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{{Infobox ET}}
{{Infobox ET}}
The '''94 equal divisions of the octave''' ('''94edo'''), or the '''94(-tone) equal temperament''' ('''94tet''', '''94et''') when viewed from a [[regular temperament]] perspective, results from dividing the [[octave]] into 94 [[equal]]ly-sized steps, where each step is about 12.8 [[cent]]s.
{{EDO intro|94}}


== Theory ==
== Theory ==
Line 7: Line 7:
The list of 23-limit commas it tempers out is huge, but it is worth noting that it tempers out [[32805/32768]] and is thus a [[schismatic]] system, that it tempers out [[225/224]] and [[385/384]] and so is a [[marvel]] system, and that it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[275/273]], and for a number of other temperaments, such as [[isis]].
The list of 23-limit commas it tempers out is huge, but it is worth noting that it tempers out [[32805/32768]] and is thus a [[schismatic]] system, that it tempers out [[225/224]] and [[385/384]] and so is a [[marvel]] system, and that it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[275/273]], and for a number of other temperaments, such as [[isis]].


94edo is an excellent EDO for [[Carlos Beta]] scale, since the difference between 5 steps of 94edo and 1 step of Carlos Beta is only -0.00314534 cents.
94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 5 steps of 94edo and 1 step of Carlos Beta is only -0.00314534 cents.


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|94|columns=11}}
{{Harmonics in equal|94|columns=11}}


== Intervals ==
== Intervals ==
{{main| Table of 94edo intervals }}
{{Main| Table of 94edo intervals }}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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|}
|}


94et is lower in relative error than any previous equal temperaments in the 23-limit, and the next ET that does better in this subgroup is 193.  
94et is lower in relative error than any previous equal temperaments in the 23-limit, and the equal temperament that does better in this subgroup is 193.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all right-3 left-5"
! Periods<br>per octave
! Periods<br>per 8ve
! Generator
! Generator
! Cents
! Cents
! Associated<br>ratio
! Associated<br>Ratio
! Temperament
! Temperament
|-
|-
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* [[Garibaldi17]]
* [[Garibaldi17]]


Since 94edo has a step of 12.766 cents, it also allows one to use its MOS scales as circulating temperaments and is the first edo to allows one to use a Mohajira, Pajara or Miracle MOS scale a as circulating temperament{{clarify}}.
Since 94edo has a step of 12.766 cents, it also allows one to use its mos scales as circulating temperaments and is the first edo to allows one to use a nohajira, pajara or miracle mos scale a as circulating temperament{{clarify}}.
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+Circulating temperaments in 94edo
|+ style=white-space:nowrap | Circulating temperaments in 94edo
!Tones
!Tones
!Pattern
!Pattern
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|}
|}


[[Category:94edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Garibaldi]]
[[Category:Garibaldi]]
[[Category:Marvel]]
[[Category:Marvel]]

Revision as of 08:42, 14 February 2023

← 93edo 94edo 95edo →
Prime factorization 2 × 47
Step size 12.766 ¢ 
Fifth 55\94 (702.128 ¢)
(semiconvergent)
Semitones (A1:m2) 9:7 (114.9 ¢ : 89.36 ¢)
Consistency limit 23
Distinct consistency limit 13

Template:EDO intro

Theory

94edo is a remarkable all-around utility tuning system, good from low prime limit to very high prime limit situations. It is the first edo to be consistent through the 23-odd-limit, and no other edo is so consistent until 282 and 311 make their appearance.

The list of 23-limit commas it tempers out is huge, but it is worth noting that it tempers out 32805/32768 and is thus a schismatic system, that it tempers out 225/224 and 385/384 and so is a marvel system, and that it also tempers out 3125/3087, 4000/3969, 5120/5103 and 540/539. It provides the optimal patent val for the rank-5 temperament tempering out 275/273, and for a number of other temperaments, such as isis.

94edo is an excellent edo for Carlos Beta scale, since the difference between 5 steps of 94edo and 1 step of Carlos Beta is only -0.00314534 cents.

Prime harmonics

Approximation of prime harmonics in 94edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.17 -3.33 +1.39 -2.38 +2.03 -2.83 -3.90 -2.74 +4.47 +3.90
Relative (%) +0.0 +1.4 -26.1 +10.9 -18.7 +15.9 -22.2 -30.5 -21.5 +35.0 +30.6
Steps
(reduced) 		94
(0) 		149
(55) 		218
(30) 		264
(76) 		325
(43) 		348
(66) 		384
(8) 		399
(23) 		425
(49) 		457
(81) 		466
(90)

