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67edo: Difference between revisions

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**Imported revision 276663346 - 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:Kosmorsky|Kosmorsky]] and made on <tt>2011-11-17 15:02:06 UTC</tt>.<br>
: The original revision id was <tt>276663346</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


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 a light meantone 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.


0: 1/1 0.000 unison, perfect prime
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]].
1: 17.910 cents 17.910
 
2: 35.821 cents 35.821
=== Prime harmonics ===
3: 53.731 cents 53.731
{{Harmonics in equal|67|columns=13}}
4: 71.642 cents 71.642
 
5: 89.552 cents 89.552
=== Subsets and supersets ===
6: 107.463 cents 107.463
67edo is the 19th [[prime edo]], following [[61edo]] and before [[71edo]].
7: 125.373 cents 125.373
 
8: 143.284 cents 143.284
== Intervals ==
9: 161.194 cents 161.194
{{Interval table}}
10: 179.104 cents 179.104
 
11: 197.015 cents 197.015
== Notation ==
12: 214.925 cents 214.925
=== Stein–Zimmermann–Gould notation ===
13: 232.836 cents 232.836
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
14: 250.746 cents 250.746
{{Sharpness-sharp5-szg}}
15: 268.657 cents 268.657
 
16: 286.567 cents 286.567
=== Kite's ups and downs notation ===
17: 304.478 cents 304.478
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).
18: 322.388 cents 322.388
{{Sharpness-sharp5a}}
19: 340.299 cents 340.299
 
20: 358.209 cents 358.209
=== Sagittal notation ===
21: 376.119 cents 376.119
==== Evo flavor ====
22: 394.030 cents 394.030
<imagemap>
23: 411.940 cents 411.940
File:67-EDO_Evo_Sagittal.svg
24: 429.851 cents 429.851
desc none
25: 447.761 cents 447.761
rect 80 0 300 50 [[Sagittal_notation]]
26: 465.672 cents 465.672
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
27: 483.582 cents 483.582
rect 20 80 160 106 [[896/891]]
28: 501.493 cents 501.493
rect 160 80 280 106 [[36/35]]
29: 519.403 cents 519.403
rect 280 80 440 106 [[1053/1024]]
30: 537.313 cents 537.313
default [[File:67-EDO_Evo_Sagittal.svg]]
31: 555.224 cents 555.224
</imagemap>
32: 573.134 cents 573.134
 
33: 591.045 cents 591.045
==== Revo flavor ====
34: 608.955 cents 608.955
<imagemap>
35: 626.866 cents 626.866
File:67-EDO_Revo_Sagittal.svg
36: 644.776 cents 644.776
desc none
37: 662.687 cents 662.687
rect 80 0 300 50 [[Sagittal_notation]]
38: 680.597 cents 680.597
rect 300 0 759 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
39: 698.507 cents 698.507
rect 20 80 160 106 [[896/891]]
40: 716.418 cents 716.418
rect 160 80 280 106 [[36/35]]
41: 734.328 cents 734.328
rect 280 80 440 106 [[1053/1024]]
42: 752.239 cents 752.239
default [[File:67-EDO_Revo_Sagittal.svg]]
43: 770.149 cents 770.149
</imagemap>
44: 788.060 cents 788.060
 
45: 805.970 cents 805.970
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.
46: 823.881 cents 823.881
 
47: 841.791 cents 841.791
== Scales ==
48: 859.701 cents 859.701
{{Idiosyncratic terms}}
49: 877.612 cents 877.612
 
50: 895.522 cents 895.522
=== Mos scales ===
51: 913.433 cents 913.433
* Meantone[5]: 11 11 17 11 17
52: 931.343 cents 931.343
* Meantone[7]: 11 11 6 11 11 11 6
53: 949.254 cents 949.254
* Barbados[5], Bustling Docks (original/default tuning): 14 14 11 14 14
54: 967.164 cents 967.164
* Barbados[9]: 11 3 11 3 11 3 11 3 11
55: 985.075 cents 985.075
 
