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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 47 (prime)
{{ED intro}}
| Step size = 25.532¢
| Fifth = 27\47 (689¢)
| Major 2nd = 7\47 (179¢)
| Minor 2nd = 6\47 (153¢)
| Augmented 1sn = 1\47 (26¢)
}}


== Theory ==
== Theory ==
'''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 first edo that has two [[5L 2s|diatonic]] perfect fifths, as both fall between {{nowrap|4\7 {{=}} 686{{c}}}} and {{nowrap|3\5 {{=}} 720{{c}}}}. The fifth closest to [[3/2]] is 12.593-cent flat, unless you use the alternative fifth which is 12.939-cent sharp, similar to [[35edo]]. The soft diatonic scale generated from its flat fifth is so soft, with {{nowrap|L:s {{=}} 7:6}}, that it stops sounding like [[meantone]] or even a [[flattone]] system like [[26edo]] or [[40edo]], but just sounds like a [[circulating temperament]] of [[7edo]]. The hard diatonic scale generated from its sharp fifth is extremely hard, with {{nowrap|L:s {{=}} 9:1}}. It has therefore not aroused much interest, but its best approximation to [[9/8]] is actually quite good, one-third-of-a-cent sharp.


47edo is the 15th [[prime EDO]], following [[43edo]] and preceding [[53edo]].
47edo is one of the most difficult diatonic edos to notate in [[native fifth notation|native fifths]], because no other diatonic edo's fifth is as 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).
=== Odd harmonics ===
 
{{Harmonics in equal|47}}
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#<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:


D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D
47edo 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 [[baldy]] and [[silver]] and their relatives. It also provides a good tuning for the [[baseball]] temperament.


D# is next to E. This notation requires quadruple, quintuple, and even sextuple ups and downs, as well as single sharps and flats.
47edo can be treated as a [[dual-fifth system]] in the 2.3+.3-.5.7.13 subgroup, or the 3+.3-.5.7.11+.11-.13 subgroup for those who aren’t intimidated by lots of [[basis element]]s. As a dual-fifth system, it really shines, as both of its fifths have low enough [[harmonic entropy]] to sound [[consonant]] to many listeners, giving two consonant intervals for the price of one.


