Superpyth: Difference between revisions

| Comma basis = [[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (7-limit); <br>[[64/63]], [[100/99]], [[245/243]] (11-limit)

| MOS scales = [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[5L&nbsp;17s]]

'''Superpyth''', sometimes called '''archy''' in the [[2.3.7 subgroup]], is a [[regular temperament|temperament]] where the [[generator]] is a [[3/2|perfect fifth]], tuned sharp such that a stack of two perfect fifths [[octave reduction|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. [[22edo|13\22]] (~1/4 septimal comma) and [[27edo|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.

If intervals of [[5/1|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♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. 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 [[7/1|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.

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 but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group | ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. 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 [[91/90]]. 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 (known as [[uberpyth]]), or a doubly diminished octave (C–C𝄫), by tempering out [[144/143]]. This has a more flexible range, but the 13 tends to be tuned very sharp except in 27edo, as [[13/8]] is equated with [[18/11]]. 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. Alternatively, one can keep the superpyth mappings of 5 and 11 to get [[archytas clan #Thomas|thomas]].

[[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.

For more technical data, see [[Archytas clan #Superpyth]].  

In these tables, odd harmonics and subharmonics 1–11 are in '''bold'''.

<div><div style="display: inline-grid; margin-right: 25px;">

|+ style="font-size: 105%;" | Archy (2.3.7)

| 1 || 709.4 || '''3/2'''

| 9 || 391.1 || '''5/4''' || || 27/22

|-

| 10 || 1101.2 || 15/8, 40/21 || ||

| 12 || 121.4 || 15/14 || ||

|-

| 13 || 831.6 || 45/28 || 44/27 ||

|-

| 15 || 1051.8 || 50/27 || 11/6 || 20/11

 

 

 

 

Another approach takes account of the fact that, 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 1–7/6–4/3 ([[6:7:8]]) and its inversion 1–8/7–4/3 ([[21:24:28]]). 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 1–7/4–3 (4:7:12) and 1–12/7–3 (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.

Therefore, it may be helpful to also consider the [[9-odd-limit]] [[anomalous saturated suspension|saturated suspensions]], 1–7/6–3/2–7/4 ([[12:14:18:21]]) and 1–9/7–3/2–12/7 ([[14:18:21:24]]), which extend the chords above and are good for creating tensions and resolutions: 1–9/7–3/2–12/7 on the fifth degree creates a leading tone that wants to go to the tonic; 1–7/6–3/2–7/4 on the fourth degree creates a flat sixth that wants to go to the fifth.

In meantone, the dominant seventh chord, a tempering of [[20:25:30:36|1–5/4–3/2–9/5]], is often used on the dominant to resolve to the tonic. A similar chord in superpyth is [[28:36:42:49|1–9/7–3/2–7/4]], with a leading tone at 9/7 above the perfect fifth, or [[27/14]]. This chord contains a [[49/36]] [[tritone]] between the 9/7 and 7/4, which creates tension in the chord. (However, 11-limit superpyth maps it to [[15/11]], making it a [[15-odd-limit]] [[swetismic chords|swetismic essentially tempered chord]].)

== Scales ==

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. See [[Pentatonic Functional Just System]] for a further explanation of what such a system would look like.

; 7-note mos ([[5L&nbsp;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 220{{c}} in size. The small steps (minor seconds) are thus narrower, being around 50{{c}} (a quartertone) wide. 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&nbsp;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 [[140:180:210:252:315|1/(9:7:6:5:4)]] chord.

* [[12-22a]] – in 22edo tuning

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 tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, between [[52edo|52b-edo]] and [[57edo|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{{c}}, 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{{c}}, 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 meantone|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{{c}}) and [[93edo]] with its sharp fifth of 709.677{{c}} 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{{c}} 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 (temperament)|dominant]]), which map [[7/5]] wider than [[10/7]] in the superpyth mapping. Furthermore, the [[11-limit]] canonical extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11 as well as 7/5 and 10/7, and 27e-edo conflating 11/8 with 7/5. Suprapyth, on the other hand, only works in 22edo, as sharper tunings map [[11/10]] wider than [[12/11]].

