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Undo revision 106233 by CompactStar (talk) No, it is correct. Flattest patent val diatonic 5th. (after all, non patent ones can be any size if you're willing to go to high enough edos)
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|47}}
{{ED intro}}
 
== Theory ==
== Theory ==
47edo has a fifth which 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 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]]. 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 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).
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.  


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:
=== Odd harmonics ===
{{Harmonics in equal|47}}


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 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.
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.


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:
=== 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.
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.
=== Harmonics ===
{{Harmonics in equal|47}}
=== Subsets and supersets ===
47edo is the 15th [[prime EDO]], following [[43edo]] and preceding [[53edo]].
== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
Line 30: Line 26:
|-
|-
| 0
| 0
| 0.0000
| 0.0
| perfect unison
| perfect unison
| P1
| P1
Line 36: Line 32:
|-
|-
| 1
| 1
| 25.5319
| 25.5
| aug 1sn
| aug 1sn
| A1
| A1
Line 42: Line 38:
|-
|-
| 2
| 2
| 51.0638
| 51.1
| double-aug 1sn
| double-aug 1sn
| AA1
| AA1
Line 48: 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 54: Line 50:
|-
|-
| 4
| 4
| 102.1277
| 102.1
| double-dim 2nd
| double-dim 2nd
| dd2
| dd2
Line 60: Line 56:
|-
|-
| 5
| 5
| 127.6596
| 127.7
| dim 2nd
| dim 2nd
| d2
| d2
Line 66: Line 62:
|-
|-
| 6
| 6
| 153.1915
| 153.2
| minor 2nd
| minor 2nd
| m2
| m2
Line 72: Line 68:
|-
|-
| 7
| 7
| 178.7234
| 178.7
| major 2nd
| major 2nd
| M2
| M2
Line 78: Line 74:
|-
|-
| 8
| 8
| 204.2553
| 204.3
| aug 2nd
| aug 2nd
| A2
| A2
Line 84: Line 80:
|-
|-
| 9
| 9
| 229.7872
| 229.8
| double-aug 2nd
| double-aug 2nd
| AA2
| AA2
Line 90: 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 96: Line 92:
|-
|-
| 11
| 11
| 280.8511
| 280.9
| double-dim 3rd
| double-dim 3rd
| dd3
| dd3
Line 102: Line 98:
|-
|-
| 12
| 12
| 306.3830
| 306.4
| dim 3rd
| dim 3rd
| d3
| d3
Line 108: Line 104:
|-
|-
| 13
| 13
| 331.9149
| 331.9
| minor 3rd
| minor 3rd
| m3
| m3
Line 114: Line 110:
|-
|-
| 14
| 14
| 357.4468
| 357.4
| major 3rd
| major 3rd
| M3
| M3
Line 120: Line 116:
|-
|-
| 15
| 15
| 382.9787
| 383.0
| aug 3rd
| aug 3rd
| A3
| A3
Line 126: Line 122:
|-
|-
| 16
| 16
| 408.5106
| 408.5
| double-aug 3rd
| double-aug 3rd
| AA3
| AA3
Line 132: 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 138: Line 134:
|-
|-
| 18
| 18
| 459.5745
| 459.6
| double-dim 4th
| double-dim 4th
| dd4
| dd4
Line 144: Line 140:
|-
|-
| 19
| 19
| 485.1064
| 485.1
| dim 4th
| dim 4th
| d4
| d4
Line 150: Line 146:
|-
|-
| 20
| 20
| 510.6383
| 510.6
| perfect 4th
| perfect 4th
| P4
| P4
Line 156: Line 152:
|-
|-
| 21
| 21
| 536.1702
| 536.2
| aug 4th
| aug 4th
| A4
| A4
Line 162: Line 158:
|-
|-
| 22
| 22
| 561.7021
| 561.7
| double-aug 4th
| double-aug 4th
| AA4
| AA4
Line 168: 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 174: 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 180: Line 176:
|-
|-
| 25
| 25
| 638.2979
| 638.3
| double-dim 5th
| double-dim 5th
| dd5
| dd5
Line 186: Line 182:
|-
|-
| 26
| 26
| 663.8298
| 663.8
| dim 5th
| dim 5th
| d5
| d5
Line 192: Line 188:
|-
|-
| 27
| 27
| 689.3617
| 689.4
| perfect 5th
| perfect 5th
| P5
| P5
Line 198: Line 194:
|-
|-
| 28
| 28
| 714.8936
| 714.9
| aug 5th
| aug 5th
| A5
| A5
Line 204: Line 200:
|-
|-
| 29
| 29
| 740.4255
| 740.4
| double-aug 5th
| double-aug 5th
| AA5
| AA5
Line 210: 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 216: Line 212:
|-
|-
| 31
| 31
| 791.4894
| 791.5
| double-dim 6th
| double-dim 6th
| dd6
| dd6
Line 222: Line 218:
|-
|-
| 32
| 32
| 817.0213
| 817.0
| dim 6th
| dim 6th
| d6
| d6
Line 228: Line 224:
|-
|-
| 33
| 33
| 842.5532
| 842.6
| minor 6th
| minor 6th
| m6
| m6
Line 234: Line 230:
|-
|-
| 34
| 34
| 868.0851
| 868.1
| major 6th
| major 6th
| M6
| M6
Line 240: Line 236:
|-
|-
| 35
| 35
| 893.6170
| 893.6
| aug 6th
| aug 6th
| A6
| A6
Line 246: Line 242:
|-
|-
| 36
| 36
| 919.1489
| 919.1
| double-aug 6th
| double-aug 6th
| AA6
| AA6
Line 252: 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 258: Line 254:
|-
|-
| 38
| 38
| 970.2128
| 970.2
| double-dim 7th
| double-dim 7th
| dd7
| dd7
Line 264: Line 260:
|-
|-
| 39
| 39
| 995.7447
| 995.7
| dim 7th
| dim 7th
| d7
| d7
Line 270: Line 266:
|-
|-
| 40
| 40
| 1021.2766
| 1021.3
| minor 7th
| minor 7th
| m7
| m7
Line 276: Line 272:
|-
|-
| 41
| 41
| 1046.8085
| 1046.8
| major 7th
| major 7th
| M7
| M7
Line 282: Line 278:
|-
|-
| 42
| 42
| 1072.3404
| 1072.3
| aug 7th
| aug 7th
| A7
| A7
Line 288: Line 284:
|-
|-
| 43
| 43
| 1097.8723
| 1097.9
| double-aug 7th
| double-aug 7th
| AA7
| AA7
Line 294: 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 300: Line 296:
|-
|-
| 45
| 45
| 1148.9362
| 1148.9
| double-dim 8ve
| double-dim 8ve
| dd8
| dd8
Line 306: Line 302:
|-
|-
| 46
| 46
| 1174.4681
| 1174.5
| dim 8ve
| dim 8ve
| d8
| d8
Line 312: Line 308:
|-
|-
| 47
| 47
| 1200.0000
| 1200.0
| perfect 8ve
| perfect 8ve
| P8
| P8
Line 318: 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]]
 
[[Category:Dual-fifth]]
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"