [[12L 17s]] (pythagotonic) | |||
* [[5L 7s]] (p-chromatic) | * [[12L 29s]] (pythamystonic) | ||
* [[12L 41s]] (antipythomerc) | |||
* [[12L 5s]] (p-enharmonic) | * [[53L 12s]] (m-chro antipythomerc) | ||
* [[12L 17s]] (pythagotonic) | |||
* [[12L 29s]] (pythamystonic) | |||
* [[12L 41s]] (antipythomerc) | |||
* [[53L 12s]] (m-chro antipythomerc) | |||
=== Scala files === | === Scala files === | ||
Revision as of 13:03, 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 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
- 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
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 701.7187 ¢ | CWE: ~3/2 = 701.7308 ¢ | POTE: ~3/2 = 701.7359 ¢ |
Target 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 |
| 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