Superpyth
| Archy; superpyth |
64/63, 245/243 (2.3.5.7)
(9-odd limit) 13.63 ¢
(9-odd limit) 27 notes
Superpyth, sometimes called archy in the 2.3.7 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 canonically 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, a name coined by Mike Battaglia in 2011[1]. 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,
octave reduced
| # | Cents* | Approximate ratios | ||
|---|---|---|---|---|
| 7-limit | 11-limit extensions | |||
| 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, octave reduced
Tunings
Tuning considerations and optima
The fifth of superpyth is supposed to be tuned sharp of just for the accuracy of the overall temperament. Roughly speaking, it ranges from as flat as Pythagorean (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, between 52b-edo and 57b-edo), with 22edo and 27edo being typical endpoints of superpyth's optimal range.
Despite superpyth being seen as the "counterpart" of meantone for sharp fifths and septimal thirds, it is actually of considerably higher damage than meantone. The tempered comma, 64/63, is not only larger than 81/80, but it must be split over only three intervals (one minor seventh and two perfect fifths), rather than five as in meantone (one major third and four fifths). This can be shown by the fact that 1/5-comma meantone, the meantone tuning with the minimum damage to harmonics 3 and 5, has a tuning error on 3 and 5 of 4.3 ¢, while 1/3-comma superpyth, the superpyth tuning with the minimum damage to harmonics 3 and 7 (the minimax tuning for the no-5 7-odd-limit tonality diamond) has a tuning error on 3 and 7 of 9.09 ¢, over twice as much. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing others. An additional consideration is the use of tertian triads in conventional diatonic harmony, whereby the 9/7 supermajor third may be more important than it looks from the bare math.
If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of quarter-comma meantone, treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma and 1/4-comma turn out to be logical solutions. In 1/3-comma superpyth, the whole tone leans towards 8/7 so that 3 and 7 are equally sharp and the minor third is tuned to exactly 7/6; 27edo is extremely close to a closed system of 1/3-comma. In 1/4-comma tuning, which is the minimax tuning for the no-5 9-odd-limit, the whole tone is midway between 8/7 and 9/8 so that the 7 is twice as sharp as 3 and that the major third is exactly 9/7; 22edo is very close to a closed circle of 1/4-comma.
In general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of 1/5-comma. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard CTE and CWE metrics. In fact, 22edo is slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; 71edo (709.859 ¢) and 93edo with its sharp fifth of 709.677 ¢ come very close to forming closed systems of 2/7-comma.
27edo is also the point where superpyth tunes 5/4 to the familiar 400 ¢ major third of 12edo, and in sharper tunings different mappings of 5/4 arise with more accuracy (see quasiultra and ultrapyth), somewhat analogous to 19edo (which represents 1/3-comma meantone and is on the edge between septimal meantone and flattone). The same goes for flatter tunings than 22edo (see quasisuper and dominant). Furthermore, the 11-limit canonical 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.
Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to 39edo), 1/6-comma, … are analogous to the historical "modified meantones" (1/6-comma, 1/7-comma, …), as they prioritize the tuning of 3/2 more than the accuracy of septimal harmony. The alternative 11-limit extension, suprapyth, and an alternative extension to 5, quasisuper, work best for tunings in the range of 17edo to 22edo.
A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for 2/7-comma meantone, treating 6:7:8 as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where 49/48 is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by 32edo. Unlike in the case of meantone, CEE optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth.
Finally, it may be noted that the plastic number has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. This can be explained since archy 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 | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~3/2 = 712.8606 ¢ (2/5-comma) |
CSEE: ~3/2 = 711.9997 ¢ (7/19-comma) |
POEE: ~3/2 = 709.6343 ¢ |
| Tenney | CTE: ~3/2 = 709.5948 ¢ | CWE: ~3/2 = 709.3901 ¢ | POTE: ~3/2 = 709.3213 ¢ |
| Benedetti, Wilson |
CBE: ~3/2 = 707.7286 ¢ (18/85-comma) |
CSBE: ~3/2 = 707.9869 ¢ (25/113-comma) |
POBE: ~3/2 = 708.6428 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~3/2 = 709.7805 ¢ | CSEE: ~3/2 = 710.2428 ¢ | POEE: ~3/2 = 710.4936 ¢ |
| Tenney | CTE: ~3/2 = 709.5907 ¢ | CWE: ~3/2 = 710.1193 ¢ | POTE: ~3/2 = 710.2910 ¢ |
| Benedetti, Wilson |
CBE: ~3/2 = 709.4859 ¢ | CSBE: ~3/2 = 710.0321 ¢ | POBE: ~3/2 = 710.2421 ¢ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
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 | ||
| 35\59 | 711.864 | 59cc val | |
| 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
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.
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.