Syntonic–kleismic equivalence continuum: Difference between revisions
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== Graywood == | == Graywood == | ||
Named by [[CompactStar]] in 2024, graywood tempers out the [[19-comma]], corresponding to {{nowrap| ''n'' {{=}} 0 }}. | Named by [[CompactStar]] in 2024, graywood tempers out the [[19-comma]], corresponding to {{nowrap| ''n'' {{=}} 0 }}. It takes [[19edo]]'s closed [[circle of fifths]], but adds an independent generator for [[prime interval|prime]] [[5/1|5]]. 19 is the only equal temperament that makes it to the optimal ET sequence as all the small edo tunings, e.g. [[38edo|38c-edo]] or [[57edo|57c-edo]], are not nearly as accurate as 19 itself. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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: error map: {{val| 0.000 -7.218 -3.525 }} | : error map: {{val| 0.000 -7.218 -3.525 }} | ||
{{Optimal ET sequence|legend=1| 19 }} | |||
[[Badness]] (Sintel): 32.4 | [[Badness]] (Sintel): 32.4 | ||
Revision as of 15:00, 24 March 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The syntonic–kleismic equivalence continuum (or syntonic–enneadecal equivalence continuum) is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the 19-comma ([-30 19⟩).
All temperaments in the continuum satisfy (81/80)n ~ [-30 19⟩. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 19edo (due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them). The just value of n is approximately 6.376…, and temperaments having n near this value tend to be the most accurate ones.
This continuum can also be expressed as the relationship between 81/80 and the enneadeca ([-14 -19 19⟩). That is, (81/80)k ~ [-14 -19 19⟩. In this case, k = 3n − 19.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Graywood | 1162261467/1073741824 | [-30 19⟩ |
| 1 | 7c & 12c | 71744535/67108864 | [-26 15 1⟩ |
| 2 | Hogzilla | 4428675/4194304 | [-22 11 2⟩ |
| 3 | Stump | 273375/262144 | [-18 7 3⟩ |
| 4 | Negri | 16875/16384 | [-14 3 4⟩ |
| 5 | Magic | 3125/3072 | [-10 -1 5⟩ |
| 6 | Hanson | 15625/15552 | [-6 -5 6⟩ |
| 7 | Sensipent | 78732/78125 | [2 9 -7⟩ |
| 8 | Unicorn | 1594323/1562500 | [-2 13 -8⟩ |
| 9 | 19 & 51c | 129140163/125000000 | [-6 17 -9⟩ |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of k:
| Temperament | n | Comma |
|---|---|---|
| Unsmate | 9/2 = 4.5 | [-24 2 9⟩ |
| Sycamore | 11/2 = 5.5 | [-16 -6 11⟩ |
| Counterhanson | 25/4 = 6.25 | [-20 -24 25⟩ |
| Enneadecal | 19/3 = 6.3 | [-14 -19 19⟩ |
| Egads | 51/8 = 6.375 | [-36 -52 51⟩ |
| Acrokleismic | 32/5 = 6.4 | [22 33 -32⟩ |
| Parakleismic | 13/2 = 6.5 | [8 14 -13⟩ |
| Countermeantone | 20/3 = 6.6 | [10 23 -20⟩ |
| Mowgli | 15/2 = 7.5 | [0 22 -15⟩ |
Graywood
Named by CompactStar in 2024, graywood tempers out the 19-comma, corresponding to n = 0. It takes 19edo's closed circle of fifths, but adds an independent generator for prime 5. 19 is the only equal temperament that makes it to the optimal ET sequence as all the small edo tunings, e.g. 38c-edo or 57c-edo, are not nearly as accurate as 19 itself.
Subgroup: 2.3.5
Comma list: 1162261467/1073741824
Mapping: [⟨19 30 0], ⟨0 0 1]]
- mapping generators: ~2187/2048, ~5
- WE: ~2187/2048 = 63.2773 ¢, ~5/4 = 381.7568 ¢
- error map: ⟨+2.268 -3.637 -0.020]
- CWE: ~2187/2048 = 63.1579 ¢, ~5/4 = 382.7889 ¢
- error map: ⟨0.000 -7.218 -3.525]
Badness (Sintel): 32.4
Hogzilla
- For extensions, see Semaphoresmic clan #Helayo.
