47edo: Difference between revisions

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{{Infobox ET
{{Infobox ET
| Prime factorization = 47 is prime
| Prime factorization = 47 (prime)
| Step size = 25.53191¢
| Step size = 25.532¢
| Fifth = 27\47 (689¢)
| Fifth = 27\47 (689¢)
| Major 2nd = 7\47 (179¢)
| Major 2nd = 7\47 (179¢)
Line 9: Line 9:


== Theory ==
== Theory ==
'''47-EDO''' divides the octave into 47 equal parts of 25.5 cents each. It has a fifth which is 12.5933 cents flat, unless you use the alternative fifth which is 12.9386 cents sharp, similar to 35edo. It has therefore not aroused much interest, but its best approximation to 9/8 is actually quite good, one-third of a cent sharp. It does a good job of approximating the 2.9.5.7.33.13.17.57.69 23-limit [[k*N_subgroups|2*47 subgroup]] of the [[23-limit|23-limit]], on which it tempers out the same commas as [[94edo|94edo]]. It provides a good tuning for [[Chromatic_pairs#Baldy|baldy]] and [[Chromatic_pairs#Silver|silver]] temperaments and relatives.
'''47edo''' divides the octave into 47 equal parts of 25.532{{cent}} each. It has a fifth which is 12.593{{cent}} flat, unless you use the alternative fifth which is 12.939{{cent}} sharp, similar to 35edo. It has therefore not aroused much interest, but its best approximation to 9/8 is actually quite good, one-third of a cent sharp. It does a good job of approximating the 2.9.5.7.33.13.17.57.69 23-limit [[k*N_subgroups|2*47 subgroup]] of the [[23-limit]], on which it tempers out the same commas as [[94edo]]. It provides a good tuning for [[Chromatic_pairs#Baldy|baldy]] and [[Chromatic_pairs#Silver|silver]] temperaments and relatives.


47edo is the 15th [[prime_numbers|prime]] edo, following [[43edo|43edo]] and preceding [[53edo|53edo]].
47edo is the 15th [[prime EDO]], following [[43edo]] and preceding [[53edo]].


47edo is a diatonic edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. (Its alternate 5th does as well.) 47edo is one of the most difficult diatonic edos to notate, because no other diatonic edo's 5th is as flat (see [[42edo]] for the opposite extreme).
47edo is a diatonic edo because its 5th falls between 4\7 = 686{{cent}} and 3\5 = 720{{cent}}, as does its alternate 5th as well. 47edo is one of the most difficult diatonic edos to notate, because no other diatonic edos 5th is as flat (see [[42edo]] for the opposite extreme).


A notation using the best 5th has major and minor 2nds of 7 and 6 edosteps respectively, with the naturals creating a 7edo-like scale:
A notation using the best 5th has major and minor 2nds of 7 and 6 edosteps respectively, with the naturals creating a 7edo-like scale:
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D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D
D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D


D# is next to D. This notation requires triple, quadruple and in some keys, quintuple or more sharps and flats. For example, a 0-15-27-38 chord (an approximate 4:5:6:7) on the note three edosteps above D would be spelled either as D#<span style="font-size: 90%; vertical-align: super;">3</span> - F#<span style="font-size: 90%; vertical-align: super;">5</span> - A#<span style="font-size: 90%; vertical-align: super;">3</span> - C# or as Eb<span style="font-size: 90%; vertical-align: super;">4</span> - Gbb - Ab<span style="font-size: 90%; vertical-align: super;">4</span> - Db<span style="font-size: 90%; vertical-align: super;">6</span>. This is an aug-three double-dim-seven chord, written D#<span style="font-size: 90%; vertical-align: super;">3</span>(A3)dd7 or Eb<span style="font-size: 90%; vertical-align: super;">4</span>(A3)dd7. It could also be called a sharp-three triple-flat-seven chord, written D#<span style="font-size: 90%; vertical-align: super;">3</span>(#3)b<span style="font-size: 90%; vertical-align: super;">3</span>7 or Eb<span style="font-size: 90%; vertical-align: super;">4</span>(#3)b<span style="vertical-align: super;">3</span>7.
D# is next to D. This notation requires triple, quadruple and in some keys, quintuple or more sharps and flats. For example, a 0-15-27-38 chord (an approximate 4:5:6:7) on the note three edosteps above D would be spelled either as D#<sup>3</sup> - F#<sup>5</sup> - A#<sup>3</sup> - C# or as Eb<sup>4</sup> - Gbb - Ab<sup>4</sup> - Db<sup>6</sup>. This is an aug-three double-dim-seven chord, written D#<sup>3</sup>(A3)dd7 or Eb<sup>4</sup>(A3)dd7. It could also be called a sharp-three triple-flat-seven chord, written D#<sup>3</sup>(#3)b<sup>3</sup>7 or Eb<sup>4</sup>(#3)b<sup>3</sup>7.


