Superpyth: Difference between revisions
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Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the [[4:5:6|1–5/4–3/2]] and [[10:12:15|1–6/5–3/2]] triads in meantone. However, the contrast between these triads isn't as expressive as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180{{c}} in size rather than the ideal ~60–80{{c}} semitone in meantone. | Superpyth contains a version of the [[5L 2s|diatonic]] scale where the major third represents [[9/7]], and the minor third represents [[7/6]]. {{W|Tertian harmony}} can thus be used, with the major and minor triads representing [[14:18:21|1–9/7–3/2]] and [[6:7:9|1–7/6–3/2]] respectively, rather than the [[4:5:6|1–5/4–3/2]] and [[10:12:15|1–6/5–3/2]] triads in meantone. However, the contrast between these triads isn't as expressive as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180{{c}} in size rather than the ideal ~60–80{{c}} semitone in meantone. | ||
If one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in [[meantone]] cannot be used in superpyth, as superpyth doesn't temper out [[81/80]]. For example, the major triad on C is written as C – D♯ – G rather than C – E – G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C – Ev – G and the minor triad as C – E♭^ – G. The 81/80 comma is equated to [[28/27]], [[36/35]], and [[256/243]] in superpyth, leading to the enharmonic equivalences C^ = D♭, E = Fv, etc. The limma (C – D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C – C♯) in meantone, which is a submajor second in size in superpyth. | If one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in [[meantone]] cannot be used in superpyth, as superpyth doesn't temper out [[81/80]]. For example, the major triad on C is written as C – D♯ – G rather than C – E – G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C – Ev – G and the minor triad as C – E♭^ – G. The 81/80 comma is equated to [[28/27]], [[36/35]], and [[256/243]] in superpyth, leading to the enharmonic equivalences C^ = D♭, E = Fv, etc. The limma (C – D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C – C♯) as in meantone, which is a submajor second in size in superpyth. | ||
Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 ([[6:7:8:9]]) and its utonal inverse 1–9/8–9/7–3/2 (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in [[mavila]]. Chromatic alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2. | Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 ([[6:7:8:9]]) and its utonal inverse 1–9/8–9/7–3/2 (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in [[mavila]]. Chromatic alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2. | ||
Revision as of 00:12, 24 December 2025
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 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. For a tuning flat of 22edo, the 5th harmonic can be mapped to -13 generators to get quasisuper.
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. Note that the only reasonable tuning for suprapyth is 22edo, as sharper tunings swap the sizes of 11/10 and 12/11, and flatter tunings swap 11/10 and 10/9, as well as 7/5 and 10/7. However, by keeping the 2.3.7.11 mapping of suprapyth (simply called supra) and using the quasisuper mapping of 5, we get quasisupra, which has a flexible tuning range.
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. In practice, however, this mapping only works in 27edo, as flatter tunings swap the sizes of 13/12 and 14/13. An alternative mapping is -14 generators, or a doubly diminished octave (C-C𝄫), by tempering out 9604/9477. This has a more flexible range, but the 13 tends to be tuned very sharp except in 27edo. A more practical option is to split the sharp ~3/2 into two ~16/13's, which results in beatles, and has an alternative mapping of primes 5 and 11.
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 and subharmonics 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 |
| 18 | 782.1 | 25/16 | 11/7 | |
| 19 | 292.3 | 25/21 | 33/28 | |
| 20 | 1002.4 | 25/14 | ||
| 21 | 512.5 | 75/56 | ||
| 22 | 22.6 | 50/49, 225/224 | 99/98 | |
* In 7-limit CWE tuning, octave reduced
Chords and harmony
Superpyth contains a version of the diatonic scale where the major third represents 9/7, and the minor third represents 7/6. Tertian harmony can thus be used, with the major and minor triads representing 1–9/7–3/2 and 1–7/6–3/2 respectively, rather than the 1–5/4–3/2 and 1–6/5–3/2 triads in meantone. However, the contrast between these triads isn't as expressive as the contrast between the meantone triads, as the interval between 7/6 and 9/7 is too wide, being ~140–180 ¢ in size rather than the ideal ~60–80 ¢ semitone in meantone.
If one wishes to use the 5-limit triads as bases for harmony, then much of the logic which is used in meantone cannot be used in superpyth, as superpyth doesn't temper out 81/80. For example, the major triad on C is written as C – D♯ – G rather than C – E – G as in meantone, which is awkward to notate and conceptualize. To solve this, one may want to adopt a pair of accidentals (such as ^ and v) to represent modifications by 81/80, thus notating the major triad as C – Ev – G and the minor triad as C – E♭^ – G. The 81/80 comma is equated to 28/27, 36/35, and 256/243 in superpyth, leading to the enharmonic equivalences C^ = D♭, E = Fv, etc. The limma (C – D♭) thus becomes the most important interval for note alterations, being around a quartertone in size and representing so many important ratios, rather than the apotome (C – C♯) as in meantone, which is a submajor second in size in superpyth.
Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 (6:7:8:9) and its utonal inverse 1–9/8–9/7–3/2 (representing 14:16:18:21 as 64/63 is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the mu chord of Steely Dan fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in mavila. Chromatic alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2.
Another approach is also possible. In the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us 6:7:8 and its inversion 21:24:28, which are built with the intervals 7/6 and 8/7. These intervals contrast by 49/48, similarly to how 5-limit thirds contrast by 25/24. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 4:7:12 and 7:12:21, solve both of these issues. These triads span a twelfth. In terms of the chain of fifths, these chords are simpler in superpyth than the 5-limit triads in meantone.
Scales
- 5-note mos (2L 3s, proper)
In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around 5edo. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~28/27 becoming a chroma.
- 7-note mos (5L 2s, improper)
In contrast to the meantone diatonic scale, the superpyth diatonic is improper. Since the fifth is sharp rather than flat in meantone, the large steps (major seconds) are wider, being around 212–222 ¢ depending on the tuning. The small steps (minor seconds) are thus narrower, being around 44–71 ¢. This has the effect of large and small steps being more distinct compared to meantone diatonic, as well as stronger leading tones due to narrower small steps, though one may want to bend the leading tone down by a small step to avoid it being too close to the tonic.
- 12-note mos (5L 7s, improper)
The superpyth chromatic scale is also improper (the boundary of propriety is 17edo). This scale is the first that contains much beyond the 2.3.7-subgroup, with three 4:5:6 chords, three 10:12:15 chords, one 4:5:6:7:9 chord, and one 1/(9:7:6:5:4) chord.
Scala files
- Archy5 – archy in 472edo tuning
- Archy7 – archy in 472edo tuning
- Archy12 – archy in 472edo tuning
- Supra7 – supra in 56edo tuning
- Supra12 – supra in 56edo tuning
- 12-22a – superpyth in 22edo tuning
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.
Without tempered octaves, superpyth is of considerably higher damage than meantone, despite it being seen as the "counterpart" of meantone for sharp fifths and septimal thirds. The vanishing 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 9.1 ¢, 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.
Norm-based 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 | ||
| 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 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 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
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.