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Ultrapyth: Difference between revisions

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{{Infobox regtemp
| Title = Ultrapyth
| Subgroups = 2.3.5.7, 2.3.5.7.13
| Comma basis = [[64/63]], [[6860/6561]] (2.3.5.7)<br>[[64/63]], [[91/90]], [[6125/6084]] (2.3.5.7.13)
| Edo join 1 = 5 | Edo join 2 = 32
| Mapping = 1; 1 14 -2 18
| Generators = 3/2
| Generators tuning = 713.6
| Optimization method = CWE
| MOS scales = [[5L 7s]], [[5L 12s]], [[5L 17s]], [[5L 22s]]
| Pergen = (P8, P5)
| Odd limit 1 = 7 | Mistuning 1 = 11.4 | Complexity 1 = 17
| Odd limit 2 = 2.3.5.7.13 21 | Mistuning 2 = 22.8 | Complexity 2 = 22
}}
'''Ultrapyth''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], tempering out [[64/63]]. The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as '''oceanfront''') by mapping the ultramajor third [[13/10]] to +4 generators (which is also the diatonic major third), tempering out [[91/90]]. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and [[7/6]]) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. [[5/4]], [[13/8]]), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a double-augmented unison) and 13 to +18 fifths (a double-augmented third). The best tunings for ultrapyth are between 712 and 714 cents.
'''Ultrapyth''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], tempering out [[64/63]]. The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as '''oceanfront''') by mapping the ultramajor third [[13/10]] to +4 generators (which is also the diatonic major third), tempering out [[91/90]]. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and [[7/6]]) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. [[5/4]], [[13/8]]), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a double-augmented unison) and 13 to +18 fifths (a double-augmented third). The best tunings for ultrapyth are between 712 and 714 cents.


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|}
|}
<nowiki/>* In 2.3.7.13/5-subgroup [[CWE]] tuning, <br>octave reduced
<nowiki/>* In 2.3.7.13/5-subgroup [[CWE]] tuning, <br>octave reduced
</div>
</div></div>
<div style="display: inline-grid;">
<div><div style="display: inline-grid;">
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Ultrapyth
|+ style="font-size: 105%;" | Ultrapyth
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== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
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| POTE: ~3/2 = 713.6509{{c}}
| POTE: ~3/2 = 713.6509{{c}}
|}
|}
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 3/2
| 701.955
| Pythagorean tuning
|-
|
| 9/7
| 708.771
|
|-
| [[22edo|13\22]]
|
| 709.091
| 22ccff val
|-
|
| 7/6
| 711.043
|
|-
| [[27edo|16\27]]
|
| 711.111
| 27cf val
|-
| '''[[32edo|19\32]]'''
|
| '''712.500'''
| '''Lower bound of 7- and 9-odd-limit diamond monotone'''
|-
|
| 15/8
| 712.551
|
|-
|
| 15/14
| 712.908
|
|-
|
| 5/4
| 713.308
| 7- and 9-odd-limit minimax
|-
|
|13/8
|713.363
| 2.3.5.7.13 13- to 21-odd-limit minimax
|-
| '''[[37edo|22\37]]'''
|
| '''713.514'''
| '''Lower bound of 2.3.5.7.13 13-odd-limit diamond monotone<br>2.3.5.7.13 15- and 21-odd-limit diamond monotone (singleton)
|-
|
|13/10
|713.553
|
|-
|
|14/13
|713.585
|
|-
|
| 7/5
| 713.593
|
|-
|
|13/12
|714.034
|
|-
|
| 5/3
| 714.181
|
|-
|
|21/13
|714.197
|
|-
| [[42edo|25\42]]
|
| 714.286
| 42f val
|-
|
| 21/20
| 714.369
|
|-
|
|13/9
|714.789
|
|-
| [[47edo|28\47]]
|
| 714.894
| 47bcff val
|-
|
| 9/5
| 715.200
|
|-
|
| 7/4
| 715.587
|
|-
|
|15/13
|717.420
|
|-
| '''[[5edo|3\5]]'''
|
| '''720.000'''
| '''Upper bound of 7- and 9-odd-limit,<br>2.3.5.7.13 13-odd-limit diamond monotone'''
|-
|
| 21/16
| 729.219
|
|}
<nowiki/>* Besides the octave


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/Ultrapyth"