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67edo: Difference between revisions

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**Imported revision 288887153 - Original comment: **
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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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-12-31 02:06:45 UTC</tt>.<br>
: The original revision id was <tt>288887153</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">67 equal divisions of the octave divides the octave into 67 equal parts of 17.910 cents each. It tempers out 81/80, supporting meantone temperament, with a tuning which is approximately 1/6 comma (the tuning favored by Mozart and contemporaries), or 0.16 comma, meantone. In the 7-limit the patent val tempers out 1029/1024 and 1728/1715, so that it supports [[Meantone family|mothra temperament]]. In the 11-limit it tempers out 176/175 and 540/539, supporting mosura, an alternative 11-limit mothra. In the 13-limit it tempers out 144/143 and 196/195, supporting 13-limit mosura. It tempers out the orgonisma, and on the 2.7.11 subgroup it supports [[Orgonia|orgone temperament]].


A promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the first edo to have both meantone (26 could be called meantone, but it's more of a flattone) and an orgone temperament. It has relatively good approximations of the 3rd, 7th, 11th, 13th, 15th, 17th harmonics, although the 5th, 9th, and 19th as well as certain higher ones are workable as well. 33+34 can be used to construct this temperament explaining some of its properties.
== Theory ==
67edo [[tempering out|tempers out]] [[81/80]], [[support]]ing [[meantone]], with a tuning which is slightly sharp of [[1/6-comma meantone|1/6-comma]] (the tuning favored by {{w|Wolfgang Amadeus Mozart|Mozart}} and contemporaries, though they suggested the flatter and composite [[55edo]] as an approximation). It is indistinguishable from {{nowrap|{{frac|4|25}} {{=}} 0.16-comma}} meantone. In the 7-limit the [[patent val]] tempers out [[1029/1024]] and [[1728/1715]], so that it supports [[mothra]]. In the 11-limit it tempers out [[176/175]] and [[540/539]], supporting [[mosura]], an alternative 11-limit mothra. In the 13-limit it tempers out [[144/143]] and [[196/195]], supporting 13-limit mosura. It tempers out the [[orgonisma]], and on the 2.7.11 subgroup it supports the [[orgone]] temperament.


67edo is the 19th [[prime numbers|prime]] edo.
It is a promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the second edo after [[26edo]] to have both meantone and an orgone temperament. It has relatively good approximations of the [[3/1|3rd]], [[7/1|7th]], [[11/1|11th]], [[13/1|13th]], [[15/1|15th]], [[17/1|17th]] [[harmonic]]s, although the [[5/1|5th]], [[9/1|9th]], and [[19/1|19th]] as well as certain higher ones are workable as well. {{nowrap|33 + 34}} can be used to construct this temperament explaining some of its properties. It does well on the 2.3.7.11.13.17.23.31.37.41 [[subgroup]].


Music:
=== Prime harmonics ===
[[http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3|Beginning of a piece in 67 tone]], [[Peter Kosmorsky]]
{{Harmonics in equal|67|columns=13}}


Cents
=== Subsets and supersets ===
67edo is the 19th [[prime edo]], following [[61edo]] and before [[71edo]].


0: 1/1 0.000 unison, perfect prime
== Intervals ==
1: 17.910 cents 17.910
{{Interval table}}
2: 35.821 cents 35.821
 
3: 53.731 cents 53.731
== Notation ==
4: 71.642 cents 71.642
=== Stein–Zimmermann–Gould notation ===
5: 89.552 cents 89.552
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
6: 107.463 cents 107.463
{{Sharpness-sharp5-szg}}
7: 125.373 cents 125.373
 
8: 143.284 cents 143.284
=== Kite's ups and downs notation ===
9: 161.194 cents 161.194
67edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
10: 179.104 cents 179.104
{{Sharpness-sharp5a}}
11: 197.015 cents 197.015
 
12: 214.925 cents 214.925
=== Sagittal notation ===
13: 232.836 cents 232.836
==== Evo flavor ====
14: 250.746 cents 250.746
<imagemap>
15: 268.657 cents 7/6
File:67-EDO_Evo_Sagittal.svg
16: 286.567 cents 286.567
desc none
17: 304.478 cents 304.478
rect 80 0 300 50 [[Sagittal_notation]]
18: 322.388 cents 322.388
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
19: 340.299 cents 340.299
rect 20 80 160 106 [[896/891]]
20: 358.209 cents 358.209
rect 160 80 280 106 [[36/35]]
21: 376.119 cents 5/4 -
rect 280 80 440 106 [[1053/1024]]
22: 394.030 cents 5/4 +
default [[File:67-EDO_Evo_Sagittal.svg]]
23: 411.940 cents 411.940
</imagemap>
24: 429.851 cents 429.851
 