Hogzilla is similar to godzilla in that it is generated by a semitwelfth. It corresponds to n = 2.
Subgroup: 2.3.5
Comma list: 4428675/4194304
Mapping: [⟨1 0 11], ⟨0 2 -11]]
- mapping generators: ~2, ~2048/1215
- WE: ~2 = 1202.5490 ¢, ~2048/1215 = 949.3637 ¢
- error map: ⟨+2.549 -3.228 -1.275]
- CWE: ~2 = 1200.0000 ¢, ~2048/1215 = 947.2462 ¢
- error map: ⟨0.000 -7.463 -6.022]
Badness (Sintel): 9.96
Stump
- For extensions, see Marvel temperaments #Triton and Sensamagic clan #Pycnic.
Stump splits the 3rd harmonic into three equal parts, each for ~64/45. It corresponds to n = 3.
Subgroup: 2.3.5
Comma list: 273375/262144
Mapping: [⟨1 0 6], ⟨0 3 -7]]
- WE: ~2 = 1202.6288 ¢, ~64/45 = 633.1214 ¢
- error map: ⟨+2.629 -2.591 -2.391]
- CWE: ~2 = 1200.0000 ¢, ~64/45 = 631.6779 ¢
- error map: ⟨0.000 -6.921 -8.059]
Optimal ET sequence: 17, 19, 207bbccc
Badness (Sintel): 4.71
Negri (5-limit)
- For extensions, see Semaphoresmic clan #Negri.
The 5-limit version of negri tempers out the negri comma, spliting a perfect fourth into four ~16/15 generators. It corresponds to n = 4. The only 7-limit extension that make any sense to use is to map the hemifourth to 7/6~8/7.
Subgroup: 2.3.5
Comma list: 16875/16384
Mapping: [⟨1 2 2], ⟨0 -4 3]]
- mapping generators: ~2, ~16/15
- WE: ~2 = 1202.3403 ¢, ~16/15 = 126.0002 ¢
- error map: ⟨+2.340 -1.275 -3.633]
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 125.6610 ¢
- error map: ⟨0.000 -4.599 -9.331]
Optimal ET sequence: 9, 10, 19, 67c, 86c, 105c
Badness (Sintel): 2.04
Lalasepyo (8c & 11)
Subgroup: 2.3.5
Comma list: 4613203125/4294967296
Mapping: [⟨1 -1 6], ⟨0 7 -10]]
- mapping generators: ~2, ~675/512
- WE: ~2 = 1202.5641 ¢, ~675/512 = 443.2124 ¢
- error map: ⟨+2.564 -2.033 -3.053]
- CWE: ~2 = 1200.0000 ¢, ~675/512 = 442.2692 ¢
- error map: ⟨0.000 -6.071 -9.006]
Optimal ET sequence: 8c, 11, 19
Badness (Sintel): 24.9
The temperament finder - 5-limit 19 & 8c
Unsmate
Subgroup: 2.3.5
Comma list: 17578125/16777216
Mapping: [⟨1 -6 4], ⟨0 9 -2]]
- mapping generators: ~2, ~1875/1024
- WE: ~2 = 1201.8892 ¢, ~1875/1024 = 1012.5428 ¢
- error map: ⟨+1.889 -0.405 -3.843]
- CWE: ~2 = 1200.0000 ¢, ~1875/1024 = 1011.0348 ¢
- error map: ⟨0.000 -2.642 -8.383]
Optimal ET sequence: 6b, 13, 19, 89c, 108c, 127c, 146cc
Badness (Sintel): 10.8
Parakleismic (5-limit)
- For extensions, see Ragismic microtemperaments #Parakleismic.
The 5-limit version of parakleismic tempers out the parakleisma. It corresponds to n = 13/2, and 13 generator steps give the interval class of 3.
