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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro}}
{{ED intro}}


== Theory ==
== 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 & composite [[55edo]] as an approximation). It is indistinguishable from 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 [[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 & 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.


It is 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 and an orgone temperament ([[26edo]] could be called meantone, but it is more of a [[flattone]]). 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. 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]].
It is 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 and an orgone temperament ([[26edo]] could be called meantone, but it is more of a [[flattone]]). 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]].


=== Prime harmonics ===
=== Prime harmonics ===
Line 17: Line 17:


== Notation ==
== Notation ==
=== Ups and downs notation ===
[[Ups and downs notation]] uses free-standing arrows, but the arrows can be connected to conventional accidentals using [[Helmholtz–Ellis notation|Helmholtz–Ellis]] accidentals:
{{Sharpness-sharp5}}
Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with triple arrows.


===Sagittal notation===
=== Sagittal notation ===
====Evo flavor====
==== Evo flavor ====
 
<imagemap>
<imagemap>
File:67-EDO_Evo_Sagittal.svg
File:67-EDO_Evo_Sagittal.svg
Line 32: Line 35:
</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:67-EDO_Revo_Sagittal.svg
File:67-EDO_Revo_Sagittal.svg
Line 117: Line 119:
* Apple Mint (approximated from [[80afdo]]): 9 11 9 9 10 9 10
* Apple Mint (approximated from [[80afdo]]): 9 11 9 9 10 9 10


==Music==
== Music ==
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3 Beginning of a piece in 67 tone], [[Peter Kosmorsky]] {{dead link}}
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3 Beginning of a piece in 67 tone], [[Peter Kosmorsky]] {{dead link}}
* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation], [[Budjarn Lambeth]]
* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation], [[Budjarn Lambeth]]

Revision as of 15:49, 20 February 2025

← 66edo 67edo 68edo →
Prime factorization 67 (prime)
Step size 17.9104 ¢ 
Fifth 39\67 (698.507 ¢)
Semitones (A1:m2) 5:6 (89.55 ¢ : 107.5 ¢)
Consistency limit 3
Distinct consistency limit 3

67 equal divisions of the octave (abbreviated 67edo or 67ed2), also called 67-tone equal temperament (67tet) or 67 equal temperament (67et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 67 equal parts of about 17.9 ¢ each. Each step represents a frequency ratio of 21/67, or the 67th root of 2.

Theory

67edo tempers out 81/80, supporting meantone, with a tuning which is slightly sharp of 1/6-comma (the tuning favored by Mozart and contemporaries, though they suggested the flatter & composite 55edo as an approximation). It is indistinguishable from 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.

It is 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 and an orgone temperament (26edo could be called meantone, but it is more of a flattone). 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. It does well on the 2.3.7.11.13.17.23.31.37.41 subgroup.

Prime harmonics

Approximation of prime harmonics in 67edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41
Error Absolute (¢) +0.00 -3.45 +7.72 -1.66 +3.91 +1.26 +2.51 +6.96 -1.41 -8.68 +1.23 -0.60 +0.79
Relative (%) +0.0 -19.2 +43.1 -9.3 +21.8 +7.1 +14.0 +38.9 -7.9 -48.5 +6.9 -3.3 +4.4
Steps
(reduced) 		67
(0) 		106
(39) 		156
(22) 		188
(54) 		232
(31) 		248
(47) 		274
(6) 		285
(17) 		303
(35) 		325
(57) 		332
(64) 		349
(14) 		359
(24)