25: 447.761 cents 447.761
==== Revo flavor ====
26: 465.672 cents 21/16
<imagemap>
27: 483.582 cents 483.582
File:67-EDO_Revo_Sagittal.svg
28: 501.493 cents 501.493
desc none
29: 519.403 cents 519.403
rect 80 0 300 50 [[Sagittal_notation]]
30: 537.313 cents 537.313
rect 300 0 759 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
31: 555.224 cents 11/8
rect 20 80 160 106 [[896/891]]
32: 573.134 cents 573.134
rect 160 80 280 106 [[36/35]]
33: 591.045 cents 591.045
rect 280 80 440 106 [[1053/1024]]
34: 608.955 cents 608.955
default [[File:67-EDO_Revo_Sagittal.svg]]
35: 626.866 cents 626.866
</imagemap>
36: 644.776 cents 644.776
 
37: 662.687 cents 662.687
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation #Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.
38: 680.597 cents 680.597
 
39: 698.507 cents 3/2
== Scales ==
40: 716.418 cents 716.418
{{Idiosyncratic terms}}
41: 734.328 cents 734.328
 
42: 752.239 cents 752.239
=== Mos scales ===
43: 770.149 cents 770.149
* Meantone[5]: 11 11 17 11 17
44: 788.060 cents 788.060
* Meantone[7]: 11 11 6 11 11 11 6
45: 805.970 cents 805.970
* Barbados[5], Bustling Docks (original/default tuning): 14 14 11 14 14
46: 823.881 cents 823.881
* Barbados[9]: 11 3 11 3 11 3 11 3 11
47: 841.791 cents 841.791
 
48: 859.701 cents 859.701
=== Modmos scales ===
49: 877.612 cents 877.612
* Cavernous (original/default tuning): 14 14 11 21 7
50: 895.522 cents 895.522
* Formicarium (original/default tuning): 14 7 18 14 14
51: 913.433 cents 913.433
* Negri Blues (original/default tuning): 14 14 3 8 14 14
52: 931.343 cents 931.343
* Negri Blues Septatonic (original/default tuning): 14 14 3 8 11 3 14
53: 949.254 cents 949.254
* Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7
54: 967.164 cents 7/4
* Understory (original/default tuning): 14 7 18 7 21
55: 985.075 cents 985.075
* Meantone Ionian Pentatonic: 22 6 11 22 6
56: 1002.985 cents 1002.985
* Meantone Minor Melodic: 11 6 11 11 11 11 6
57: 1020.896 cents 1020.896
* Meantone Minor Harmonic: 11 6 11 11 6 16 6
58: 1038.806 cents 1038.806
* Meantone Minor Hexatonic: 11 6 11 11 17 11
59: 1056.716 cents 1056.716
* Meantone Dorian Harmonic: 11 6 16 6 11 6 11
60: 1074.627 cents 1074.627
* Meantone Mixolydian Pentatonic: 22 6 11 17 11
61: 1092.537 cents 1092.537
* Meantone Phrygian Dominant: 6 16 6 11 6 11 11
62: 1110.448 cents 1110.448
* Meantone Phrygian Dominant Hexatonic: 6 16 6 11 6 22
63: 1128.358 cents 1128.358
* Meantone Phrygian Dominant Pentatonic: 22 6 11 6 22
64: 1146.269 cents 1146.269
* Meantone Phrygian Pentatonic: 6 11 22 6 22
65: 1164.179 cents 1164.179
* Meantone Double Harmonic: 6 16 6 11 6 16 6
66: 1182.090 cents 1182.090
 
