Ultrapyth: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,044 edits
m added categories
Cleanup on infobox
 
(18 intermediate revisions by 4 users not shown)
Line 1: Line 1:
'''Ultrapyth''' is an alternative extension of the [[archy]] chain of fifths to [[superpyth]]. Like superpyth, it is a temperament generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]] (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 (a supermajor third 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 doubly augmented unison) and 13 to +18 fifths (a doubly-augmented third). The best tunings for ultrapyth are between 712 and 714 cents.
{{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.


If intervals of 11 are desired, [[14/11]] may be mapped to +9 generators, implying [[16/11]] is (fittingly) mapped to +11 generators and [[11/9]] is tempered together with 6/5 (a feature common to many systems with sharp fifths).
If intervals of 11 are desired, [[14/11]] may be mapped to +9 generators, implying [[16/11]] is (fittingly) mapped to +11 generators and [[11/9]] is tempered together with 6/5 (a feature common to many systems with sharp fifths).


The oceanfront MOS scales take the form of 5L (2+5n)s, for n up to 7. Most of these scales are extremely close to [[5edo]]. [[37edo]] makes a good tuning of oceanfront or ultrapyth.
The oceanfront [[mos scale]]s take the form of {{nowrap| 5L (5''n'' + 2)s }}, for ''n'' up to 7. Most of these scales resemble [[5edo]]. [[37edo]] makes a good tuning of oceanfront or ultrapyth.


View technical temperament data [[The Biosphere#Oceanfront|here]] for oceanfront and [[Archytas clan#Ultrapyth|here]] for ultrapyth.
Both ''oceanfront'' and ''ultrapyth'' were named by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_98570.html Yahoo! Tuning Group | ''The Biosphere'']</ref>.  


== Generator chain ==
For technical data, see [[The Biosphere #Oceanfront]] and [[Archytas clan #Ultrapyth]].
{| class="wikitable"
 