56: 1002.985 cents 1002.985
=== Modmos scales ===
57: 1020.896 cents 1020.896
* Cavernous (original/default tuning): 14 14 11 21 7
58: 1038.806 cents 1038.806
* Formicarium (original/default tuning): 14 7 18 14 14
59: 1056.716 cents 1056.716
* Negri Blues (original/default tuning): 14 14 3 8 14 14
60: 1074.627 cents 1074.627
* Negri Blues Septatonic (original/default tuning): 14 14 3 8 11 3 14
61: 1092.537 cents 1092.537
* Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7
62: 1110.448 cents 1110.448
* Understory (original/default tuning): 14 7 18 7 21
63: 1128.358 cents 1128.358
* Meantone Ionian Pentatonic: 22 6 11 22 6
64: 1146.269 cents 1146.269
* Meantone Minor Melodic: 11 6 11 11 11 11 6
65: 1164.179 cents 1164.179
* Meantone Minor Harmonic: 11 6 11 11 6 16 6
66: 1182.090 cents 1182.090
* Meantone Minor Hexatonic: 11 6 11 11 17 11
67: 2/1 1200.000 octave</pre></div>
* Meantone Dorian Harmonic: 11 6 16 6 11 6 11
<h4>Original HTML content:</h4>
* Meantone Mixolydian Pentatonic: 22 6 11 17 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&lt;br /&gt;
* Meantone Phrygian Dominant: 6 16 6 11 6 11 11
&lt;br /&gt;
* Meantone Phrygian Dominant Hexatonic: 6 16 6 11 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 a light meantone 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;
* Meantone Phrygian Dominant Pentatonic: 22 6 11 6 22
&lt;br /&gt;
* Meantone Phrygian Pentatonic: 6 11 22 6 22
0: 1/1 0.000 unison, perfect prime&lt;br /&gt;
* Meantone Double Harmonic: 6 16 6 11 6 16 6
1: 17.910 cents 17.910&lt;br /&gt;
 
2: 35.821 cents 35.821&lt;br /&gt;
=== Blues scales ===
3: 53.731 cents 53.731&lt;br /&gt;
* [[Lost spirit]]  (approximated from [[31edo]]): 17 11 6 5 13 4 11
4: 71.642 cents 71.642&lt;br /&gt;
* [[Blackened skies]] (approximated from [[72edo]]): 18 10 5 6 5 18 5
5: 89.552 cents 89.552&lt;br /&gt;
* Blues Aeolian Hexatonic: 17 11 6 5 6 22
6: 107.463 cents 107.463&lt;br /&gt;
* Blues Aeolian Pentatonic I: 17 11 11 6 22
7: 125.373 cents 125.373&lt;br /&gt;
* Blues Aeolian Pentatonic II: 17 22 6 11 11
8: 143.284 cents 143.284&lt;br /&gt;
* Blues Bright Double Harmonic: 6 16 6 11 6 11 6 5
9: 161.194 cents 161.194&lt;br /&gt;
* Blues Dark Double Harmonic: 11 6 11 6 5 6 16 6
10: 179.104 cents 179.104&lt;br /&gt;
* Blues Dorian Hexatonic: 17 11 11 11 6 11
11: 197.015 cents 197.015&lt;br /&gt;
* Blues Dorian Pentatonic: 17 22 11 6 11
12: 214.925 cents 214.925&lt;br /&gt;
* Blues Dorian Septatonic: 17 11 6 5 11 6 11
13: 232.836 cents 232.836&lt;br /&gt;
* Blues Harmonic Hexatonic: 11 6 11 11 22 6
14: 250.746 cents 250.746&lt;br /&gt;
* Blues Harmonic Septatonic: 17 11 6 5 6 11 5 6
15: 268.657 cents 268.657&lt;br /&gt;
* Blues Leading: 17 11 6 5 17 6 5
16: 286.567 cents 286.567&lt;br /&gt;
* Blues Minor: 17 11 6 5 17 11
17: 304.478 cents 304.478&lt;br /&gt;
* Blues Minor Maj7: 17 11 6 5 22 6
18: 322.388 cents 322.388&lt;br /&gt;
* Blues Pentachordal: 11 6 11 5 6 28
19: 340.299 cents 340.299&lt;br /&gt;
* Greyed Skies (approximated from [[91edo]]): 17 11 5 6 6 17 5
20: 358.209 cents 358.209&lt;br /&gt;
* Akebono I: 11 6 11 11 17
21: 376.119 cents 376.119&lt;br /&gt;
* Augmented: 17 6 16 6 16 6
22: 394.030 cents 394.030&lt;br /&gt;
* Dominant Pentatonic: 11 11 17 17 11
23: 411.940 cents 411.940&lt;br /&gt;
* Hirajoshi: 11 6 12 6 22
24: 429.851 cents 429.851&lt;br /&gt;
* Javanese Pentachordal: 6 11 17 4 29
25: 447.761 cents 447.761&lt;br /&gt;
 
