Superpyth
Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). Superpyth, sometimes called archy in the no-5 subgroup, is a temperament where the generator is a perfect fifth, tuned sharp such that a stack of two perfect fifths octave-reduced gives a whole tone that represents both 9/8 and 8/7, tempering out the septimal comma, 64/63. Likewise, two perfect fourths give a minor seventh that represents both 7/4 and 16/9, so that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, three fourths reach a minor third that approximates 7/6, while four fifths reach a major third that approximates 9/7.
Since the generator is a perfect fifth, superpyth can be notated using the same standard chain-of-fifths notation that is also used for meantone, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in Pythagorean tuning, in contrast to meantone where sharps are flatter than or equal to the corresponding flats. 13\22 (~1/4 septimal comma) and 16\27 (~1/3 septimal comma) are the most common tunings of the generator.
If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out 245/243, so 5/4 is an augmented second (e.g. C–D♯, a limma-flat major third). Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including 12edo) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.
Alternatively, for a sharper tuning, the 5th harmonic can be mapped to +14 generators, resulting in ultrapyth.
If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second (C–D𝄪), tempering out 100/99. A simpler way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out 99/98. The latter is called supra, or suprapyth. The two mappings unite on 22edo.
If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out 31213/31104.
Mos scales of superpyth have cardinalities of 5, 7, 12, 17, or 22.
For more technical data, see Archytas clan #Superpyth.
Interval chains
In these tables, odd harmonics 1–11 are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 709.4 | 3/2 |
| 2 | 218.8 | 8/7, 9/8 |
| 3 | 928.2 | 12/7 |
| 4 | 437.6 | 9/7 |
| 5 | 1147.0 | 27/14 |
| 6 | 656.3 | 72/49, 81/56 |
| 7 | 165.7 | 54/49 |
* In 2.3.7-subgroup CWE tuning, octave reduced
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 707.5 | 3/2 |
| 2 | 215.0 | 8/7, 9/8 |
| 3 | 922.5 | 12/7 |
| 4 | 430.0 | 9/7, 14/11 |
| 5 | 1137.5 | 21/11, 27/14, 64/33 |
| 6 | 645.0 | 16/11 |
| 7 | 152.5 | 12/11 |
* In 2.3.7.11-subgroup CWE tuning
| # | Cents* | Approximate ratios | ||
|---|---|---|---|---|
| 7-limit | 11-limit extension | |||
| Superpyth | Suprapyth | |||
| 0 | 0.0 | 1/1 | ||
| 1 | 710.1 | 3/2 | ||
| 2 | 220.2 | 8/7, 9/8 | ||
| 3 | 930.4 | 12/7 | ||
| 4 | 440.5 | 9/7 | 14/11 | |
| 5 | 1150.6 | 27/14, 35/18 | 88/45 | 21/11, 64/33 |
| 6 | 660.7 | 35/24, 40/27 | 22/15 | 16/11 |
| 7 | 170.8 | 10/9 | 11/10 | 12/11 |
| 8 | 881.0 | 5/3 | 33/20 | 18/11 |
| 9 | 391.1 | 5/4 | 27/22 | |
| 10 | 1101.2 | 15/8, 40/21 | ||
| 11 | 611.3 | 10/7 | ||
| 12 | 121.4 | 15/14 | ||
| 13 | 831.6 | 45/28 | 44/27 | |
| 14 | 341.7 | 60/49 | 11/9 | 40/33 |
| 15 | 1051.8 | 50/27 | 11/6 | 20/11 |
| 16 | 561.9 | 25/18 | 11/8 | 15/11 |
| 17 | 72.0 | 25/24 | 22/21, 33/32 | 45/44 |
* In 7-limit CWE tuning
Scales
- 5-note mos (2L 3s, proper)
- Archy5 – archy in 472edo tuning
- 7-note mos (5L 2s, improper)
In contrast to the meantone diatonic scale, the superpyth diatonic is improper.
- 12-note mos (5L 7s, borderline improper)
The boundary of propriety is 17edo.