Subsets and supersets

67edo is the 19th prime edo, following 61edo and before 71edo.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 17.9 ^D, E♭♭
2 35.8 ^^D, ^E♭♭
3 53.7 31/30, 32/31, 33/32, 34/33, 35/34 vvD♯, ^^E♭♭
4 71.6 24/23 vD♯, vvE♭
5 89.6 20/19 D♯, vE♭
6 107.5 17/16, 33/31 ^D♯, E♭
7 125.4 14/13, 29/27 ^^D♯, ^E♭
8 143.3 vvD𝄪, ^^E♭
9 161.2 11/10, 23/21, 34/31 vD𝄪, vvE
10 179.1 31/28 D𝄪, vE
11 197 E
12 214.9 17/15, 26/23 ^E, F♭
13 232.8 8/7 ^^E, ^F♭
14 250.7 15/13, 22/19 vvE♯, ^^F♭
15 268.7 7/6 vE♯, vvF
16 286.6 13/11, 33/28 E♯, vF
17 304.5 31/26 F
18 322.4 ^F, G♭♭
19 340.3 28/23 ^^F, ^G♭♭
20 358.2 16/13 vvF♯, ^^G♭♭
21 376.1 36/29 vF♯, vvG♭
22 394 F♯, vG♭
23 411.9 19/15, 33/26 ^F♯, G♭
24 429.9 ^^F♯, ^G♭
25 447.8 22/17 vvF𝄪, ^^G♭
26 465.7 17/13 vF𝄪, vvG
27 483.6 F𝄪, vG
28 501.5 4/3 G
29 519.4 23/17, 31/23 ^G, A♭♭
30 537.3 15/11 ^^G, ^A♭♭
31 555.2 11/8, 29/21 vvG♯, ^^A♭♭
32 573.1 32/23 vG♯, vvA♭
33 591 31/22 G♯, vA♭
34 609 ^G♯, A♭
35 626.9 23/16, 33/23 ^^G♯, ^A♭
36 644.8 16/11 vvG𝄪, ^^A♭
37 662.7 22/15 vG𝄪, vvA
38 680.6 34/23 G𝄪, vA
39 698.5 3/2 A
40 716.4 ^A, B♭♭
41 734.3 26/17 ^^A, ^B♭♭
42 752.2 17/11 vvA♯, ^^B♭♭
43 770.1 vA♯, vvB♭
44 788.1 30/19 A♯, vB♭
45 806 35/22 ^A♯, B♭
46 823.9 29/18 ^^A♯, ^B♭
47 841.8 13/8 vvA𝄪, ^^B♭
48 859.7 23/14 vA𝄪, vvB
49 877.6 A𝄪, vB
50 895.5 B
51 913.4 22/13 ^B, C♭
52 931.3 12/7 ^^B, ^C♭
53 949.3 19/11, 26/15 vvB♯, ^^C♭
54 967.2 7/4 vB♯, vvC
55 985.1 23/13, 30/17 B♯, vC
56 1003 C
57 1020.9 ^C, D♭♭
58 1038.8 20/11, 31/17 ^^C, ^D♭♭
59 1056.7 35/19 vvC♯, ^^D♭♭
60 1074.6 13/7 vC♯, vvD♭
61 1092.5 32/17 C♯, vD♭
62 1110.4 19/10 ^C♯, D♭
63 1128.4 23/12 ^^C♯, ^D♭
64 1146.3 31/16, 33/17 vvC𝄪, ^^D♭
65 1164.2 vC𝄪, vvD
66 1182.1 C𝄪, vD
67 1200 2/1 D

Notation

Ups and downs notation

Ups and downs notation uses free-standing arrows, but the arrows can be connected to conventional accidentals using Helmholtz–Ellis accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
Flat symbol

Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with triple arrows.

Sagittal notation

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation896/89136/351053/1024

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation896/89136/351053/1024

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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.