|+
== Interval chain ==
!#
<div><div style="display: inline-grid; margin-right: 25px;">
!Cents
{| class="wikitable center-1 right-2"
!Approximate ratios (Oceanfront)
|+ style="font-size: 105%;" | Oceanfront (2.3.7.13/5)
!Approximate ratios (added by Ultrapyth)
|-
!Approximate ratios (added in the 11-limit)
! # !! Cents* !! Approximate ratios
|-
| 0 || 0.0 || '''1/1'''
|-
| 1 || 711.7 || '''3/2'''
|-
| 2 || 223.5 || '''8/7''', '''9/8'''
|-
| 3 || 935.2 || 12/7, 26/15
|-
| 4 || 447.0 || 9/7, 13/10
|-
| 5 || 1158.7 || 27/14, 39/20
|-
| 6 || 670.4 || 52/35, 72/49
|-
| 7 || 182.2 || 39/35, 54/49
|}
<nowiki/>* In 2.3.7.13/5-subgroup [[CWE]] tuning, <br>octave reduced
</div></div>
<div><div style="display: inline-grid;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Ultrapyth
|-
! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
|-
! rowspan="2" | 2.3.5.7.13 subgroup !! colspan="2" | Full 13-limit extensions
|-
! Ultrapyth !! Ultramarine
|-
| 0 || 0.0 || '''1/1''' ||  ||
|-
| 1 || 713.6 || '''3/2''' ||  ||
|-
| 2 || 227.3 || '''8/7''', '''9/8''' ||  ||
|-
| 3 || 940.9 || 12/7, 26/15 ||  ||
|-
| 4 || 454.5 || 9/7, 13/10 ||  ||
|-
| 5 || 1168.1 || 27/14, 39/20 ||  ||
|-
| 6 || 681.8 || 52/35, 72/49 ||  ||
|-
| 7 || 195.4 || 39/35, 54/49 ||  ||
|-
| 8 || 909.0 || 81/49, 117/70 || 56/33 || 22/13
|-
| 9 || 422.7 || 35/27 || 14/11 || 33/26
|-
| 10 || 1136.3 || 25/13, 35/18 || 21/11, 64/33 || 88/45
|-
| 11 || 649.9 || 35/24, 40/27 || '''16/11''' || 22/15
|-
| 12 || 163.5 || 10/9 || 12/11 || 11/10
|-
| 13 || 877.2 || 5/3 || 18/11 || 33/20
|-
|-
|0
| 14 || 390.8 || '''5/4''' ||  ||
|0
|-
|1/1
| 15 || 1104.4 || 15/8, 40/21, 52/27 ||  ||
|
|-
|
| 16 || 618.1 || 10/7, 13/9 ||  ||
|-
| 17 || 131.7 || 13/12, 15/14 ||  ||
|-
| 18 || 845.3 || '''13/8''' ||  ||
|}
<nowiki/>* In 2.3.5.7.13-subgroup CWE tuning, octave reduced
</div></div>
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
|-
|1
! Tenney
|713.4
| CTE: ~3/2 = 713.2179{{c}}
|'''3/2'''
| CWE: ~3/2 = 713.5430{{c}}
|
| POTE: ~3/2 = 713.6509{{c}}
|
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
|2
! Edo<br>generator
|226.8
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
|'''8/7''', '''9/8''', 91/80
! Generator (¢)
|
! Comments
|
|-
|-
|3
|  
|940.2
| 3/2
|12/7, 26/15
| 701.955
|
| Pythagorean tuning
|
|-
|-
|4
|  
|453.6
| 9/7
|13/10, 9/7
| 708.771
|
|  
|
|-
|-
|5
| [[22edo|13\22]]
|1167
|  
|39/20, 27/14
| 709.091
|
| 22ccff val
|
|-
|-
|6
|  
|680.4
| 7/6
|72/49, 52/35
| 711.043
|
|  
|
|-
|-
|7
| [[27edo|16\27]]
|193.8
|  
|54/49
| 711.111
|
| 27cf val
|112/99, 160/143
|-
|-
|8
| '''[[32edo|19\32]]'''
|907.2
|  
|117/70
| '''712.500'''
|
| '''Lower bound of 7- and 9-odd-limit diamond monotone'''
|56/33
|-
|-
|9
|  
|420.6
| 15/8
|
| 712.551
|
|  
|14/11
|-
|-
|10
|  
|1134
| 15/14
|
| 712.908
|160/81
|  
|64/33
|-
|-
|11
|  
|647.4
| 5/4
|
| 713.308
|40/27
| 7- and 9-odd-limit minimax
|'''16/11'''
|-
|-
|12
|160.8
|
|
|10/9
|13/8
|12/11
|713.363
| 2.3.5.7.13 13- to 21-odd-limit minimax
|-
|-
|13
| '''[[37edo|22\37]]'''
|874.2
|  
|
| '''713.514'''
|5/3
| '''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)
|128/77
|-
|-
|14
|387.6
|
|
|'''5/4'''
|13/10
|96/77
|713.553
|-
|15
|1101
|
|
|'''15/8''', 52/27
|144/77
|-
|-
|16
|614.4
|
|
|45/32, 13/9
|14/13
|713.585
|
|
|-
|-
|17
|  
|127.8
| 7/5
| 713.593
|
|-
|
|
|13/12, 135/128
|13/12
|714.034
|
|
|-
|-
|18
|  
|841.2
| 5/3
| 714.181
|
|-
|
|
|'''13/8'''
|21/13
|714.197
|
|
|-
|-
|19
| [[42edo|25\42]]
|354.6
|
| 714.286
| 42f val
|-
|  
| 21/20
| 714.369
|
|-
|
|
|39/32
|13/9
|714.789
|
|
|-
|-
|20
| [[47edo|28\47]]
|1068
|
| 714.894
| 47bcff val
|-
|
| 9/5
| 715.200
|
|-
|
| 7/4
| 715.587
|
|-
|
|
|117/64
|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 ==
* [[Oceanfront scales]]
== References ==
<references/>