Subgroup: 2.3.5
Comma list: 1224440064/1220703125
Mapping: [⟨1 -8 -8], ⟨0 13 14]]
- mapping generators: ~2, ~5/3
- WE: ~2 = 1199.971 ¢, ~5/3 = 884.7383 ¢
- error map: ⟨-0.029 -0.127 +0.253]
- CWE: ~2 = 1200.000 ¢, ~5/3 = 884.8576 ¢
- error map: ⟨0.000 -0.106 +0.293]
Optimal ET sequence: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Badness (Sintel): 1.02
Mowgli
- For extensions, see Hemimean clan #Mowglic.
TE, CTE and POTE coincide at 126.7237 ¢ with pure octaves since prime 2 is not involved in the comma to begin with.
Subgroup: 2.3.5
Comma list: 31381059609/30517578125
Mapping: [⟨1 0 0], ⟨0 15 22]]
- mapping generators: ~2, ~27/25
- WE: ~2 = 1199.9478 ¢, ~27/25 = 126.7236 ¢
- error map: ⟨-0.001 -1.100 +1.606]
- CWE: ~2 = 1200.0000 ¢, ~27/25 = 126.7237 ¢
- error map: ⟨0.000 -1.100 +1.607]
Optimal ET sequence: 19, 85c, 104c, 123, 142, 161, 303
Badness (Sintel): 15.3
Enneadecal (5-limit)
- For extensions, see Ragismic microtemperaments #Enneadecal.
The 5-limit version of enneadecal tempers out the enneadeca, which simply equates a stack of nineteen 6/5 minor thirds with five octaves. It corresponds to n = 19/3, with a 19th-octave period and a generator of a perfect fifth.
Subgroup: 2.3.5
Comma list: 19073486328125/19042491875328
Mapping: [⟨19 0 14], ⟨0 1 1]]
- mapping generators: ~648/625, ~3
- WE: ~648/625 = 63.1579 ¢, ~3/2 = 701.9861 ¢
- error map: ⟨+0.013 +0.044 -0.095]
- CWE: ~648/625 = 63.1579 ¢, ~3/2 = 701.9900 ¢
- error map: ⟨0.000 -0.035 -0.113]
Optimal ET sequence: 19, 95, 114, 133, 152, 171, 323, 494, 665, 1159, 1824, 2983, 7125c
Badness (Sintel): 1.12
Countermeantone
Subgroup: 2.3.5
Comma list: [10 23 -20⟩
Mapping: [⟨1 -10 -11], ⟨0 20 23]]
- mapping generators: ~2, ~78125/52488
- WE: ~2 = 1199.9478 ¢, ~78125/52488 = 695.0566 ¢
- error map: ⟨-0.052 -0.301 +0.562]
- CWE: ~2 = 1200.0000 ¢, ~78125/52488 = 695.0846 ¢
- error map: ⟨0.000 -0.264 +0.631]
Optimal ET sequence: 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c
Badness (Sintel): 8.76
Counterhanson
- For extensions, see Ragismic microtemperaments #Counterkleismic.
Subgroup: 2.3.5
Comma list: [-20 -24 25⟩
Mapping: [⟨1 -5 -4], ⟨0 25 24]]
- mapping generators: ~2, ~6/5
- WE: ~2 = 1200.0419 ¢, ~6/5 = 316.0916 ¢
- error map: ⟨+0.042 +0.126 -0.282]
- CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0021 ¢
- error map: ⟨0.000 +0.097 -0.344]
Optimal ET sequence: 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c
Badness (Sintel): 7.45
Oviminor
Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.
Subgroup: 2.3.5
Comma list: [-134 -185 184⟩
Mapping: [⟨1 -134 -134], ⟨0 184 185]]
- mapping generators: ~2, ~5/3
- WE: ~2 = 1200.0094 ¢, ~5/3 = 884.2568 ¢
- error map: ⟨+0.009 +0.033 -0.069]
- CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.2499 ¢
- error map: ⟨0.000 +0.026 -0.083]
Optimal ET sequence: 19, …, 1600, 3219, 4819
Badness (Sintel): 751