Scales

Mos scales

  • Meantone[5]: 11 11 17 11 17
  • Meantone[7]: 11 11 6 11 11 11 6
  • Barbados[5], Bustling Docks (original/default tuning): 14 14 11 14 14

Modmos scales

  • Cavernous (original/default tuning): 14 14 11 21 7
  • Formicarium (original/default tuning): 14 7 18 14 14
  • Negri Blues (original/default tuning): 14 14 3 8 14 14
  • Negri Blues Septatonic (original/default tuning): 14 14 3 8 11 3 14
  • Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7
  • Understory (original/default tuning): 14 7 18 7 21
  • Meantone Ionian Pentatonic: 22 6 11 22 6
  • Meantone Minor Melodic: 11 6 11 11 11 11 6
  • Meantone Minor Harmonic: 11 6 11 11 6 16 6
  • Meantone Minor Hexatonic: 11 6 11 11 17 11
  • Meantone Dorian Harmonic: 11 6 16 6 11 6 11
  • Meantone Mixolydian Pentatonic: 22 6 11 17 11
  • Meantone Phrygian Dominant: 6 16 6 11 6 11 11
  • Meantone Phrygian Dominant Hexatonic: 6 16 6 11 6 22
  • Meantone Phrygian Dominant Pentatonic: 22 6 11 6 22
  • Meantone Phrygian Pentatonic: 6 11 22 6 22
  • Meantone Double Harmonic: 6 16 6 11 6 16 6

Blues scales

  • Lost spirit (approximated from 31edo): 17 11 6 5 13 4 11
  • Blackened skies (approximated from 72edo): 18 10 5 6 5 18 5
  • Blues Aeolian Hexatonic: 17 11 6 5 6 22
  • Blues Aeolian Pentatonic I: 17 11 11 6 22
  • Blues Aeolian Pentatonic II: 17 22 6 11 11
  • Blues Bright Double Harmonic: 6 16 6 11 6 11 6 5
  • Blues Dark Double Harmonic: 11 6 11 6 5 6 16 6
  • Blues Dorian Hexatonic: 17 11 11 11 6 11
  • Blues Dorian Pentatonic: 17 22 11 6 11
  • Blues Dorian Septatonic: 17 11 6 5 11 6 11
  • Blues Harmonic Hexatonic: 11 6 11 11 22 6
  • Blues Harmonic Septatonic: 17 11 6 5 6 11 5 6
  • Blues Leading: 17 11 6 5 17 6 5
  • Blues Minor: 17 11 6 5 17 11
  • Blues Minor Maj7: 17 11 6 5 22 6
  • Blues Pentachordal: 11 6 11 5 6 28
  • Greyed Skies (approximated from 91edo): 17 11 5 6 6 17 5
  • Akebono I: 11 6 11 11 17
  • Augmented: 17 6 16 6 16 6
  • Dominant Pentatonic: 11 11 17 17 11
  • Hirajoshi: 11 6 12 6 22
  • Javanese Pentachordal: 6 11 17 4 29

Others

  • Approximation of Pelog lima: 6 10 22 7 22
  • Arcade (approximated from 32afdo): 22 4 13 15 13
  • Cosmic (approximated from 32afdo): 29 10 6 11 11
  • Mechanical (approximated from 16afdo): 17 5 17 15 13
  • Moonbeam (approximated from 16afdo): 11 6 12 22 6
  • Springwater (approximated from 8afdo): 11 11 17 15 13
  • Volcanic (approximated from 16afdo): 6 16 17 15 13
  • Deja Vu (approximated from 101afdo): 18 21 6 12 10
  • Freeway (approximated from 6afdo): 15 12 11 11 9 8
  • Mushroom (approximated from 30afdo): 15 12 11 4 24
  • Underpass (approximated from 10afdo): 18 21 12 6 10
  • Sourgummy (approximated from 51afdo): 14 12 14 14 13
  • Bubblegum/Cola (approximated from 60afdo/99afdo): 14 13 13 13 14
  • Spearmint/Whitechocolate (approximated from 62afdo/90afdo): 13 14 13 14 13
  • Lemonade (approximated from 79afdo): 14 13 13 14 13
  • Candycorn (approximated from 91afdo): 11 12 11 10 12 11
  • Trailmix (approximated from 97afdo): 11 11 11 12 11 11
  • Liquorice (approximated from 101afdo): 11 11 12 10 12 11
  • Apple Mint (approximated from 80afdo): 9 11 9 9 10 9 10

Music

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