{{Harmonics in equal|47}}
=== Subsets and supersets ===
47edo is the 15th [[prime edo]], following [[43edo]] and preceding [[53edo]], so it does not contain any nontrivial subset edos, though it contains [[47ed4]]. [[94edo]], which doubles it, corrects its approximations of harmonics 3 and 11 to near-just qualities.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
|-
|-
! Degree
! [[Degree|#]]
! Size ({{cent}})
! [[Cent]]s
! colspan="2" | Relative notation
! colspan="2" | Relative notation
! Absolute notation
! Absolute notation
|-
|-
| 0
| 0
| 0.0000
| 0.0
| perfect unison
| perfect unison
| P1
| P1
Line 44: Line 32:
|-
|-
| 1
| 1
| 25.5319
| 25.5
| aug 1sn
| aug 1sn
| A1
| A1
Line 50: Line 38:
|-
|-
| 2
| 2
| 51.0638
| 51.1
| double-aug 1sn
| double-aug 1sn
| AA1
| AA1
Line 56: Line 44:
|-
|-
| 3
| 3
| 76.5957
| 76.6
| triple-aug 1sn, triple-dim 2nd
| triple-aug 1sn, triple-dim 2nd
| A<sup>3</sup>1, d<sup>3</sup>2
| A<sup>3</sup>1, d<sup>3</sup>2
Line 62: Line 50:
|-
|-
| 4
| 4
| 102.1277
| 102.1
| double-dim 2nd
| double-dim 2nd
| dd2
| dd2
Line 68: Line 56:
|-
|-
| 5
| 5
| 127.6596
| 127.7
| dim 2nd
| dim 2nd
| d2
| d2
Line 74: Line 62:
|-
|-
| 6
| 6
| 153.1915
| 153.2
| minor 2nd
| minor 2nd
| m2
| m2
Line 80: Line 68:
|-
|-
| 7
| 7
| 178.7234
| 178.7
| major 2nd
| major 2nd
| M2
| M2
Line 86: Line 74:
|-
|-
| 8
| 8
| 204.2553
| 204.3
| aug 2nd
| aug 2nd
| A2
| A2
Line 92: Line 80:
|-
|-
| 9
| 9
| 229.7872
| 229.8
| double-aug 2nd
| double-aug 2nd
| AA2
| AA2
Line 98: Line 86:
|-
|-
| 10
| 10
| 255.3191
| 255.3
| 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
Line 104: Line 92:
|-
|-
| 11
| 11
| 280.8511
| 280.9
| double-dim 3rd
| double-dim 3rd
| dd3
| dd3
Line 110: Line 98:
|-
|-
| 12
| 12
| 306.3830
| 306.4
| dim 3rd
| dim 3rd
| d3
| d3
Line 116: Line 104:
|-
|-
| 13
| 13
| 331.9149
| 331.9
| minor 3rd
| minor 3rd
| m3
| m3
Line 122: Line 110:
|-
|-
| 14
| 14
| 357.4468
| 357.4
| major 3rd
| major 3rd
| M3
| M3
Line 128: Line 116:
|-
|-
| 15
| 15
| 382.9787
| 383.0
| aug 3rd
| aug 3rd
| A3
| A3
Line 134: Line 122:
|-
|-
| 16
| 16
| 408.5106
| 408.5
| double-aug 3rd
| double-aug 3rd
| AA3
| AA3
Line 140: Line 128:
|-
|-
| 17
| 17
| 434.0426
| 434.0
| 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
Line 146: Line 134:
|-
|-
| 18
| 18
| 459.5745
| 459.6
| double-dim 4th
| double-dim 4th
| dd4
| dd4
Line 152: Line 140:
|-
|-
| 19
| 19
| 485.1064
| 485.1
| dim 4th
| dim 4th
| d4
| d4
Line 158: Line 146:
|-
|-
| 20
| 20
| 510.6383
| 510.6
| perfect 4th
| perfect 4th
| P4
| P4
Line 164: Line 152:
|-
|-
| 21
| 21
| 536.1702
| 536.2
| aug 4th
| aug 4th
| A4
| A4
Line 170: Line 158:
|-
|-
| 22
| 22
| 561.7021
| 561.7
| double-aug 4th
| double-aug 4th
| AA4
| AA4
Line 176: Line 164:
|-
|-
| 23
| 23
| 587.2340
| 587.2
| triple-aug 4th
| triple-aug 4th
| A<sup>3</sup>4
| A<sup>3</sup>4
Line 182: Line 170:
|-
|-
| 24
| 24
| 612.7660
| 612.8
| triple-dim 5th
| triple-dim 5th
| d<sup>3</sup>5
| d<sup>3</sup>5
Line 188: Line 176:
|-
|-
| 25
| 25
| 638.2979
| 638.3
| double-dim 5th
| double-dim 5th
| dd5
| dd5
Line 194: Line 182:
|-
|-
| 26
| 26
| 663.8298
| 663.8
| dim 5th
| dim 5th
| d5
| d5
Line 200: Line 188:
|-
|-
| 27
| 27
| 689.3617
| 689.4
| perfect 5th
| perfect 5th
| P5
| P5
Line 206: Line 194:
|-
|-
| 28
| 28
| 714.8936
| 714.9
| aug 5th
| aug 5th
| A5
| A5
Line 212: Line 200:
|-
|-
| 29
| 29
| 740.4255
| 740.4
| double-aug 5th
| double-aug 5th
| AA5
| AA5
Line 218: Line 206:
|-
|-
| 30
| 30
| 765.9574
| 766.0
| 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
Line 224: Line 212:
|-
|-
| 31
| 31
| 791.4894
| 791.5
| double-dim 6th
| double-dim 6th
| dd6
| dd6
Line 230: Line 218:
|-
|-
| 32
| 32
| 817.0213
| 817.0
| dim 6th
| dim 6th
| d6
| d6
Line 236: Line 224:
|-
|-
| 33
| 33
| 842.5532
| 842.6
| minor 6th
| minor 6th
| m6
| m6
Line 242: Line 230:
|-
|-
| 34
| 34
| 868.0851
| 868.1
| major 6th
| major 6th
| M6
| M6
Line 248: Line 236:
|-
|-
| 35
| 35
| 893.6170
| 893.6
| aug 6th
| aug 6th
| A6
| A6
Line 254: Line 242:
|-
|-
| 36
| 36
| 919.1489
| 919.1
| double-aug 6th
| double-aug 6th
| AA6
| AA6
Line 260: Line 248:
|-
|-
| 37
| 37
| 944.6809
| 944.7
| 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
Line 266: Line 254:
|-
|-
| 38
| 38
| 970.2128
| 970.2
| double-dim 7th
| double-dim 7th
| dd7
| dd7
Line 272: Line 260:
|-
|-
| 39
| 39
| 995.7447
| 995.7
| dim 7th
| dim 7th
| d7
| d7
Line 278: Line 266:
|-
|-
| 40
| 40
| 1021.2766
| 1021.3
| minor 7th
| minor 7th
| m7
| m7
Line 284: Line 272:
|-
|-
| 41
| 41
| 1046.8085
| 1046.8
| major 7th
| major 7th
| M7
| M7
Line 290: Line 278:
|-
|-
| 42
| 42
| 1072.3404
| 1072.3
| aug 7th
| aug 7th
| A7
| A7
Line 296: Line 284:
|-
|-
| 43
| 43
| 1097.8723
| 1097.9
| double-aug 7th
| double-aug 7th
| AA7
| AA7
Line 302: Line 290:
|-
|-
| 44
| 44
| 1123.4043
| 1123.4
| triple-aug 7th, triple-dim 8ve
| triple-aug 7th, triple-dim 8ve
| A<sup>3</sup>7, d<sup>3</sup>8
| A<sup>3</sup>7, d<sup>3</sup>8
Line 308: Line 296:
|-
|-
| 45
| 45
| 1148.9362
| 1148.9
| double-dim 8ve
| double-dim 8ve
| dd8
| dd8
Line 314: Line 302:
|-
|-
| 46
| 46
| 1174.4681
| 1174.5
| dim 8ve
| dim 8ve
| d8
| d8
Line 320: Line 308:
|-
|-
| 47
| 47
| 1200.0000
| 1200.0
| perfect 8ve
| perfect 8ve
| P8
| P8
Line 326: Line 314:
|}
|}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
== Notation ==
[[Category:Prime EDO]]
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:
[[Category:Subgroup]]
 
