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240edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-07 16:29:22 UTC</tt>.<br>
: The original revision id was <tt>601655574</tt>.<br>
: The revision comment was: <tt></tt><br>
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>
<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">The 240edo divides the octave into 240 steps of exactly five cents each. One important use for it is in tuning marvel temperament and marvel's extension to spectacle temperament.


If we round off to the nearest five cents, we end up with a [[Vals and Tuning Space|val]] (mapping to primes) for 240edo of &lt;[[tel/240 380 557|240 380 557]] 674|. This tempers out the [[http://en.wikipedia.org/wiki/Septimal_kleisma|septimal kleisma]] of 225/224, with low resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4.) Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning. However [[197edo]], despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 197&amp;240 temperament.
== Theory ==
240edo notably provides the [[optimal patent val]] for the 5-limit [[compton]] temperament, the rank-2 temperament associated with the [[Pythagorean comma]]. However, it is only [[consistent]] in the [[5-odd-limit]]. Its mapping for [[harmonic]] [[3/1|3]] is not well approximated, meaning it is a [[dual-fifth system]]; its alternative mapping for 3/2 is the 705{{c}} sharp fifth inherited from [[80edo]].  


For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have:
Although no longer consistent to the higher limits, 240edo's [[patent val]] [[tempering out|tempers out]] the [[225/224]] in the 7-limit, [[support]]ing [[marvel]] with harmonics 3, [[5/1|5]], [[7/1|7]] having less than two cents of error. Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning.  


3 ~ 2 (11/9)^2
240edo is similar to [[197edo]] in terms of intonation in the 7-limit. The main difference is that 197edo, despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the {{nowrap| 43 & 197 }} temperament, which has a comma basis {225/224, {{monzo| -49 19 -10 15 }}} in the 7-limit.
5 = 2^2 (5/4)
7 ~ 2 (11/9)^4 (5/4)^2
11 ~ 2^2 (11/9)^5
13 ~ 2^3 (11/9)^(-2) (5/4)^4
17 ~ 2^4 (11/9)^(-3) (5/4)^3


It should be noted that the exponents of 5/4 above are all positive and go no higher than 4.
For higher limits, 240edo tempers out [[243/242]] in the 11-limit, [[351/350]] in the 13-limit, and [[375/374]] in the 17-limit, and adding these to the mix converts marvel temperament into [[spectacle]] temperament. This is still a [[rank-3 temperament]], but more complex as two undecimal neutral thirds of [[11/9]] make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350-cent interval often employed in [[24edo]] versions of [[Arabic, Turkish, Persian music|Arabic music]].)


==Scales==  
=== Odd harmonics ===
{{Harmonics in equal|240}}


Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament.
=== Subsets and supersets ===
240edo is the 12th [[highly composite edo]], with subset edos {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120 }}. In addition, as every fifth step of [[1200edo]], it is the largest highly composite edo expressible in integer cents.


! duodene.scl
== Interval table ==
!
See [[Table of 240edo intervals]].
Ellis's Duodene : genus [33355]
12
!
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1


! duodene240.scl
== Regular temperament properties ==
!
{| class="wikitable center-4 center-5 center-6"
Ellis's Duodene : genus [33355] retuned to 240edo
|-
12
! rowspan="2" | [[Subgroup]]
!
! rowspan="2" | [[Comma list]]
115.
! rowspan="2" | [[Mapping]]
200.
! rowspan="2" | Optimal<br>8ve stretch (¢)
315.
! colspan="2" | Tuning error
385.
|-
500.
! [[TE error|Absolute]] (¢)
585.
! [[TE simple badness|Relative]] (%)
700.
|-
815.
| 2.3.5
885.
| 531441/524288, {{monzo| -29 -11 20 }}
1015.
| {{Mapping| 240 380 557 }}
1085.
| 0.5998
1200.
| 0.5044
| 10.09
|}


=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 7\240
| 35.00
| 45/44
| [[Gammy]]
|-
| 1
| 101\240
| 505.00
| 104976/78125
| [[Countermeantone]]
|-
| 12
| 77\240<br>(3\240)
| 385.00<br>(15.00)
| 5/4<br>(81/80)
| [[Compton]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


! lumma5.scl
== Scales ==
!
; Scales derived from marvel and spectacle temperaments
Carl Lumma's scale, 5-limit just version, TL 19-2-99
* 23 17 23 14 23 17 23 23 14 26 14 23 – [[Alexander Ellis|Ellis]]'s [[Duodene]] genus [33355] retuned to 240edo
! Also diadie1, prism, Fokker 12-tone just
* 23 17 14 23 23 17 23 23 14 17 23 23 – [[Carl Lumma]]'s scale
12
* 14 9 14 17 23 23 23 17 14 9 14 23 17 23 – Pum[14] scale
!
* 16 10 7 7 16 7 7 16 7 10 7 16 7 7 16 7 7 10 16 7 7 16 7 – Ellis duodene union [[11/9]] times the duodene
16/15
9/8
75/64
5/4
4/3
45/32
3/2
8/5
5/3
225/128
15/8
2/1


