View source for Superpyth

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{{Infobox regtemp
| Title = Archy; superpyth
| Subgroups = 2.3.7, 2.3.5.7
| Comma basis = [[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (2.3.5.7)
| Mapping = 1; 1 9 -2
| Edo join 1 = 5 | Edo join 2 = 22
| Generator = 3/2
| Generator tuning = 710.1
| Optimization method = CWE
| Pergen = (P8, P5)
| Color name = Ruti
| MOS scales = [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[5L&nbsp;17s]]
| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = 9.09{{c}} | Complexity 1 = 12
| Odd limit 2 = 9 | Mistuning 2 = ? | Complexity 2 = 27
}}
'''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 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<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group | ''A few full 11-limit 896/891 temperaments'']</ref>. 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 scale]]s 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'''.

<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Archy (2.3.7)
|-
! # !! 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
|}
<nowiki/>* In 2.3.7-subgroup [[CWE]] tuning, <br>octave reduced
</div>
<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2 right-4"
|+ style="font-size: 105%;" | Supra (2.3.7.11)
|-
! # !! 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
|}
<nowiki/>* In 2.3.7.11-subgroup CWE tuning, <br>octave reduced
</div>
<div style="display: inline-grid;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Full 7-limit superpyth
|-
! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
|-
! rowspan="2" | 7-limit !! colspan="2" | 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
|}
<nowiki/>* In 7-limit CWE tuning, octave reduced
</div></div>

== 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 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. 

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 (meantone's 5-[[integer limit|integer-limit]] [[minimax tuning]]) 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 9.09{{c}}, over twice as much as meantone. 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}}) is very close to a closed system 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]]). 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.

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 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 {{w|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 {{nowrap| ~9 + ~12 {{=}} ~21 }} the generator becomes the plastic number.

=== Prime-optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7-subgroup prime-optimized tunings
|-
! rowspan="2" | 
! colspan="2" | Euclidean
|-
! Unskewed
! Skewed
|-
! Equilateral
| CEE: ~3/2 = 712.8606¢<br>(2/5-comma)
| CSEE: ~3/2 = 711.9997¢<br>(7/19-comma)
|-
! Tenney
| CTE: ~3/2 = 709.5948¢
| CWE: ~3/2 = 709.3901¢
|-
! Benedetti, <br>Wilson
| CBE: ~3/2 = 707.7286¢<br>(18/85-comma)
| CSBE: ~3/2 = 707.9869¢<br>(25/113-comma)
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|-
! rowspan="2" | 
! colspan="2" | 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, <br>Wilson
| CBE: ~3/2 = 709.4859¢
| CSBE: ~3/2 = 710.0321¢
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| 
| 3/2
| 701.955
| Pythagorean tuning
|-
| '''[[17edo|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
|-
| [[39edo|23\39]]
| 
| 707.692
| 39cd val
|-
| 
| 9/7
| 708.771
| 1/4 comma, {{nowrap|<nowiki>{1, 3, 7, 9}</nowiki>}} minimax
|-
| 
| 15/8
| 708.807
| 
|-
| [[22edo|13\22]]
| 
| 709.091
| 
|-
| 
| 5/4
| 709.590
| 9-odd-limit minimax
|-
| 
| 49/27
| 709.745
| 2/7 comma
|-
| [[71edo|42\71]]
| 
| 709.859
| 71d val
|-
| 
| 15/14
| 709.954
| 
|-
| 
| 25/24
| 710.040
| 
|-
| [[49edo|29\49]]
| 
| 710.204
| 
|-
| [[76edo|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, {{nowrap|<nowiki>{1, 3, 7}</nowiki>}} minimax
|-
| [[27edo|16\27]]
| 
| 711.111
| 
|-
| 
| 21/20
| 711.553
| 
|-
| 
| 9/5
| 711.772
| 
|-
| [[59edo|35\59]]
| 
| 711.864
| 59cc val
|-
| [[32edo|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
|-
| [[37edo|22\37]]
| 
| 713.514
| 37cc val
|-
| [[42edo|25\42]]
| 
| 714.286
| 42cc val
|-
| 
| 7/4
| 715.587
| 1/2 comma
|-
| '''[[5edo|3\5]]'''
| 
| '''720.000'''
| '''Upper bound of 7- and 9-odd-limit diamond monotone'''
|-
| 
| 21/16
| 729.219
| Full comma
|}
<nowiki/>* Besides the octave

=== Other tunings ===
* [[DKW theory|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&nbsp;3s]], proper)
* [[Archy5]] – archy in 472edo tuning

; 7-note mos ([[5L&nbsp;2s]], improper)
* [[Archy7]] – archy in 472edo tuning
* [[Supra7]] – supra in 56edo tuning
In contrast to the meantone diatonic scale, the superpyth diatonic is improper.

; 12-note mos ([[5L&nbsp;7s]], borderline improper)
* [[Archy12]] – archy in 472edo tuning
* [[Supra12]] – supra in 56edo tuning
* [[12-22a]] – superpyth in 22edo tuning
The boundary of propriety is [[17edo]].

== Music ==
; [[User:Lhearne|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)
: Both in 22edo tuning

; [[Joel Grant Taylor]]
* [https://web.archive.org/web/20201127013613/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 ''12of22studyPentUp4thsMstr'']
* [https://web.archive.org/web/20201127013450/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 ''12of22gamelan1b'']
* [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'']
: All in Superpyth[12], 22edo tuning.

== Notes ==

[[Category:Superpyth| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Archytas clan]]
[[Category:Sensamagic clan]]
[[Category:Orwellismic temperaments]]