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 meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of 3/2 more than the accuracy of septimal harmony. The [[supra]] mapping of the [[2.3.7.11 subgroup]], and the quasisuper mapping of 5, work best for tunings in the range of 17c-edo to 22-edo.

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 {{w|plastic ratio}} 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. In fact, it is the tuning that makes ~6:7:8 become +1+1 [[delta-rational]].

{| class="wikitable mw-collapsible mw-collapsed"

! rowspan="2" |

| CEE: ~3/2 = 712.8606{{c}}<br>(2/5-comma)

| CSEE: ~3/2 = 711.9997{{c}}<br>(7/19-comma)

| POEE: ~3/2 = 709.6343{{c}}

|-

| CTE: ~3/2 = 709.5948{{c}}

| CWE: ~3/2 = 709.3901{{c}}

| POTE: ~3/2 = 709.3213{{c}}

|-

| CBE: ~3/2 = 707.7286{{c}}<br>(18/85-comma)

| CSBE: ~3/2 = 707.9869{{c}}<br>(25/113-comma)

| POBE: ~3/2 = 708.6428{{c}}

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings

| CBE: ~3/2 = 709.4859{{c}}

| CSBE: ~3/2 = 710.0321{{c}}

| POBE: ~3/2 = 710.2421{{c}}

{| class="wikitable mw-collapsible mw-collapsed"

! rowspan="2" |  

! colspan="3" | Euclidean

|-

| CTE: ~3/2 = 709.5143{{c}}

| CWE: ~3/2 = 710.0129{{c}}

| POTE: ~3/2 = 710.1747{{c}}

=== Other tunings ===

* DKW (2.3.7 archy): ~2 = 1200.000{{c}}, ~3/2 = 712.585{{c}}

{| class="wikitable center-all left-4 left-5"

! [[Eigenmonzo|Unchanged&nbsp;interval<br>(eigenmonzo)]]*

| '''[[7edo|4\7]]'''

| 3/2

| [[17edo|10\17]]

|

| 81/56

|

| 1/6 comma

|

| 27/14

| 707.408

|

| 1/5 comma

| 9/7

| 708.771

|

| 1/4 comma, 2.3.7-subgroup 9-odd-limit minimax

|-

|

| 49/27

| 2/7 comma

|-

| [[49edo|29\49]]

| 1/3 comma, 2.3.7-subgroup 7-odd-limit minimax

|-

| [[32edo|19\32]]

| 49/48

|

| 2/5 comma, 2.3.7-subgroup CEE tuning

|

| 7/4

| 715.587

| 1/2 comma

| '''[[5edo|3\5]]'''

|

| '''720.000'''

| ↑&nbsp;Ultrapyth

| '''Upper bound of 2.3.7-subgroup 7- and 9-odd-limit diamond monotone'''

|-

|

| 21/16

|

| Full comma

==== Superpyth ====

{| class="wikitable center-all left-4"

| '''[[5edo|3\5]]'''

|  

| '''720.000'''

| 5e val, '''upper bound of 7- and 9-odd-limit diamond monotone'''

|  

| 21/16

| 729.219

|  

== Music ==

; [[Flora Canou]]

* [https://soundcloud.com/floracanou/prelude-the-triad-challenge?in=floracanou/sets/totmc-suite "Prelude: the Triad Challenge"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in superpyth, [[70ed6]] tuning

; [[Lillian Hearne]]

* [https://soundcloud.com/lillianhearne/superpyth12-chromatic-riff ''Superpyth{{lbrack}}12{{rbrack}} chromatic riff''] (2015)

* [https://soundcloud.com/lillianhearne/trio-in-superpyth-temperament-for-irish-whistle-cello-and-piano ''Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello''] (2015)

; [[Joel Grant Taylor]]

* [https://web.archive.org/web/20201127015919/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 ''12of22study3 (children's story)'']

* [https://web.archive.org/web/20201127012539/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 ''12of22study7'']

[[Category:Superpyth| ]] <!-- Main article -->

[[Category:Rank-2 temperaments]]

[[Category:Archytas clan]]

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[[Category:Orwellismic temperaments]]