68edo: Difference between revisions

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{{Infobox ET
{{Infobox ET
| Prime factorization = 2<sup>2</sup> × 17
| Prime factorization = 2<sup>2</sup> × 17
| Step size = 17.64706¢
| Step size = 17.6471¢
| Fifth = 40\68 (705.88¢) (→ [[17edo|10\17]])
| Fifth = 40\68 (705.9¢) (→ [[17edo|10\17]])
| Semitones = 8:4 (141.18¢ : 70.59¢)
| Semitones = 8:4 (141.2¢ : 70.6¢)
| Consistency = 9
| Consistency = 9
| Monotonicity = 27
}}
}}
The '''68 equal divisions of the octave''' ('''68edo'''), or the '''68(-tone) equal temperament''' ('''68tet''', '''68et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 68 [[equal]]ly-sized steps. Each step represents a frequency ratio of 17.65 [[cent]]s.  
The '''68 equal divisions of the octave''' ('''68edo'''), or the '''68(-tone) equal temperament''' ('''68tet''', '''68et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 68 [[equal]]ly-sized steps. Each step represents a frequency ratio of 17.65 [[cent]]s.  
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68edo's step is half of the step size of [[34edo]], which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of [[17edo]], which does well in the [[3-limit]], but not so well in the [[5-limit]]. The luck continues: 68 is a strong [[7-limit]] system, but does not do as well for in [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly.
68edo's step is half of the step size of [[34edo]], which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of [[17edo]], which does well in the [[3-limit]], but not so well in the [[5-limit]]. The luck continues: 68 is a strong [[7-limit]] system, but does not do as well for in [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly.


As a 7-limit system it tempers out [[Diaschisma|2048/2025]], [[245/243]], 4000/3969, [[15625/15552]], [[3136/3125]], [[6144/6125]] and [[2401/2400]]. It [[support]]s [[octacot]], [[shrutar]], [[hemiwürschmidt]], [[hemikleismic]], [[clyde]] and [[neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]]. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp.
As a 7-limit system it tempers out [[2048/2025]], [[245/243]], 4000/3969, [[15625/15552]], [[3136/3125]], [[6144/6125]] and [[2401/2400]]. It [[support]]s [[octacot]], [[shrutar]], [[hemiwürschmidt]], [[hemikleismic]], [[clyde]] and [[neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]]. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp.


The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching|stretched 23edo]]. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents).
The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching|stretched 23edo]]. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents).


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|68}}
{{Harmonics in equal|68}}


== Intervals ==
== Intervals ==
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| 1200.00
| 1200.00
| 2/1
| 2/1
|}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 245/243, 2048/2025, 2401/2400
| [{{val| 68 108 158 191 }}]
| -0.983
| 0.915
| 5.19
|}
|}


Revision as of 13:41, 23 April 2022

← 67edo 68edo 69edo →
Prime factorization 22 × 17
Step size 17.6471 ¢ 
Fifth 40\68 (705.882 ¢) (→ 10\17)
Semitones (A1:m2) 8:4 (141.2 ¢ : 70.59 ¢)
Consistency limit 9
Distinct consistency limit 9

The 68 equal divisions of the octave (68edo), or the 68(-tone) equal temperament (68tet, 68et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 68 equally-sized steps. Each step represents a frequency ratio of 17.65 cents.

Theory

68edo's step is half of the step size of 34edo, which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of 17edo, which does well in the 3-limit, but not so well in the 5-limit. The luck continues: 68 is a strong 7-limit system, but does not do as well for in 11-limit; though it's certainly usable for that purpose, it does not represent the 11-limit diamond consistently.

As a 7-limit system it tempers out 2048/2025, 245/243, 4000/3969, 15625/15552, 3136/3125, 6144/6125 and 2401/2400. It supports octacot, shrutar, hemiwürschmidt, hemikleismic, clyde and neptune temperaments, and supplies the optimal patent val for 11-limit hemikleismic. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp.

The 3rd degree of 68edo can be used as a generator for stretched 23edo. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents).

Prime harmonics

Approximation of prime harmonics in 68edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +3.93 +1.92 +1.76 -4.26 +6.53 +0.93 +2.49 +7.02 -6.05 +2.02
Relative (%) +0.0 +22.3 +10.9 +10.0 -24.1 +37.0 +5.3 +14.1 +39.8 -34.3 +11.5
Steps
(reduced) 		68
(0) 		108
(40) 		158
(22) 		191
(55) 		235
(31) 		252
(48) 		278
(6) 		289
(17) 		308
(36) 		330
(58) 		337
(65)

Intervals

Degrees Cents Approximate Ratios
0 0.00 1/1
1 17.65 64/63, 126/125, 225/224
2 35.29 81/80, 49/48, 50/49
3 52.94 28/27, 36/35, 33/32
4 70.59 25/24, 22/21
5 88.24 21/20, 19/18, 20/19
6 105.88 16/15, 17/16, 18/17
7 123.53 15/14, 14/13
8 141.18 13/12
9 158.82 12/11, 11/10
10 176.47 10/9
11 194.12 28/25, 19/17
12 211.76 9/8
13 229.41 8/7
14 247.06 15/13
15 264.71 7/6
16 282.35 20/17
17 300.00 13/11, 19/16
18 317.65 6/5
19 335.29 11/9, 40/33, 17/14
20 352.94 16/13, 39/32
21 370.59 27/22, 26/21, 21/17
22 388.24 5/4
23 405.88 24/19, 19/15
24 423.53 14/11
25 441.18 9/7
26 458.82 13/10, 17/13
27 476.47 21/16
28 494.12 4/3
29 511.76 75/56
30 529.41 27/20, 19/14
31 547.06 11/8, 15/11
32 564.71 25/18, 18/13, 26/19
33 582.35 7/5
34 600.00 17/12, 24/17
35 617.65 10/7
36 635.29 36/25, 13/9, 19/13
37 652.94 16/11, 22/15
38 670.59 40/27, 28/19
39 688.24 112/75
40 705.88 3/2
41 723.53 32/21
42 741.18 16/13, 26/17
43 758.82 14/9
44 776.47 11/7
45 794.12 19/12, 30/19
46 811.76 8/5
47 829.41 44/27, 21/13, 34/21
48 847.06 13/8, 64/39
49 864.71 18/11, 33/20, 28/17
50 882.35 5/3
51 900.00 22/13, 32/19
52 917.65 17/10
53 935.29 12/7
54 952.94 26/15
55 970.59 7/4
56 988.24 16/9
57 1005.88 25/14, 34/19
58 1023.53 9/5
59 1041.18 11/6, 20/11
60 1058.82 24/13
61 1076.47 28/15, 13/7
62 1094.12 15/8, 32/17, 17/9
63 1111.76 40/21, 36/19, 19/10
64 1129.41 48/25, 21/11
65 1147.06 27/14, 35/18, 64/33
66 1164.71 160/81, 96/49, 49/25
67 1182.35 63/32, 125/64, 448/225
68 1200.00 2/1

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5.7 245/243, 2048/2025, 2401/2400 [⟨68 108 158 191]] -0.983 0.915 5.19

Scales

Diatonic scales

Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C5)

Superpyth: 12 12 4 12 12 12 4

Superpyth quarter octave: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

Flattone: 10 10 9 10 10 10 9

Inverse: 8 8 14 8 8 8 14

Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7

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