Tunings
The fifth of superpyth is supposed to be tuned sharp of just. Roughly speaking, it ranges from as flat as Pythagorean (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, close to 57b-edo), with 22edo and 27edo being typical endpoints of superpyth's optimal range.
22edo can be viewed as a closed form of 1/4-comma superpyth, where the whole tone is midway between 8/7 and 9/8, so that the 7 is as sharp as the 9 and that the 9/7 major third is tuned just. 27edo can be viewed as a closed form of 1/3-comma superpyth, where the whole tone leans towards 8/7 a bit, so that the 7 is as sharp as the 3 and that the 7/6 minor third is tuned just.
27edo is also the point where 5/4 is tuned to the familiar 400 cents of 12edo, and in sharper tunings, there are different mappings of 5/4 with more accuracy (see quasiultra and ultrapyth). The same goes for flatter tunings than 22edo (see quasisuper and dominant). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.
The plastic number has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes a variety of superpyth. This can be explained since superpyth equates 21/16 and 4/3, making the 9:12:16:21 chord evenly spaced by ~4/3, and when keeping ~9 + ~12 = ~21 the generator becomes the plastic number.
Prime-optimized tunings
| Euclidean | ||
|---|---|---|
| Unskewed | Skewed | |
| Equilateral | CEE: ~3/2 = 712.8606¢ (2/5-comma) |
CSEE: ~3/2 = 711.9997¢ (7/19-comma) |
| Tenney | CTE: ~3/2 = 709.5948¢ | CWE: ~3/2 = 709.3901¢ |
| Benedetti, Wilson |
CBE: ~3/2 = 707.7286¢ (18/85-comma) |
CSBE: ~3/2 = 707.9869¢ (25/113-comma) |
| Euclidean | ||
|---|---|---|
| Unskewed | Skewed | |
| Equilateral | CEE: ~3/2 = 709.7805¢ | CSEE: ~3/2 = 710.2428¢ |
| Tenney | CTE: ~3/2 = 709.5907¢ | CWE: ~3/2 = 710.1193¢ |
| Benedetti, Wilson |
CBE: ~3/2 = 709.4859¢ | CSBE: ~3/2 = 710.0321¢ |
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 3/2 | 701.955 | Pythagorean tuning | |
| 10\17 | 705.882 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 81/56 | 706.499 | 1/6 comma | |
| 27/14 | 707.408 | 1/5 comma | |
| 23\39 | 707.692 | 39cd val | |
| 9/7 | 708.771 | 1/4 comma, {1, 3, 7, 9} minimax | |
| 15/8 | 708.807 | ||
| 13\22 | 709.091 | ||
| 5/4 | 709.590 | 9-odd-limit minimax | |
| 49/27 | 709.745 | 2/7 comma | |
| 42\71 | 709.859 | 71d val | |
| 15/14 | 709.954 | ||
| 25/24 | 710.040 | ||
| 29\49 | 710.204 | ||
| 45\76 | 710.526 | 76bcd val | |
| 5/3 | 710.545 | ||
| 7/5 | 710.681 | 7-odd-limit minimax | |
| 7/6 | 711.043 | 1/3 comma, {1, 3, 7} minimax | |
| 16\27 | 711.111 | ||
| 21/20 | 711.553 | ||
| 9/5 | 711.772 | ||
| 19\32 | 712.500 | 32c val | |
| 55/32 | 712.544 | Suprapyth mapping | |
| 49/48 | 712.861 | 2/5 comma, 2.3.7 subgroup CEE tuning | |
| 22\37 | 713.514 | 37cc val | |
| 25\42 | 714.286 | 42cc val | |
| 7/4 | 715.587 | 1/2 comma | |
| 3\5 | 720.000 | Upper bound of 7- and 9-odd-limit diamond monotone | |
| 21/16 | 729.219 | Full comma |
* Besides the octave
Other tunings
- DKW (2.3.5 superpyth): ~2 = 1200.000, ~3/2 = 709.758
- DKW (2.3.7 archy): ~2 = 1200.000, ~3/2 = 712.585
Music
- Superpyth[12] chromatic riff (2015)
- Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello (2015)
- Both in 22edo tuning
- All in Superpyth[12], 22edo tuning.