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'''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]]
Retrieved from "https://en.xen.wiki/w/Ultrapyth"