=== Other scales ===
* 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 – [[Compton]][24]
* 23 31 80 23 83 – [[Indonesian|Balinese]] pentatonic [[pelog]] scale; [[Tolgahan Çoğulu]]'s tuning


! lumma5_240.scl
== Instruments ==
!
A [[Lumatone mapping for 240edo]] is now available.
Carl Lumma's scale aka diadie1, 240edo version
12
!
115.
200.
270.
385.
500.
585.
700.
815.
885.
970.
1085.
1200.
! marvel chords
! [-1, -1, 2]-&gt;[-1, 0, -2]||[0, -1, -1]-&gt;[0, 0, -1]-&gt;[0, 0, 0]-&gt;[0, 0, 1]-&gt;[0, 0, 2]


! pum14.scl
== Music ==
pum14 scale
; [[Chris Charles]] (via [https://www.youtube.com/@microtonalguitar Microtonal Guitar - Tolgahan Çoğulu])
14
* [https://www.youtube.com/watch?v=6GoGlj5IyZc ''Balinese Gamelan Music on Microtonal Guitar - Chris Charles''] (2017) (Uses a 5-tone subset of 240edo for all three pieces performed in the recording—as explained in the video description: "''The scale we used in the piece: Pelog Selisir: D, Eb +30 F -15 A -30 Bb -15''".)
!
25/24
16/15
10/9
75/64
5/4
4/3
64/45
3/2
25/16
8/5
5/3
16/9
15/8
2


! pum14_240.scl
; [[Bryan Deister]]
pum14 in 240edo
* [https://www.youtube.com/shorts/Nu-xBrsd8_o ''microtonal improvisation in 240edo''] (2025)
14
!
70.
115.
185.
270.
385.
500.
615.
700.
770.
815.
885.
1000.
1085.
1200.
! tetrads [[0, -1, 0], [0, -1, 1], [1, -1, 1], [1, -1, 2], ! [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]]


! doubleduo.scl
== Trivia ==
Ellis duodene union 11/9 times the duodene in 240et
[[Shaahin Mohajeri]], an [[Arabic, Turkish, Persian music|Iranian]] Tombak player and composer, calls his personal [https://sites.google.com/site/240edo/ Google site] "240edo", where he makes the point that five cents is a size close to the [[just-noticeable difference]] between pitches.
24
!
35.
115.
165.
200.
235.
315.
350.
385.
465.
500.
550.
585.
665.
700.
735.
815.
850.
885.
935.
1015.
1050.
1085.
1165.
1200.