Using the 2nd best 5th is even more awkward. The major 2nd is 9 edosteps and the minor is only one. The naturals create a 5edo-like scale, with two of the notes inflected by a comma-sized edostep:
Using the 2nd best 5th is even more awkward. The major 2nd is 9 edosteps and the minor is only one. The naturals create a 5edo-like scale, with two of the notes inflected by a comma-sized edostep:
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D# is next to E. This notation requires quadruple, quintuple, and even sextuple ups and downs, as well as single sharps and flats.
D# is next to E. This notation requires quadruple, quintuple, and even sextuple ups and downs, as well as single sharps and flats.
{{Odd harmonics in edo|edo=47}}
 
{{Harmonics in equal|47}}


== Intervals ==
== Intervals ==
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|-
|-
| 10
| 10
| 255.31915
| 255.3191
| triple-aug 2nd, triple-dim 3rd
| triple-aug 2nd, triple-dim 3rd
| A<sup>3</sup>2, d<sup>3</sup>3
| A<sup>3</sup>2, d<sup>3</sup>3
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|-
|-
| 12
| 12
| 306.383
| 306.3830
| dim 3rd
| dim 3rd
| d3
| d3
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|-
|-
| 17
| 17
| 434.04255
| 434.0426
| triple-aug 3rd, triple-dim 4th
| triple-aug 3rd, triple-dim 4th
| A<sup>3</sup>3, d<sup>3</sup>4
| A<sup>3</sup>3, d<sup>3</sup>4
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|-
|-
| 30
| 30
| 765.95745
| 765.9574
| triple-aug 5th, triple-dim 6th
| triple-aug 5th, triple-dim 6th
| A<sup>3</sup>5, d<sup>3</sup>6
| A<sup>3</sup>5, d<sup>3</sup>6
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|-
|-
| 35
| 35
| 893.617
| 893.6170
| aug 6th
| aug 6th
| A6
| A6
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|-
|-
| 37
| 37
| 944.68085
| 944.6809
| triple-aug 6th, triple-dim 7th
| triple-aug 6th, triple-dim 7th
| A<sup>3</sup>6, d<sup>3</sup>7
| A<sup>3</sup>6, d<sup>3</sup>7
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|-
|-
| 47
| 47
| 1200
| 1200.0000
| perfect 8ve
| perfect 8ve
| P8
| P8

Revision as of 03:40, 2 July 2022

← 46edo 47edo 48edo →
Prime factorization 47 (prime)
Step size 25.5319 ¢ 
Fifth 27\47 (689.362 ¢)
Semitones (A1:m2) 1:6 (25.53 ¢ : 153.2 ¢)
Dual sharp fifth 28\47 (714.894 ¢)
Dual flat fifth 27\47 (689.362 ¢)
Dual major 2nd 8\47 (204.255 ¢)
(convergent)
Consistency limit 5
Distinct consistency limit 5

Theory

47edo divides the octave into 47 equal parts of 25.532 ¢ each. It has a fifth which is 12.593 ¢ flat, unless you use the alternative fifth which is 12.939 ¢ sharp, similar to 35edo. It has therefore not aroused much interest, but its best approximation to 9/8 is actually quite good, one-third of a cent sharp. It does a good job of approximating the 2.9.5.7.33.13.17.57.69 23-limit 2*47 subgroup of the 23-limit, on which it tempers out the same commas as 94edo. It provides a good tuning for baldy and silver temperaments and relatives.

47edo is the 15th prime EDO, following 43edo and preceding 53edo.

47edo is a diatonic edo because its 5th falls between 4\7 = 686 ¢ and 3\5 = 720 ¢, as does its alternate 5th as well. 47edo is one of the most difficult diatonic edos to notate, because no other diatonic edos 5th is as flat (see 42edo for the opposite extreme).

