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40edo: Difference between revisions

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Wikispaces>genewardsmith
**Imported revision 214948148 - Original comment: **
 
→Music: Stephen Weigel's ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!'': Add live performance in Munich, Germany (2026)
 
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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>2011-03-29 02:45:02 UTC</tt>.<br>
== Theory ==
: The original revision id was <tt>214948148</tt>.<br>
Up to this point, all the multiples of 5 have had the 720{{c}} [[blackwood]] fifth as their best approximation of [[3/2]]. 35edo combined the small circles of blackwood and whitewood fifths, almost equally far from just, requiring the use of both to reach all keys. 40edo adds a diatonic fifth that's closer to just. However, it is still the second flattest diatonic fifth, only exceeded by 47edo in error, which results in it being inconsistent in the [[5-limit]] - combining the best 5/4 (390{{c}}) and the best 6/5 (330{{c}}) will result in the blackwood fifth instead. So some may not consider it a valid perfect fifth.
: 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">The //40 equal division// divides the octave into 40 equal parts of exactly 30 cents each. It has a generally flat tendency, with fifths 12 cents flat. It tempers out 648/625 in the 5-limit; 225/224 and in the 7-limit; 99/98, 121/120 and 176/175 in the 11-limit; and 66/65 in the 13-limit.  


40edo is more accurate on the 2.9.5.21.33.13.51.19 subgroup, where it offers the same tuning as 80edo, and tempers out the same commas.</pre></div>
Despite all keys being reachable by stacking this fifth, it does not qualify as meantone either. Instead, it supports [[deeptone]], which tempers out [[177147/163840]] and [[1053/1024]] in the patent val instead of [[81/80]], meaning that four fifths make a near perfect [[16/13|tridecimal neutral third (16/13)]] and it takes a full 11 fifths (i.e. at the augmented third) to reach the 5th harmonic.
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;40edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;40 equal division&lt;/em&gt; divides the octave into 40 equal parts of exactly 30 cents each. It has a generally flat tendency, with fifths 12 cents flat. It tempers out 648/625 in the 5-limit; 225/224 and in the 7-limit; 99/98, 121/120 and 176/175 in the 11-limit; and 66/65 in the 13-limit. &lt;br /&gt;
40edo tempers out [[648/625]] in the 5-limit; [[225/224]] and [[16807/16384]] in the [[7-limit]]; [[99/98]], [[121/120]] and [[176/175]] in the [[11-limit]] - tuning [[orwell]] though highly suboptimally; and [[66/65]] in the 13-limit.
&lt;br /&gt;
 
40edo is more accurate on the 2.9.5.21.33.13.51.19 subgroup, where it offers the same tuning as 80edo, and tempers out the same commas.&lt;/body&gt;&lt;/html&gt;</pre></div>
81/80 is only tempered out in the 40c alternative [[val]] where the aforementioned high neutral third is equated with 5/4 instead of the much more accurate 390-cent interval. The resulting [[5L 2s]] scale has large steps of 6 intervals and small ones of 5, putting sharps and flats right next to letters without any ups or downs in between and requiring a lot of them to notate more distant keys.
 
=== Odd harmonics ===
40edo is most accurate on the 2.9.5.21.33.13.51.19.23 [[k*N_subgroups| 2*40 subgroup]], where it offers the same tuning as [[80edo|80edo]], and tempers out the same commas. It is also the first equal temperament to approximate both the 23rd and 19th harmonic, by tempering out the 9 cent comma to 4-edo, with 10 divisions therein.
 
40edo can be treated as a [[dual-fifth system]] in the 2.3+.3-.5.7.11 subgroup, or the 2.3+.3-.5.7.11.13.19.23 subgroup for those who aren’t intimidated by lots of [[basis element]]s. Both of its fifths can sound [[consonant]] to many listeners.
{{harmonics in equal|40}}
 
