75edo: Difference between revisions

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== Theory ==
== Theory ==
75et tempers out 20000/19683 ([[tetracot comma]]) and 2109375/2097152 ([[semicomma]]) in the 5-limit, and provides a good tuning for the [[tetracot]] temperament. It tempers out [[225/224]] and [[1728/1715]] in the 7-limit, [[support]]ing [[bunya]] and [[orwell]], and providing the [[optimal patent val]] for [[fog]].  
75et [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) and 2109375/2097152 ([[semicomma]]) in the 5-limit, and provides a good tuning for the [[tetracot]] temperament. It tempers out [[225/224]] and [[1728/1715]] in the 7-limit, [[support]]ing [[bunya]] and [[orwell]], and providing the [[optimal patent val]] for [[fog]].  


In the 11-limit, 75e [[val]] {{val| 75 119 174 211 '''260''' }} scores lower in [[TE error|error]], and tempers [[100/99]] and [[243/242]], whereas the [[patent val]] {{val| 75 119 174 211 '''259''' }} tempers [[99/98]] and [[121/120]]. In the 13-limit, it tempers [[325/324]] and [[512/507]], 17-limit [[120/119]] and [[256/255]] and 19-limit 190/189 and 250/247.
In the 11-limit, 75e [[val]] {{val| 75 119 174 211 '''260''' }} scores lower in [[TE error|error]], and tempers [[100/99]] and [[243/242]], whereas the [[patent val]] {{val| 75 119 174 211 '''259''' }} tempers [[99/98]] and [[121/120]]. In the 13-limit, it tempers [[325/324]] and [[512/507]], 17-limit [[120/119]] and [[256/255]] and 19-limit 190/189 and 250/247.


Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates the [[peppermint]] temperament. The size of its fifth is exactly 704 {{cent}}, which is very close to the peppermint fifth of 704.096 {{cent}}. This makes it suitable for neo-Gothic tunings. It also approximates the [[Carlos Beta]] scale well (<code>4\75 ≈ 1\[Carlos Beta]</code>).
Since 75 is part of the {{w|Fibonacci sequence}} beginning with 5 and 12, it closely approximates the [[peppermint]] temperament. The size of its fifth is exactly 704 cents, which is very close to the peppermint fifth of 704.096 cents. This makes it suitable for neo-Gothic tunings. It also approximates the [[Carlos Beta]] scale well (4\75 ≈ 1\Carlos Beta).


=== Odd harmonics ===
=== Odd harmonics ===
Line 28: Line 28:
| 2.3
| 2.3
| {{monzo| 119 -75 }}
| {{monzo| 119 -75 }}
| [{{val| 75 119 }}]
| {{mapping| 75 119 }}
| -0.645
| -0.645
| 0.645
| 0.645
Line 35: Line 35:
| 2.3.5
| 2.3.5
| 20000/19683, 2109375/2097152
| 20000/19683, 2109375/2097152
| [{{val| 75 119 174 }}]
| {{mapping| 75 119 174 }}
| -0.099
| -0.099
| 0.936
| 0.936
Line 42: Line 42:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1728/1715, 15625/15309
| 225/224, 1728/1715, 15625/15309
| [{{val| 75 119 174 211 }}]
| {{mapping| 75 119 174 211 }}
| -0.713
| -0.713
| 1.337
| 1.337
Line 49: Line 49:


== Music ==
== Music ==
* [https://www.youtube.com/watch?v=oL6K6O4FBxc Fugue on The Lick] by [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=oL6K6O4FBxc ''Fugue on The Lick''] (2019)


[[Category:Listen]]
[[Category:Listen]]

Revision as of 10:00, 27 June 2024

← 74edo 75edo 76edo →
Prime factorization 3 × 52
Step size 16 ¢ 
Fifth 44\75 (704 ¢)
Semitones (A1:m2) 8:5 (128 ¢ : 80 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

75et tempers out 20000/19683 (tetracot comma) and 2109375/2097152 (semicomma) in the 5-limit, and provides a good tuning for the tetracot temperament. It tempers out 225/224 and 1728/1715 in the 7-limit, supporting bunya and orwell, and providing the optimal patent val for fog.

In the 11-limit, 75e val ⟨75 119 174 211 260] scores lower in error, and tempers 100/99 and 243/242, whereas the patent val ⟨75 119 174 211 259] tempers 99/98 and 121/120. In the 13-limit, it tempers 325/324 and 512/507, 17-limit 120/119 and 256/255 and 19-limit 190/189 and 250/247.

Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates the peppermint temperament. The size of its fifth is exactly 704 cents, which is very close to the peppermint fifth of 704.096 cents. This makes it suitable for neo-Gothic tunings. It also approximates the Carlos Beta scale well (4\75 ≈ 1\Carlos Beta).

Odd harmonics

Approximation of odd harmonics in 75edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.04 -2.31 +7.17 +4.09 -7.32 +7.47 -0.27 +7.04 +6.49 -6.78 -4.27
Relative (%) +12.8 -14.5 +44.8 +25.6 -45.7 +46.7 -1.7 +44.0 +40.5 -42.4 -26.7
Steps
(reduced) 		119
(44) 		174
(24) 		211
(61) 		238
(13) 		259
(34) 		278
(53) 		293
(68) 		307
(7) 		319
(19) 		329
(29) 		339
(39)

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 16 ^D, v4E♭
2 32 ^^D, v3E♭
3 48 35/34, 36/35, 37/36, 38/37 ^3D, vvE♭
4 64 27/26, 28/27 ^4D, vE♭
5 80 23/22 v3D♯, E♭
6 96 18/17, 19/18, 37/35 vvD♯, ^E♭
7 112 16/15 vD♯, ^^E♭
8 128 14/13 D♯, ^3E♭
9 144 25/23, 37/34, 38/35 ^D♯, v4E
10 160 34/31 ^^D♯, v3E
11 176 21/19, 31/28 ^3D♯, vvE
12 192 19/17 ^4D♯, vE
13 208 35/31 E
14 224 25/22, 33/29 ^E, v4F
15 240 23/20, 31/27 ^^E, v3F
16 256 29/25, 36/31 ^3E, vvF
17 272 ^4E, vF
18 288 F
19 304 31/26, 37/31 ^F, v4G♭
20 320 ^^F, v3G♭
21 336 17/14 ^3F, vvG♭
22 352 38/31 ^4F, vG♭
23 368 21/17, 26/21 v3F♯, G♭
24 384 5/4 vvF♯, ^G♭
25 400 29/23, 34/27 vF♯, ^^G♭
26 416 F♯, ^3G♭
27 432 9/7 ^F♯, v4G
28 448 35/27 ^^F♯, v3G
29 464 17/13 ^3F♯, vvG
30 480 29/22, 33/25, 37/28 ^4F♯, vG
31 496 4/3 G
32 512 35/26 ^G, v4A♭
33 528 19/14 ^^G, v3A♭
34 544 26/19, 37/27 ^3G, vvA♭
35 560 18/13 ^4G, vA♭
36 576 v3G♯, A♭
37 592 38/27 vvG♯, ^A♭
38 608 27/19, 37/26 vG♯, ^^A♭
39 624 33/23 G♯, ^3A♭
40 640 13/9, 29/20 ^G♯, v4A
41 656 19/13, 35/24 ^^G♯, v3A
42 672 28/19, 31/21 ^3G♯, vvA
43 688 ^4G♯, vA
44 704 3/2 A
45 720 ^A, v4B♭
46 736 26/17 ^^A, v3B♭
47 752 37/24 ^3A, vvB♭
48 768 14/9 ^4A, vB♭
49 784 v3A♯, B♭
50 800 27/17 vvA♯, ^B♭
51 816 8/5 vA♯, ^^B♭
52 832 21/13, 34/21 A♯, ^3B♭
53 848 31/19 ^A♯, v4B
54 864 28/17, 33/20 ^^A♯, v3B
55 880 ^3A♯, vvB
56 896 ^4A♯, vB
57 912 B
58 928 ^B, v4C
59 944 31/18 ^^B, v3C
60 960 ^3B, vvC
61 976 ^4B, vC
62 992 C
63 1008 34/19 ^C, v4D♭
64 1024 38/21 ^^C, v3D♭
65 1040 31/17 ^3C, vvD♭
66 1056 35/19 ^4C, vD♭
67 1072 13/7 v3C♯, D♭
68 1088 15/8 vvC♯, ^D♭
69 1104 17/9, 36/19 vC♯, ^^D♭
70 1120 C♯, ^3D♭
71 1136 27/14 ^C♯, v4D
72 1152 35/18, 37/19 ^^C♯, v3D
73 1168 ^3C♯, vvD
74 1184 ^4C♯, vD
75 1200 2/1 D

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [119 -75⟩ [⟨75 119]] -0.645 0.645 4.03
2.3.5 20000/19683, 2109375/2097152 [⟨75 119 174]] -0.099 0.936 5.85
2.3.5.7 225/224, 1728/1715, 15625/15309 [⟨75 119 174 211]] -0.713 1.337 8.36

Music

Claudi Meneghin
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