A notation using the best 5th has major and minor 2nds of 7 and 6 edosteps respectively, with the naturals creating a 7edo-like scale:

D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D

D# is next to D. This notation requires triple, quadruple and in some keys, quintuple or more sharps and flats. For example, a 0-15-27-38 chord (an approximate 4:5:6:7) on the note three edosteps above D would be spelled either as D#3 - F#5 - A#3 - C# or as Eb4 - Gbb - Ab4 - Db6. This is an aug-three double-dim-seven chord, written D#3(A3)dd7 or Eb4(A3)dd7. It could also be called a sharp-three triple-flat-seven chord, written D#3(#3)b37 or Eb4(#3)b37.

Using the 2nd best 5th is even more awkward. The major 2nd is 9 edosteps and the minor is only one. The naturals create a 5edo-like scale, with two of the notes inflected by a comma-sized edostep:

D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D

D# is next to E. This notation requires quadruple, quintuple, and even sextuple ups and downs, as well as single sharps and flats.


Approximation of odd harmonics in 47edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -12.6 -3.3 +1.4 +0.3 +10.4 +2.0 +9.6 -2.8 +8.9 -11.2 +10.0
Relative (%) -49.3 -13.1 +5.4 +1.4 +40.7 +7.9 +37.6 -11.1 +34.7 -43.9 +39.3
Steps
(reduced) 		74
(27) 		109
(15) 		132
(38) 		149
(8) 		163
(22) 		174
(33) 		184
(43) 		192
(4) 		200
(12) 		206
(18) 		213
(25)

Intervals

Degree Size ( ¢) Relative notation Absolute notation
0 0.0000 perfect unison P1 D
1 25.5319 aug 1sn A1 D#
2 51.0638 double-aug 1sn AA1 Dx
3 76.5957 triple-aug 1sn, triple-dim 2nd A31, d32 D#3, Eb4
4 102.1277 double-dim 2nd dd2 Eb3
5 127.6596 dim 2nd d2 Ebb
6 153.1915 minor 2nd m2 Eb
7 178.7234 major 2nd M2 E
8 204.2553 aug 2nd A2 E#
9 229.7872 double-aug 2nd AA2 Ex
10 255.3191 triple-aug 2nd, triple-dim 3rd A32, d33 E#3, Fb3
11 280.8511 double-dim 3rd dd3 Fbb
12 306.3830 dim 3rd d3 Fb
13 331.9149 minor 3rd m3 F
14 357.4468 major 3rd M3 F#
15 382.9787 aug 3rd A3 Fx
16 408.5106 double-aug 3rd AA3 F#3
17 434.0426 triple-aug 3rd, triple-dim 4th A33, d34 F#4, Gb3
18 459.5745 double-dim 4th dd4 Gbb
19 485.1064 dim 4th d4 Gb
20 510.6383 perfect 4th P4 G
21 536.1702 aug 4th A4 G#
22 561.7021 double-aug 4th AA4 Gx
23 587.2340 triple-aug 4th A34 G#3
24 612.7660 triple-dim 5th d35 Ab3
25 638.2979 double-dim 5th dd5 Abb
26 663.8298 dim 5th d5 Ab
27 689.3617 perfect 5th P5 A
28 714.8936 aug 5th A5 A#
29 740.4255 double-aug 5th AA5 Ax
30 765.9574 triple-aug 5th, triple-dim 6th A35, d36 A#3, Bb4
31 791.4894 double-dim 6th dd6 Bb3
32 817.0213 dim 6th d6 Bbb
33 842.5532 minor 6th m6 Bb
34 868.0851 major 6th M6 B
35 893.6170 aug 6th A6 B#
36 919.1489 double-aug 6th AA6 Bx
37 944.6809 triple-aug 6th, triple-dim 7th A36, d37 B#3, Cb3
38 970.2128 double-dim 7th dd7 Cbb
39 995.7447 dim 7th d7 Cb
40 1021.2766 minor 7th m7 C
41 1046.8085 major 7th M7 C#
42 1072.3404 aug 7th A7 Cx
43 1097.8723 double-aug 7th AA7 C#3
44 1123.4043 triple-aug 7th, triple-dim 8ve A37, d38 C#4, Db3
45 1148.9362 double-dim 8ve dd8 Dbb
46 1174.4681 dim 8ve d8 Db
47 1200.0000 perfect 8ve P8 D
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