Ultrapyth: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Cleanup (1/2)
Cleanup (2/2)
Line 7: Line 7:
For technical data, see [[The Biosphere #Oceanfront]] and [[Archytas clan #Ultrapyth]].  
For technical data, see [[The Biosphere #Oceanfront]] and [[Archytas clan #Ultrapyth]].  


== Generator chain ==
== Interval chain ==
{| class="wikitable"
<div><div style="display: inline-grid; margin-right: 25px;">
|+
{| class="wikitable center-1 right-2"
!#
|+ style="font-size: 105%;" | Oceanfront (2.3.7.13/5)
!Cents
!Approximate ratios (Oceanfront)
!Approximate ratios (added by Ultrapyth)
!Approximate ratios (added in the 11-limit)
|-
|-
|0
! # !! Cents* !! Approximate ratios
|0
|1/1
|
|
|-
|-
|1
| 0 || 0.0 || '''1/1'''
|713.4
|'''3/2'''
|
|
|-
|-
|2
| 1 || 711.7 || '''3/2'''
|226.8
|'''8/7''', '''9/8''', 91/80
|
|
|-
|-
|3
| 2 || 223.5. || '''8/7''', '''9/8'''
|940.2
|12/7, 26/15
|
|
|-
|-
|4
| 3 || 935.2 || 12/7, 26/15
|453.6
|13/10, 9/7
|
|
|-
|-
|5
| 4 || 447.0 || 9/7, 13/10
|1167
|39/20, 27/14
|
|
|-
|-
|6
| 5 || 1158.7 || 27/14, 39/20
|680.4
|72/49, 52/35
|
|
|-
|-
|7
| 6 || 670.4 || 52/35, 72/49
|193.8
|54/49
|
|112/99, 160/143
|-
|-
|8
| 7 || 182.2 || 39/35, 54/49
|907.2
|}
|117/70
<nowiki/>* In 2.3.7.13/5-subgroup [[CWE]] tuning, <br>octave reduced
|
</div>
|56/33
<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 ||  ||
|-
|-
|9
| 7 || 195.4 || 39/35, 54/49 ||  ||  
|420.6
|
|
|14/11
|-
|-
|10
| 8 || 909.0 || 81/49, 117/70 || 56/33 || 22/13
|1134
|
|160/81
|64/33
|-
|-
|11
| 9 || 422.7 || 35/27 || 14/11 || 33/26
|647.4
|
|40/27
|'''16/11'''
|-
|-
|12
| 10 || 1136.3 || 25/13, 35/18 || 21/11, 64/33 || 88/45
|160.8
|
|10/9
|12/11
|-
|-
|13
| 11 || 649.9 || 35/24, 40/27 || '''16/11''' || 22/15
|874.2
|
|5/3
|128/77
|-
|-
|14
| 12 || 163.5 || 10/9 || 12/11 || 11/10
|387.6
|
|'''5/4'''
|96/77
|-
|-
|15
| 13 || 877.2 || 5/3 || 18/11 || 33/20
|1101
|
|'''15/8''', 52/27
|144/77
|-
|-
|16
| 14 || 390.8 || '''5/4''' || ||  
|614.4
|
|45/32, 13/9
|
|-
|-
|17
| 15 || 1104.4 || 15/8, 40/21, 52/27 ||  ||  
|127.8
|
|13/12, 135/128
|
|-
|-
|18
| 16 || 618.1 || 10/7, 13/9 ||  ||  
|841.2
|
|'''13/8'''
|
|-
|-
|19
| 17 || 131.7 || 13/12, 15/14 ||  ||  
|354.6
|
|39/32
|
|-
|-
|20
| 18 || 845.3 || '''13/8''' ||  ||  
|1068
|
|117/64
|
|}
|}
<nowiki/>* In 2.3.5.7.13-subgroup CWE tuning, octave reduced
</div></div>


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

Revision as of 17:09, 9 June 2025

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 are extremely close to 5edo. 37edo makes a good tuning of oceanfront or ultrapyth.

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

Retrieved from "https://en.xen.wiki/index.php?title=Ultrapyth&oldid=199052"