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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-29 00:49:33 UTC</tt>.<br>
: The original revision id was <tt>232656270</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="color: #616161; font-size: 99%;"&gt;47 tone Equal Temperament&lt;/span&gt;=


The //47-EDO// divides the octave into 47 equal parts of 25.5319 cents each. It has a fifth which is 12.5933 cents flat, unless you use the alternative fifth which is 12.9386 cents sharp. 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]] and [[Chromatic pairs|silver]] temperaments and relatives.</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
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.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;47edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x47 tone Equal Temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #616161; font-size: 99%;"&gt;47 tone Equal Temperament&lt;/span&gt;&lt;/h1&gt;
 
&lt;br /&gt;
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.
The &lt;em&gt;47-EDO&lt;/em&gt; divides the octave into 47 equal parts of 25.5319 cents each. It has a fifth which is 12.5933 cents flat, unless you use the alternative fifth which is 12.9386 cents sharp. 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 &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;2*47 subgroup&lt;/a&gt; of the &lt;a class="wiki_link" href="/23-limit"&gt;23-limit&lt;/a&gt;, on which it tempers out the same commas as &lt;a class="wiki_link" href="/94edo"&gt;94edo&lt;/a&gt;. It provides a good tuning for &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;baldy&lt;/a&gt; and &lt;a class="wiki_link" href="/Chromatic%20pairs"&gt;silver&lt;/a&gt; temperaments and relatives.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Odd harmonics ===
{{Harmonics in equal|47}}
 
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.
 
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.
 
=== 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 ==
{| class="wikitable center-all right-2"
|-
! [[Degree|#]]
! [[Cent]]s
! colspan="2" | Relative notation
! Absolute notation
|-
| 0
| 0.0
| perfect unison
| P1
| D
|-
| 1
| 25.5
| aug 1sn
| A1
| D#
|-
| 2
| 51.1
| double-aug 1sn
| AA1
| Dx
|-
| 3
| 76.6
| triple-aug 1sn, triple-dim 2nd
| A<sup>3</sup>1, d<sup>3</sup>2
| D#<sup>3</sup>, Eb<sup>4</sup>
|-
| 4
| 102.1
| double-dim 2nd
| dd2
| Eb<sup>3</sup>
|-
| 5
| 127.7
| dim 2nd
| d2
| Ebb
|-
| 6
| 153.2
| minor 2nd
| m2
| Eb
|-
| 7
| 178.7
| major 2nd
| M2
| E
|-
| 8
| 204.3
| aug 2nd
| A2
| E#
|-
| 9
| 229.8
| double-aug 2nd
| AA2
| Ex
|-
| 10
| 255.3
| triple-aug 2nd, triple-dim 3rd
| A<sup>3</sup>2, d<sup>3</sup>3
| E#<sup>3</sup>, Fb<sup>3</sup>
|-
| 11
| 280.9
| double-dim 3rd
| dd3
| Fbb
|-
| 12
| 306.4
| dim 3rd
| d3
| Fb
|-
| 13
| 331.9
| minor 3rd
| m3
| F
|-
| 14
| 357.4
| major 3rd
| M3
| F#
|-
| 15
| 383.0
| aug 3rd
| A3
| Fx
|-
| 16
| 408.5
| double-aug 3rd
| AA3
| F#<sup>3</sup>
|-
| 17
| 434.0
| triple-aug 3rd, triple-dim 4th
| A<sup>3</sup>3, d<sup>3</sup>4
| F#<sup>4</sup>, Gb<sup>3</sup>
|-
| 18
| 459.6
| double-dim 4th
| dd4
| Gbb
|-
| 19
| 485.1
| dim 4th
| d4
| Gb
|-
| 20
| 510.6
| perfect 4th
| P4
| G
|-
| 21
| 536.2
| aug 4th
| A4
| G#
|-
| 22
| 561.7
| double-aug 4th
| AA4
| Gx
|-
| 23
| 587.2
| triple-aug 4th
| A<sup>3</sup>4
| G#<sup>3</sup>
|-
| 24
| 612.8
| triple-dim 5th
| d<sup>3</sup>5
| Ab<sup>3</sup>
|-
| 25
| 638.3
| double-dim 5th
| dd5
| Abb
|-
| 26
| 663.8
| dim 5th
| d5
| Ab
|-
| 27
| 689.4
| perfect 5th
| P5
| A
|-
| 28
| 714.9
| aug 5th
| A5
| A#
|-
| 29
| 740.4
| double-aug 5th
| AA5
| Ax
|-
| 30
| 766.0
| triple-aug 5th, triple-dim 6th
| A<sup>3</sup>5, d<sup>3</sup>6
| A#<sup>3</sup>, Bb<sup>4</sup>
|-
| 31
| 791.5
| double-dim 6th
| dd6
| Bb<sup>3</sup>
|-
| 32
| 817.0
| dim 6th
| d6
| Bbb
|-
| 33
| 842.6
| minor 6th
| m6
| Bb
|-
| 34
| 868.1
| major 6th
| M6
| B
|-
| 35
| 893.6
| aug 6th
| A6
| B#
|-
| 36
| 919.1
| double-aug 6th
| AA6
| Bx
|-
| 37
| 944.7
| triple-aug 6th, triple-dim 7th
| A<sup>3</sup>6, d<sup>3</sup>7
| B#<sup>3</sup>, Cb<sup>3</sup>
|-
| 38
| 970.2
| double-dim 7th
| dd7
| Cbb
|-
| 39
| 995.7
| dim 7th
| d7
| Cb
|-
| 40
| 1021.3
| minor 7th
| m7
| C
|-
| 41
| 1046.8
| major 7th
| M7
| C#
|-
| 42
| 1072.3
| aug 7th
| A7
| Cx
|-
| 43
| 1097.9
| double-aug 7th
| AA7
| C#<sup>3</sup>
|-
| 44
| 1123.4
| triple-aug 7th, triple-dim 8ve
| A<sup>3</sup>7, d<sup>3</sup>8
| C#<sup>4</sup>, Db<sup>3</sup>
|-
| 45
| 1148.9
| double-dim 8ve
| dd8
| Dbb
|-
| 46
| 1174.5
| dim 8ve
| d8
| Db
|-
| 47
| 1200.0
| perfect 8ve
| P8
| D
|}
 
== Notation ==
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
 
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"