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#<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:
 
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.
 
=== Ups and downs notation ===
Using [[Helmholtz–Ellis]] accidentals and the sharp fifth, 47edo can be notated using [[ups and downs notation|ups and downs]]:
{{Sharpness-sharp8}}
 
With the flat fifth, notation is identical to standard notation:
{{Sharpness-sharp1}}
 
=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[42edo#Second-best fifth notation|42b]].
 
<imagemap>
File:47-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 519 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:47-EDO_Sagittal.svg]]
</imagemap>
 
==== Second-best fifth notation ====
===== Evo and Revo flavors =====
<imagemap>
File:47b_Sagittal.svg
desc none
rect 80 0 280 50 [[Sagittal_notation]]
rect 280 0 440 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:47b_Sagittal.svg]]
</imagemap>
 
===== Alternative Evo flavor =====
<imagemap>
File:47b_Alternative_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 389 0 549 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 389 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:47b_Alternative_Evo_Sagittal.svg]]
</imagemap>
 
===== Evo-SZ flavor =====
<imagemap>
File:47b_Evo-SZ_Sagittal.svg
desc none
rect 80 0 335 50 [[Sagittal_notation]]
rect 335 0 495 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:47b_Evo-SZ_Sagittal.svg]]
</imagemap>
 
== Scales ==
* [[Negri in zeta-stretched 47edo]]
* Quasi-equal [[equiheptatonic]] (Dorian): 7 6 7 7 7 6 7
** Quasi-equiheptatonic minor pentatonic: 13 7 7 13 7
* Quasi-equal [[equiheptatonic]] (Mixolydian): 7 7 6 7 7 6 7
* Quasi-equal [[equipentatonic]]: 9 10 9 10 9
* Sabertooth hexatonic: 3 9 3 13 12 7 (this is the original/default tuning; [[scalesmith|designed]] for the "gold" and "platinum" timbres in [[Scale Workshop]])
** Sabertooth pentatonic: 3 9 3 13 19 (this is the original/default tuning)
** Sabertooth neutral: 3 11 14 11 8 (this is the original/default tuning)
 
== Instruments ==
=== Lumatone ===
* [[Lumatone mapping for 47edo]]
 
=== Skip fretting ===
'''Skip fretting system 47 3 11''' is a [[skip-fretting]] system for [[47edo]] where strings are 11\47 and frets are 3\47. This is effectively 15.6666...-edo. All examples of this system on this page are for 5-string bass.
 
; Chords
Neutral-dominant 7th: 1 0 1 2 2
 
== Music ==
* [https://youtu.be/_TqaWw7tv_E Improvisation in 47edo (octave-compressed tuning, 7-note subset of Negri[9<nowiki>]</nowiki>)] by [[Budjarn Lambeth]], Jan 2024
 
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/47edo"