26: 465.672 cents 465.672&lt;br /&gt;
=== Others ===
27: 483.582 cents 483.582&lt;br /&gt;
* Approximation of ''[[Pelog]] lima'': 6 10 22 7 22
28: 501.493 cents 501.493&lt;br /&gt;
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] ''(octave-reduced: 9 6 23 16 13)''
29: 519.403 cents 519.403&lt;br /&gt;
* Arcade (approximated from [[32afdo]]): 22 4 13 15 13
30: 537.313 cents 537.313&lt;br /&gt;
* Cosmic (approximated from [[32afdo]]): 29 10 6 11 11
31: 555.224 cents 555.224&lt;br /&gt;
* Mechanical (approximated from [[16afdo]]): 17 5 17 15 13
32: 573.134 cents 573.134&lt;br /&gt;
* Moonbeam (approximated from [[16afdo]]): 11 6 12 22 6
33: 591.045 cents 591.045&lt;br /&gt;
* Springwater (approximated from [[8afdo]]): 11 11 17 15 13
34: 608.955 cents 608.955&lt;br /&gt;
* Volcanic (approximated from [[16afdo]]): 6 16 17 15 13
35: 626.866 cents 626.866&lt;br /&gt;
* Deja Vu (approximated from [[101afdo]]): 18 21 6 12 10
36: 644.776 cents 644.776&lt;br /&gt;
* Freeway (approximated from [[6afdo]]): 15 12 11 11 9 8
37: 662.687 cents 662.687&lt;br /&gt;
* Mushroom (approximated from [[30afdo]]): 15 12 11 4 24
38: 680.597 cents 680.597&lt;br /&gt;
* Underpass (approximated from [[10afdo]]): 18 21 12 6 10
39: 698.507 cents 698.507&lt;br /&gt;
* Sourgummy (approximated from [[51afdo]]): 14 12 14 14 13
40: 716.418 cents 716.418&lt;br /&gt;
* Bubblegum/Cola (approximated from [[60afdo]]/[[99afdo]]): 14 13 13 13 14
41: 734.328 cents 734.328&lt;br /&gt;
* Tropicalpunch/Whitechocolate (approximated from [[62afdo]]/[[90afdo]]): 13 14 13 14 13
42: 752.239 cents 752.239&lt;br /&gt;
* Lemonade (approximated from [[79afdo]]): 14 13 13 14 13
43: 770.149 cents 770.149&lt;br /&gt;
* Candycorn (approximated from [[91afdo]]): 11 12 11 10 12 11
44: 788.060 cents 788.060&lt;br /&gt;
* Trailmix (approximated from [[97afdo]]): 11 11 11 12 11 11
45: 805.970 cents 805.970&lt;br /&gt;
* Liquorice (approximated from [[101afdo]]): 11 11 12 10 12 11
46: 823.881 cents 823.881&lt;br /&gt;
* Fishcracker (approximated from [[80afdo]]): 9 11 9 9 10 9 10
47: 841.791 cents 841.791&lt;br /&gt;
 
48: 859.701 cents 859.701&lt;br /&gt;
== Instruments ==
49: 877.612 cents 877.612&lt;br /&gt;
* [[Lumatone mapping for 67edo]]
50: 895.522 cents 895.522&lt;br /&gt;
 
51: 913.433 cents 913.433&lt;br /&gt;
== Music ==
52: 931.343 cents 931.343&lt;br /&gt;
; [[Bryan Deister]]
53: 949.254 cents 949.254&lt;br /&gt;
* [https://www.youtube.com/shorts/uwxey9_jINA ''microtonal improvisation in 67edo''] (2025)
54: 967.164 cents 967.164&lt;br /&gt;
* [https://www.youtube.com/shorts/L6BXGZyvK8Y ''67edo prelude''] (2025)
55: 985.075 cents 985.075&lt;br /&gt;
* [https://www.youtube.com/shorts/za_Ov95HbjQ ''improv in 67edo''] (2025)
56: 1002.985 cents 1002.985&lt;br /&gt;
 
57: 1020.896 cents 1020.896&lt;br /&gt;
; [[Delta Quartz]]
58: 1038.806 cents 1038.806&lt;br /&gt;
* [https://youtu.be/WOguarC1lEI ''Making microtonality accessible - "Keep It Tight"''] (2026) (also has a small amount of 24edo)
59: 1056.716 cents 1056.716&lt;br /&gt;
 
60: 1074.627 cents 1074.627&lt;br /&gt;
; [[Dolores Catherino]]
61: 1092.537 cents 1092.537&lt;br /&gt;
* [https://youtu.be/AYHpxeM6o_g ''Moments of Unexpected Beauty''] (2026)
62: 1110.448 cents 1110.448&lt;br /&gt;
 
63: 1128.358 cents 1128.358&lt;br /&gt;
; [[Peter Kosmorsky]]
64: 1146.269 cents 1146.269&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}}
65: 1164.179 cents 1164.179&lt;br /&gt;
 
66: 1182.090 cents 1182.090&lt;br /&gt;
; [[Budjarn Lambeth]]
67: 2/1 1200.000 octave&lt;/body&gt;&lt;/html&gt;</pre></div>
* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation] (2024)
 
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Meantone]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/67edo"