Ultrapyth: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Tags: Removed redirect Visual edit
Cleanup on infobox
 
(23 intermediate revisions by 6 users not shown)
Line 1: Line 1:
'''Oceanfront''' is an alternative extension to [[superpyth]]. Like superpyth, it is a temperament generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]]. The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it maps the ultramajor third 13/10 to +4 generators (a supermajor third which is also the diatonic major third). This makes sense because the best tunings of 2.3.7 superpyth (also called archy) are sharp of the optimal tuning for 9/7, making it 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.


== Generator chain ==
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>.
{| class="wikitable"
 
|+
For technical data, see [[The Biosphere #Oceanfront]] and [[Archytas clan #Ultrapyth]].
!#
 
!Cents
== Interval chain ==
!Approximate ratios (Oceanfront)
<div><div style="display: inline-grid; margin-right: 25px;">
!Approximate ratios (added by Ultrapyth)
{| class="wikitable center-1 right-2"
!Approximate ratios (added in the 11-limit)
|+ style="font-size: 105%;" | 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
|}
<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
|-
|-
|0
| 10 || 1136.3 || 25/13, 35/18 || 21/11, 64/33 || 88/45
|0
|-
|1/1
| 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''' ||  ||
|}
<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:Rank-2 temperaments]]
[[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"