==Links==
[[Category:Compton]]
[[Shaahin Mohajeri]], an Iranian Tombak player and composer, calls his personal [[http://sites.google.com/site/240edo/|Google site]] "240edo", where he makes the point that five cents is a size close to the [[Just noticeable difference|just noticeable difference]] between pitches.</pre></div>
[[Category:Marvel]]
<h4>Original HTML content:</h4>
<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;240edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 240edo divides the octave into 240 steps of exactly five cents each. One important use for it is in tuning marvel temperament and marvel's extension to spectacle temperament.&lt;br /&gt;
&lt;br /&gt;
If we round off to the nearest five cents, we end up with a &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; (mapping to primes) for 240edo of &amp;lt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/240%20380%20557"&gt;240 380 557&lt;/a&gt; 674|. This tempers out the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_kleisma" rel="nofollow"&gt;septimal kleisma&lt;/a&gt; of 225/224, with low resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4.) Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning. However &lt;a class="wiki_link" href="/197edo"&gt;197edo&lt;/a&gt;, despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 197&amp;amp;240 temperament.&lt;br /&gt;
&lt;br /&gt;
For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have:&lt;br /&gt;
&lt;br /&gt;
3 ~ 2 (11/9)^2&lt;br /&gt;
5 = 2^2 (5/4)&lt;br /&gt;
7 ~ 2 (11/9)^4 (5/4)^2&lt;br /&gt;
11 ~ 2^2 (11/9)^5&lt;br /&gt;
13 ~ 2^3 (11/9)^(-2) (5/4)^4&lt;br /&gt;
17 ~ 2^4 (11/9)^(-3) (5/4)^3&lt;br /&gt;
&lt;br /&gt;
It should be noted that the exponents of 5/4 above are all positive and go no higher than 4.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Scales&lt;/h2&gt;
&lt;br /&gt;
Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament.&lt;br /&gt;
&lt;br /&gt;
! duodene.scl&lt;br /&gt;
!&lt;br /&gt;
Ellis's Duodene : genus [33355]&lt;br /&gt;
12&lt;br /&gt;
!&lt;br /&gt;
16/15&lt;br /&gt;
9/8&lt;br /&gt;
6/5&lt;br /&gt;
5/4&lt;br /&gt;
4/3&lt;br /&gt;
45/32&lt;br /&gt;
3/2&lt;br /&gt;
8/5&lt;br /&gt;
5/3&lt;br /&gt;
9/5&lt;br /&gt;
15/8&lt;br /&gt;
2/1&lt;br /&gt;
&lt;br /&gt;
! duodene240.scl&lt;br /&gt;
!&lt;br /&gt;
Ellis's Duodene : genus [33355] retuned to 240edo&lt;br /&gt;
12&lt;br /&gt;
!&lt;br /&gt;
115.&lt;br /&gt;
200.&lt;br /&gt;
315.&lt;br /&gt;
385.&lt;br /&gt;
500.&lt;br /&gt;
585.&lt;br /&gt;
700.&lt;br /&gt;
815.&lt;br /&gt;
885.&lt;br /&gt;
1015.&lt;br /&gt;
1085.&lt;br /&gt;
1200.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
! lumma5.scl&lt;br /&gt;
!&lt;br /&gt;
Carl Lumma's scale, 5-limit just version, TL 19-2-99&lt;br /&gt;
! Also diadie1, prism, Fokker 12-tone just&lt;br /&gt;
12&lt;br /&gt;
!&lt;br /&gt;
16/15&lt;br /&gt;
9/8&lt;br /&gt;
75/64&lt;br /&gt;
5/4&lt;br /&gt;
4/3&lt;br /&gt;
45/32&lt;br /&gt;
3/2&lt;br /&gt;
8/5&lt;br /&gt;
5/3&lt;br /&gt;
225/128&lt;br /&gt;
15/8&lt;br /&gt;
2/1&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
! lumma5_240.scl&lt;br /&gt;
!&lt;br /&gt;
Carl Lumma's scale aka diadie1, 240edo version&lt;br /&gt;
12&lt;br /&gt;
!&lt;br /&gt;
115.&lt;br /&gt;
200.&lt;br /&gt;
270.&lt;br /&gt;
385.&lt;br /&gt;
500.&lt;br /&gt;
585.&lt;br /&gt;
700.&lt;br /&gt;
815.&lt;br /&gt;
885.&lt;br /&gt;
970.&lt;br /&gt;
1085.&lt;br /&gt;
1200.&lt;br /&gt;
! marvel chords&lt;br /&gt;
! [-1, -1, 2]-&amp;gt;[-1, 0, -2]||[0, -1, -1]-&amp;gt;[0, 0, -1]-&amp;gt;[0, 0, 0]-&amp;gt;[0, 0, 1]-&amp;gt;[0, 0, 2]&lt;br /&gt;
&lt;br /&gt;
! pum14.scl&lt;br /&gt;
pum14 scale&lt;br /&gt;
14&lt;br /&gt;
!&lt;br /&gt;
25/24&lt;br /&gt;
16/15&lt;br /&gt;
10/9&lt;br /&gt;
75/64&lt;br /&gt;
5/4&lt;br /&gt;
4/3&lt;br /&gt;
64/45&lt;br /&gt;
3/2&lt;br /&gt;
25/16&lt;br /&gt;
8/5&lt;br /&gt;
5/3&lt;br /&gt;
16/9&lt;br /&gt;
15/8&lt;br /&gt;
2&lt;br /&gt;
&lt;br /&gt;
! pum14_240.scl&lt;br /&gt;
pum14 in 240edo&lt;br /&gt;
14&lt;br /&gt;
!&lt;br /&gt;
70.&lt;br /&gt;
115.&lt;br /&gt;
185.&lt;br /&gt;
270.&lt;br /&gt;
385.&lt;br /&gt;
500.&lt;br /&gt;
615.&lt;br /&gt;
700.&lt;br /&gt;
770.&lt;br /&gt;
815.&lt;br /&gt;
885.&lt;br /&gt;
1000.&lt;br /&gt;
1085.&lt;br /&gt;
1200.&lt;br /&gt;
! tetrads [[0, -1, 0], [0, -1, 1], [1, -1, 1], [1, -1, 2], ! [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]]&lt;br /&gt;
&lt;br /&gt;
! doubleduo.scl&lt;br /&gt;
Ellis duodene union 11/9 times the duodene in 240et&lt;br /&gt;
24&lt;br /&gt;
!&lt;br /&gt;
35.&lt;br /&gt;
115.&lt;br /&gt;
165.&lt;br /&gt;
200.&lt;br /&gt;
235.&lt;br /&gt;
315.&lt;br /&gt;
350.&lt;br /&gt;
385.&lt;br /&gt;
465.&lt;br /&gt;
500.&lt;br /&gt;
550.&lt;br /&gt;
585.&lt;br /&gt;
665.&lt;br /&gt;
700.&lt;br /&gt;
735.&lt;br /&gt;
815.&lt;br /&gt;
850.&lt;br /&gt;
885.&lt;br /&gt;
935.&lt;br /&gt;
1015.&lt;br /&gt;
1050.&lt;br /&gt;
1085.&lt;br /&gt;
1165.&lt;br /&gt;
1200.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Links&lt;/h2&gt;
&lt;a class="wiki_link" href="/Shaahin%20Mohajeri"&gt;Shaahin Mohajeri&lt;/a&gt;, an Iranian Tombak player and composer, calls his personal &lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/" rel="nofollow"&gt;Google site&lt;/a&gt; &amp;quot;240edo&amp;quot;, where he makes the point that five cents is a size close to the &lt;a class="wiki_link" href="/Just%20noticeable%20difference"&gt;just noticeable difference&lt;/a&gt; between pitches.&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/240edo"