Intervals

Main article: Table of 94edo intervals

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [149 -94⟩ [⟨94 149]] -0.054 0.054 0.43
2.3.5 32805/32768, 9765625/9565938 [⟨94 149 218]] +0.442 0.704 5.52
2.3.5.7 225/224, 3125/3087, 118098/117649 [⟨94 149 218 264]] +0.208 0.732 5.74
2.3.5.7.11 225/224, 385/384, 1331/1323, 2200/2187 [⟨94 149 218 264 325]] +0.304 0.683 5.35
2.3.5.7.11.13 225/224, 275/273, 325/324, 385/384, 1331/1323 [⟨94 149 218 264 325 348]] +0.162 0.699 5.48
2.3.5.7.11.13.17 170/169, 225/224, 275/273, 289/288, 325/324, 385/384 [⟨94 149 218 264 325 348 384]] +0.238 0.674 5.28
2.3.5.7.11.13.17.19 170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384 [⟨94 149 218 264 325 348 384 399]] +0.323 0.669 5.24
2.3.5.7.11.13.17.19.23 170/169, 190/189, 209/208, 225/224, 275/273, 289/288, 300/299, 323/322 [⟨94 149 218 264 325 348 384 399 425]] +0.354 0.637 4.99

94et is lower in relative error than any previous equal temperaments in the 23-limit, and the equal temperament that does better in this subgroup is 193.

Rank-2 temperaments

Periods
per 8ve 		Generator Cents Associated
Ratio 		Temperament
1 3\94 38.30 49/48 Slender
1 5\94 63.83 25/24 Sycamore / betic
1 11\94 140.43 243/224 Tsaharuk / quanic
1 13\94 165.96 11/10 Tertiaschis
1 19\94 242.55 147/128 Septiquarter
1 39\94 497.87 4/3 Helmholtz / garibaldi / cassandra
2 2\94 25.53 64/63 Ketchup
2 11\94 140.43 27/25 Fifive
2 30\94 382.98 5/4 Wizard / gizzard
2 34\94 434.04 9/7 Pogo / supers
2 43\94 548.94 11/8 Kleischismic

Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for garibaldi temperament, shows us how to extend it to the 23-limit.

  • 46&94 ⟨⟨ 8 30 -18 -4 -28 8 -24 2 … ]]
  • 68&94 ⟨⟨ 20 28 2 -10 24 20 34 52 … ]]
  • 53&94 ⟨⟨ 1 -8 -14 23 20 -46 -3 -35 … ]] (one garibaldi)
  • 41&94 ⟨⟨ 1 -8 -14 23 20 48 -3 -35 … ]] (another garibaldi, only differing in the mappings of 17 and 23)
  • 135&94 ⟨⟨ 1 -8 -14 23 20 48 -3 59 … ]] (another garibaldi)
  • 130&94 ⟨⟨ 6 -48 10 -50 26 6 -18 -22 … ]] (a pogo extension)
  • 58&94 ⟨⟨ 6 46 10 44 26 6 -18 -22 … ]] (a supers extension)
  • 50&94 ⟨⟨ 24 -4 40 -12 10 24 22 6 … ]]
  • 72&94 ⟨⟨ 12 -2 20 -6 52 12 -36 -44 … ]] (a gizzard extension)
  • 80&94 ⟨⟨ 18 44 30 38 -16 18 40 28 … ]]
  • 94 solo ⟨⟨ 12 -2 20 -6 -42 12 -36 -44 … ]] (a rank one temperament!)

Temperaments to which 94et can be detempered:

  • Satin (94&311) ⟨⟨ 3 70 -42 69 -34 50 85 83 … ]]
  • 94&422 ⟨⟨ 8 124 -18 90 -28 102 164 96 … ]]

Scales

Since 94edo has a step of 12.766 cents, it also allows one to use its mos scales as circulating temperaments and is the first edo to allows one to use a nohajira, pajara or miracle mos scale a as circulating temperament[clarification needed].

Circulating temperaments in 94edo
Tones Pattern L:s
5 4L 1s 19:18
6 4L 2s 16:15
7 3L 4s 14:13
8 6L 2s 12:11
9 4L 5s 11:10
10 4L 6s 10:9
11 6L 5s 9:8
12 10L 2s 8:7
13 3L 10s
14 10L 4s 7:6
15 4L 11s
16 14L 2s 6:5
17 9L 8s
18 4L 14s
19 18L 1s 5:4
20 14L 6s
21 10L 11s
22 6L 16s
23 2L 21s
24 22L 2s 4:3
25 19L 6s
26 16L 10s
27 13L 14s
28 10L 18s
29 7L 22s
30 4L 22s
31 1L 30s
32 30L 2s 3:2
33 28L 5s
34 26L 8s
35 24L 11s
36 22L 14s
37 20L 17s
38 18L 20s
39 16L 23s
40 14L 26s
41 13L 28s
42 10L 32s
43 8L 35s
44 6L 38s
45 4L 41s
46 2L 44s
47 47edo equal
48 46L 2s 2:1
49 45L 4s
50 44L 6s
51 43L 8s
52 42L 10s
53 41L 12s
54 40L 14s
55 39L 16s
56 38L 18s
57 37L 20s
58 36L 22s
59 35L 24s
60 34L 26s
61 33L 28s
62 32L 30s
63 31L 32s
64 30L 34s
65 29L 36s
66 28L 38s
67 27L 40s
68 26L 42s
69 25L 44s
70 24L 46s
71 23L 48s
72 22L 50s
73 21L 52s
74 20L 54s
75 19L 56s
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