Squares: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup|2.3.7-subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore [[tempering out]] the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the 3.7.11-subgroup [[mintaka]] temperament by identifying [[2/1]] with a false octave corresponding to 99/49~243/121, in a manner similar to [[sensi]]'s relation to [[BPS]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
 
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-18 16:44:22 UTC</tt>.<br>
However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with [[meantone]] to create full [[11-limit]] squares, which additionally can be restricted to the [[7-limit]] as the temperament with comma basis [[81/80]] and [[2401/2400]]. This 11-limit temperament is considered below.
: The original revision id was <tt>246790309</tt>.<br>
 
: The revision comment was: <tt></tt><br>
There is also a natural extension adding [[prime interval|prime]] [[23/1|23]] by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[162/161]].
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
As for prime [[13/1|13]], the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[minalzidar]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">See [[Meantone family#Squares]].</pre></div>
 
<h4>Original HTML content:</h4>
See [[Meantone family #Squares]] and [[No-fives subgroup temperaments #Skwares]] for more technical data.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Squares&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See &lt;a class="wiki_link" href="/Meantone%20family#Squares"&gt;Meantone family&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== Interval chain ==
In the following table, prime harmonics and subharmonics are labelled in '''bold'''.  
 
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="4" | Approximate ratios
|-
! rowspan="2" | 11-limit
! colspan="3" | 13-limit extensions
|-
! Squares
! Squad
! Agora
|-
| 0
| 0.0
| 1/1
|
|
|
|-
| 1
| 425.6
| 9/7, 14/11
|
| 13/10
|
|-
| 2
| 851.2
| 18/11, 33/20, 44/27
| '''13/8'''
| 21/13
|
|-
| 3
| 76.8
| 21/20, 28/27
|
| 27/26
|
|-
| 4
| 502.4
| '''4/3'''
|
|
|
|-
| 5
| 928.0
| 12/7
|
| 22/13, 26/15
|
|-
| 6
| 153.6
| 11/10, 12/11
| 13/12
| 14/13
|
|-
| 7
| 579.2
| 7/5
|
| 18/13
|
|-
| 8
| 1004.8
| 9/5, 16/9
|
|
|
|-
| 9
| 230.4
| '''8/7'''
|
|
| 15/13
|-
| 10
| 656.0
| '''16/11''', 22/15
| 13/9
|
|
|-
| 11
| 1081.6
| 28/15
| 13/7
| 24/13
|
|-
| 12
| 307.2
| 6/5
| 13/11
|
|
|-
| 13
| 732.8
| 32/21
|
|
| 20/13
|-
| 14
| 1158.4
| 49/25, 64/33, 96/49
| 52/27
|
|
|-
| 15
| 384.0
| 56/45
| 26/21
| '''16/13'''
|
|-
| 16
| 809.6
| '''8/5'''
|
|
| 21/13
|-
| 17
| 35.2
| 36/35, 64/63
|
|
|
|}
<nowiki />* In 11-limit CTE tuning
 
== Scales ==
* [[Skwares8]]
* [[Skwares11]]
* [[Skwares14]]
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~14/9 = 774.3052{{c}}
| CWE: ~14/9 = 774.1560{{c}}
| POTE: ~14/9 = 774.0585{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~11/7 = 774.4005{{c}}
| CWE: ~11/7 = 774.1754{{c}}
| POTE: ~11/7 = 774.0427{{c}}
|}
 
== Music ==
; [[Joel Kivelä]]
* ''Optimum Rains'' (2023) – [https://joelkivela.bandcamp.com/album/optimum-rains Bandcamp] | [https://www.youtube.com/watch?v=NUJOVrLqtdk YouTube]
 
; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127015038/http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 ''Square 8'']
 
[[Category:Squares| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Nuwell temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Listen]]

Latest revision as of 12:44, 24 August 2025

At its most basic level, squares can be thought of as a 2.3.7-subgroup temperament (sometimes called skwares), generated by a flat ~9/7 such that four of them stack to the perfect eleventh, 8/3, therefore tempering out the comma 19683/19208. However, it is more natural to think of the temperament first as 2.3.7.11 subgroup, tempering out 99/98 so as to identify the generator with 14/11 in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, 18/11, two of which are then identified with 8/3 due to tempering out 243/242. This can also be thought of as an octavization of the 3.7.11-subgroup mintaka temperament by identifying 2/1 with a false octave corresponding to 99/49~243/121, in a manner similar to sensi's relation to BPS.

However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with meantone to create full 11-limit squares, which additionally can be restricted to the 7-limit as the temperament with comma basis 81/80 and 2401/2400. This 11-limit temperament is considered below.

There is also a natural extension adding prime 23 by equating the generator to 23/18, and so finding 23 itself seven generators down, tempering out 162/161.

As for prime 13, the way to map it is less clear. The canonical squares mapping tempers out 144/143 in order to equate the tridecimal neutral sixth, 13/8, with 18/11, finding 13 two generators up, while agora tempers out 105/104 to equate 8/7 with 15/13, finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in 31edo. Finally, squad tempers out 351/343 (which is the same as 3.7.11.13 minalzidar's tempering of that prime) so that 13 is equated with (7/3)3, and found 15 generators down.

See Meantone family #Squares and No-fives subgroup temperaments #Skwares for more technical data.

Interval chain

In the following table, prime harmonics and subharmonics are labelled in bold.

# Cents* Approximate ratios
11-limit 13-limit extensions
Squares Squad Agora
0 0.0 1/1
1 425.6 9/7, 14/11 13/10
2 851.2 18/11, 33/20, 44/27 13/8 21/13
3 76.8 21/20, 28/27 27/26
4 502.4 4/3
5 928.0 12/7 22/13, 26/15
6 153.6 11/10, 12/11 13/12 14/13
7 579.2 7/5 18/13
8 1004.8 9/5, 16/9
9 230.4 8/7 15/13
10 656.0 16/11, 22/15 13/9
11 1081.6 28/15 13/7 24/13
12 307.2 6/5 13/11
13 732.8 32/21 20/13
14 1158.4 49/25, 64/33, 96/49 52/27
15 384.0 56/45 26/21 16/13
16 809.6 8/5 21/13
17 35.2 36/35, 64/63

* In 11-limit CTE tuning

Scales

Tunings

7-limit prime-optimized tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~14/9 = 774.3052 ¢ CWE: ~14/9 = 774.1560 ¢ POTE: ~14/9 = 774.0585 ¢
11-limit prime-optimized tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~11/7 = 774.4005 ¢ CWE: ~11/7 = 774.1754 ¢ POTE: ~11/7 = 774.0427 ¢

Music

Joel Kivelä
Chris Vaisvil
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