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

Latest revision as of 10:08, 9 February 2026

Ultrapyth
Subgroups 2.3.5.7, 2.3.5.7.13
Comma basis 64/63, 6860/6561 (2.3.5.7)
64/63, 91/90, 6125/6084 (2.3.5.7.13)
Reduced mapping ⟨1; 1 14 -2 18]
ET join 5 & 32
Generators (CWE) ~3/2 = 713.6 ¢
MOS scales 5L 7s, 5L 12s, 5L 17s, 5L 22s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 7-odd-limit: 11.4 ¢;
2.3.5.7.13 21-odd-limit: 22.8 ¢
Target scale size 7-odd-limit: 17 notes;
2.3.5.7.13 21-odd-limit: 22 notes

Ultrapyth is an alternative extension of the archy chain of fifths to superpyth. Like superpyth, it is a 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.

If intervals of 11 are desired, 14/11 may be mapped to +9 generators, implying 16/11 is (fittingly) mapped to +11 generators and 11/9 is tempered together with 6/5 (a feature common to many systems with sharp fifths).

The oceanfront mos scales take the form of 5L (5n + 2)s, for n up to 7. Most of these scales resemble 5edo. 37edo makes a good tuning of oceanfront or ultrapyth.

Both oceanfront and ultrapyth were named by Mike Battaglia in 2011[1].

For technical data, see The Biosphere #Oceanfront and Archytas clan #Ultrapyth.

Interval chain

Oceanfront (2.3.7.13/5)
# Cents* Approximate ratios
0 0.0 1/1
1 711.7 3/2
2 223.5 8/7, 9/8
3 935.2 12/7, 26/15
4 447.0 9/7, 13/10
5 1158.7 27/14, 39/20
6 670.4 52/35, 72/49
7 182.2 39/35, 54/49

* In 2.3.7.13/5-subgroup CWE tuning,
octave reduced

Ultrapyth
# Cents* Approximate ratios
2.3.5.7.13 subgroup Full 13-limit extensions
Ultrapyth Ultramarine
0 0.0 1/1
1 713.6 3/2
2 227.3 8/7, 9/8
3 940.9 12/7, 26/15
4 454.5 9/7, 13/10
5 1168.1 27/14, 39/20
6 681.8 52/35, 72/49
7 195.4 39/35, 54/49
8 909.0 81/49, 117/70 56/33 22/13
9 422.7 35/27 14/11 33/26
10 1136.3 25/13, 35/18 21/11, 64/33 88/45
11 649.9 35/24, 40/27 16/11 22/15
12 163.5 10/9 12/11 11/10
13 877.2 5/3 18/11 33/20
14 390.8 5/4
15 1104.4 15/8, 40/21, 52/27
16 618.1 10/7, 13/9
17 131.7 13/12, 15/14
18 845.3 13/8

* In 2.3.5.7.13-subgroup CWE tuning, octave reduced

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 713.2179 ¢ CWE: ~3/2 = 713.5430 ¢ POTE: ~3/2 = 713.6509 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
3/2 701.955 Pythagorean tuning
9/7 708.771
13\22 709.091 22ccff val
7/6 711.043
16\27 711.111 27cf val
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
22\37 713.514 Lower bound of 2.3.5.7.13 13-odd-limit diamond monotone
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
25\42 714.286 42f val
21/20 714.369
13/9 714.789
28\47 714.894 47bcff val
9/5 715.200
7/4 715.587
15/13 717.420
3\5 720.000 Upper bound of 7- and 9-odd-limit,
2.3.5.7.13 13-odd-limit diamond monotone
21/16 729.219

* Besides the octave

See also

References

  1. ↑ Yahoo! Tuning Group | The Biosphere
Retrieved from "https://en.xen.wiki/index.php?title=Ultrapyth&oldid=223678"