67: 2/1 1200.000 octave</pre></div>
=== Blues scales ===
<h4>Original HTML content:</h4>
* [[Lost spirit]]  (approximated from [[31edo]]): 17 11 6 5 13 4 11
<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;67edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;67 equal divisions of the octave divides the octave into 67 equal parts of 17.910 cents each. It tempers out 81/80, supporting meantone temperament, with a tuning which is approximately 1/6 comma (the tuning favored by Mozart and contemporaries), or 0.16 comma, meantone. In the 7-limit the patent val tempers out 1029/1024 and 1728/1715, so that it supports &lt;a class="wiki_link" href="/Meantone%20family"&gt;mothra temperament&lt;/a&gt;. In the 11-limit it tempers out 176/175 and 540/539, supporting mosura, an alternative 11-limit mothra. In the 13-limit it tempers out 144/143 and 196/195, supporting 13-limit mosura. It tempers out the orgonisma, and on the 2.7.11 subgroup it supports &lt;a class="wiki_link" href="/Orgonia"&gt;orgone temperament&lt;/a&gt;.&lt;br /&gt;
* [[Blackened skies]] (approximated from [[72edo]]): 18 10 5 6 5 18 5
&lt;br /&gt;
* Blues Aeolian Hexatonic: 17 11 6 5 6 22
A promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the first edo to have both meantone (26 could be called meantone, but it's more of a flattone) and an orgone temperament. It has relatively good approximations of the 3rd, 7th, 11th, 13th, 15th, 17th harmonics, although the 5th, 9th, and 19th as well as certain higher ones are workable as well. 33+34 can be used to construct this temperament explaining some of its properties.&lt;br /&gt;
* Blues Aeolian Pentatonic I: 17 11 11 6 22
&lt;br /&gt;
* Blues Aeolian Pentatonic II: 17 22 6 11 11
67edo is the 19th &lt;a class="wiki_link" href="/prime%20numbers"&gt;prime&lt;/a&gt; edo.&lt;br /&gt;
* Blues Bright Double Harmonic: 6 16 6 11 6 11 6 5
&lt;br /&gt;
* Blues Dark Double Harmonic: 11 6 11 6 5 6 16 6
Music:&lt;br /&gt;
* Blues Dorian Hexatonic: 17 11 11 11 6 11
&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3" rel="nofollow"&gt;Beginning of a piece in 67 tone&lt;/a&gt;, &lt;a class="wiki_link" href="/Peter%20Kosmorsky"&gt;Peter Kosmorsky&lt;/a&gt;&lt;br /&gt;
* Blues Dorian Pentatonic: 17 22 11 6 11
&lt;br /&gt;
* Blues Dorian Septatonic: 17 11 6 5 11 6 11
Cents&lt;br /&gt;
* Blues Harmonic Hexatonic: 11 6 11 11 22 6
&lt;br /&gt;
* Blues Harmonic Septatonic: 17 11 6 5 6 11 5 6
0: 1/1 0.000 unison, perfect prime&lt;br /&gt;
* Blues Leading: 17 11 6 5 17 6 5
1: 17.910 cents 17.910&lt;br /&gt;
* Blues Minor: 17 11 6 5 17 11
2: 35.821 cents 35.821&lt;br /&gt;
* Blues Minor Maj7: 17 11 6 5 22 6
3: 53.731 cents 53.731&lt;br /&gt;
* Blues Pentachordal: 11 6 11 5 6 28
4: 71.642 cents 71.642&lt;br /&gt;
* Greyed Skies (approximated from [[91edo]]): 17 11 5 6 6 17 5
5: 89.552 cents 89.552&lt;br /&gt;
* Akebono I: 11 6 11 11 17
6: 107.463 cents 107.463&lt;br /&gt;
* Augmented: 17 6 16 6 16 6
7: 125.373 cents 125.373&lt;br /&gt;
* Dominant Pentatonic: 11 11 17 17 11
8: 143.284 cents 143.284&lt;br /&gt;
* Hirajoshi: 11 6 12 6 22
9: 161.194 cents 161.194&lt;br /&gt;
* Javanese Pentachordal: 6 11 17 4 29
10: 179.104 cents 179.104&lt;br /&gt;
 
11: 197.015 cents 197.015&lt;br /&gt;
=== Others ===
12: 214.925 cents 214.925&lt;br /&gt;
* Approximation of ''[[Pelog]] lima'': 6 10 22 7 22
13: 232.836 cents 232.836&lt;br /&gt;
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] ''(octave-reduced: 9 6 23 16 13)''
14: 250.746 cents 250.746&lt;br /&gt;
* Arcade (approximated from [[32afdo]]): 22 4 13 15 13
15: 268.657 cents 7/6&lt;br /&gt;
* Cosmic (approximated from [[32afdo]]): 29 10 6 11 11
16: 286.567 cents 286.567&lt;br /&gt;
* Mechanical (approximated from [[16afdo]]): 17 5 17 15 13
17: 304.478 cents 304.478&lt;br /&gt;
* Moonbeam (approximated from [[16afdo]]): 11 6 12 22 6
18: 322.388 cents 322.388&lt;br /&gt;
* Springwater (approximated from [[8afdo]]): 11 11 17 15 13
19: 340.299 cents 340.299&lt;br /&gt;
* Volcanic (approximated from [[16afdo]]): 6 16 17 15 13
20: 358.209 cents 358.209&lt;br /&gt;
* Deja Vu (approximated from [[101afdo]]): 18 21 6 12 10
21: 376.119 cents 5/4 -&lt;br /&gt;
* Freeway (approximated from [[6afdo]]): 15 12 11 11 9 8
22: 394.030 cents 5/4 +&lt;br /&gt;
* Mushroom (approximated from [[30afdo]]): 15 12 11 4 24
23: 411.940 cents 411.940&lt;br /&gt;
* Underpass (approximated from [[10afdo]]): 18 21 12 6 10
24: 429.851 cents 429.851&lt;br /&gt;
* Sourgummy (approximated from [[51afdo]]): 14 12 14 14 13
25: 447.761 cents 447.761&lt;br /&gt;
* Bubblegum/Cola (approximated from [[60afdo]]/[[99afdo]]): 14 13 13 13 14
26: 465.672 cents 21/16&lt;br /&gt;
* Tropicalpunch/Whitechocolate (approximated from [[62afdo]]/[[90afdo]]): 13 14 13 14 13
27: 483.582 cents 483.582&lt;br /&gt;
* Lemonade (approximated from [[79afdo]]): 14 13 13 14 13
28: 501.493 cents 501.493&lt;br /&gt;
* Candycorn (approximated from [[91afdo]]): 11 12 11 10 12 11
29: 519.403 cents 519.403&lt;br /&gt;
* Trailmix (approximated from [[97afdo]]): 11 11 11 12 11 11
30: 537.313 cents 537.313&lt;br /&gt;
* Liquorice (approximated from [[101afdo]]): 11 11 12 10 12 11
31: 555.224 cents 11/8&lt;br /&gt;
* Fishcracker (approximated from [[80afdo]]): 9 11 9 9 10 9 10
32: 573.134 cents 573.134&lt;br /&gt;
 
