Schismic
- This page is about the regular temperament sometimes known as "helmholtz". For the music theorist, see Hermann von Helmholtz.
| Schismic |
5-limit 125-odd-limit: 0.837 ¢
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
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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. |
- 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, 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