33: 591.045 cents 591.045&lt;br /&gt;
== Instruments ==
34: 608.955 cents 608.955&lt;br /&gt;
* [[Lumatone mapping for 67edo]]
35: 626.866 cents 626.866&lt;br /&gt;
 
36: 644.776 cents 644.776&lt;br /&gt;
== Music ==
37: 662.687 cents 662.687&lt;br /&gt;
; [[Bryan Deister]]
38: 680.597 cents 680.597&lt;br /&gt;
* [https://www.youtube.com/shorts/uwxey9_jINA ''microtonal improvisation in 67edo''] (2025)
39: 698.507 cents 3/2&lt;br /&gt;
* [https://www.youtube.com/shorts/L6BXGZyvK8Y ''67edo prelude''] (2025)
40: 716.418 cents 716.418&lt;br /&gt;
* [https://www.youtube.com/shorts/za_Ov95HbjQ ''improv in 67edo''] (2025)
41: 734.328 cents 734.328&lt;br /&gt;
 
42: 752.239 cents 752.239&lt;br /&gt;
; [[Delta Quartz]]
43: 770.149 cents 770.149&lt;br /&gt;
* [https://youtu.be/WOguarC1lEI ''Making microtonality accessible - "Keep It Tight"''] (2026) (also has a small amount of 24edo)
44: 788.060 cents 788.060&lt;br /&gt;
 
45: 805.970 cents 805.970&lt;br /&gt;
; [[Dolores Catherino]]
46: 823.881 cents 823.881&lt;br /&gt;
* [https://youtu.be/AYHpxeM6o_g ''Moments of Unexpected Beauty''] (2026)
47: 841.791 cents 841.791&lt;br /&gt;
 
48: 859.701 cents 859.701&lt;br /&gt;
; [[Peter Kosmorsky]]
49: 877.612 cents 877.612&lt;br /&gt;
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3 Beginning of a piece in 67 tone] (2011) {{dead link}}
50: 895.522 cents 895.522&lt;br /&gt;
 
51: 913.433 cents 913.433&lt;br /&gt;
; [[Budjarn Lambeth]]
52: 931.343 cents 931.343&lt;br /&gt;
* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation] (2024)
53: 949.254 cents 949.254&lt;br /&gt;
 
54: 967.164 cents 7/4&lt;br /&gt;
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
55: 985.075 cents 985.075&lt;br /&gt;
[[Category:Meantone]]
56: 1002.985 cents 1002.985&lt;br /&gt;
[[Category:Listen]]
57: 1020.896 cents 1020.896&lt;br /&gt;
58: 1038.806 cents 1038.806&lt;br /&gt;
59: 1056.716 cents 1056.716&lt;br /&gt;
60: 1074.627 cents 1074.627&lt;br /&gt;
61: 1092.537 cents 1092.537&lt;br /&gt;
62: 1110.448 cents 1110.448&lt;br /&gt;
63: 1128.358 cents 1128.358&lt;br /&gt;
64: 1146.269 cents 1146.269&lt;br /&gt;
65: 1164.179 cents 1164.179&lt;br /&gt;
66: 1182.090 cents 1182.090&lt;br /&gt;
67: 2/1 1200.000 octave&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/67edo"