== Intervals ==
{{Todo|cleanup|inline=1}}
{| class="wikitable center-all"
|-
! #
! style="text-align: center;" | Cents
! colspan="3" | Notation
! colspan="2" | Approximate ratios
! Difference
|-
! 0
! 0¢
| perfect unison
| P1
| D
| 1:1
| <small>''0''</small>
| 0
|-
! 1
! 30
| augmented 1sn
| A1
| D#
| 59:58
| <small>''29.5944''</small>
| 0.40553
|-
! 2
! 60
| double-aug 1sn
| AA1
| Dx
| 29:28
| <small>''60.7512''</small>
|  -0.75128
|-
! 3
! 90
| double-dim 2nd
| dd2
| D#x, Ebbb
| 20:19
| <small>''88.8006''</small>
| 1.19930
|-
! 4
! 120
| diminished 2nd
| d2
| Ebb
| 15:14
| <small>''119.4428''</small>
| 0.55719
|-
! 5
! 150
| minor 2nd
| m2
| Eb
| 12:11
| <small>''150.6370''</small>
|  -0.63705
|-
! 6
! 180
| major 2nd
| M2
| E
| 10:9
| <small>''182.4037''</small>
|  -2.40371
|-
! 7
! 210
| augmented 2nd
| A2
| E#
| 9:8
| <small>''203.9100''</small>
| 6.08999
|-
! 8
! 240
| double-aug 2nd
| AA2
| Ex
| 8:7
| <small>''231.1741''</small>
| 8.82590
|-
! 9
! 270
| double-dim 3rd
| dd3
| Fbb
| 7:6
| <small>''266.8709''</small>
| 3.12909
|-
! 10
! 300
| diminished 3rd
| d3
| Fb
| 19:16
| <small>''297.5130''</small>
| 2.48698
|-
! 11
! 330
| minor 3rd
| m3
| F
| 6:5
| <small>''315.6412''</small>
| 14.3587
|-
! 12
! 360
| major 3rd
| M3
| F#
| 16:13
| <small>''359.4723''</small>
| 0.52766
|-
! 13
! 390
| augmented 3rd
| A3
| Fx
| 5:4
| <small>''386.3137''</small>
| 3.68628
|-
! 14
! 420
| double-aug 3rd
| AA3
| F#x, Gbbb
| 14:11
| <small>''417.5079''</small>
| 2.49203
|-
! 15
! 450
| double-dim 4th
| dd4
| Gbb
| 22:17
| <small>''446.3625''</small>
| 3.63746
|-
! 16
! 480
| diminished 4th
| d4
| Gb
| 21:16
| <small>''470.781''</small>
| 9.219
|-
! 17
! 510
| perfect 4th
| P4
| G
| 4:3
| <small>''498.0449''</small>
| 11.9550
|-
! 18
! 540
| augmented 4th
| A4
| G#
| 11:8
| <small>''551.3179''</small>
|  -11.3179
|-
! 19
! 570
| double-aug 4th
| AA4
| G##
| 25:18
| <small>''568.7174''</small>
| 1.2825
|-
! 20
! 600
| triple-aug 4th,
triple-dim 5th
| AAA4,
ddd5
| Gx#, Abbb
| 7:5
| <small>''582.5121''</small>
| 17.4878
|-
! 21
! 630
| double-dim 5th
| dd5
| Abb
| 23:16
| <small>''628.2743''</small>
| 1.72565
|-
! 22
! 660
| diminished 5th
| d5
| Ab
| 16:11
| <small>''648.6820''</small>
| 11.3179
|-
! 23
! 690
| perfect 5th
| P5
| A
| 3:2
| <small>''701.9550''</small>
|  -11.9550
|-
! 24
! 720
| augmented 5th
| A5
| A#
| 32:21
| <small>''729.2191''</small>
|  -9.219
|-
! 25
! 750
| double-aug 5th
| AA5
| Ax
| 17:11
| <small>''753.6374''</small>
|  -3.63746
|-
! 26
! 780
| double-dim 6th
| dd6
| A#x, Bbbb
| 11:7
| <small>''782.4920''</small>
|  -2.49203
|-
! 27
! 810
| diminished 6th
| d6
| Bbb
| style="text-align: center;" | 8:5
| <small>''813.6862''</small>
|  -3.68628
|-
! 28
! 840
| minor 6th
| m6
| Bb
| 13:8
| <small>''840.5276''</small>
|  -0.52766
|-
! 29
! 870
| major 6th
| M6
| B
| style="text-align: center;" | 5:3
| <small>''884.3587''</small>
|  -14.3587
|-
! 30
! 900
| augmented 6th
| A6
| B#
| style="text-align: center;" | 32:19
| <small>''902.4869''</small>
|  -2.48698
|-
! 31
! 930
| double-aug 6th
| AA6
| Bx
| style="text-align: center;" | 12:7
| <small>''933.1291''</small>
|  -3.12909
|-
! 32
! 960
| double-dim 7th
| dd7
| Cbb
| style="text-align: center;" | 7:4
| <small>''968.8259''</small>
|  -8.82590
|-
! 33
! 990
| diminished 7th
| d7
| Cb
| style="text-align: center;" | 16:9
| <small>''996.0899''</small>
|  -6.08999
|-
! 34
! 1020
| minor 7th
| m7
| C
| style="text-align: center;" | 9:5
| <small>''1017.5962''</small>
| 2.40371
|-
! 35
! 1050
| major 7th
| M7
| C#
| style="text-align: center;" | 11:6
| <small>''1049.3629''</small>
| 0.63705
|-
! 36
! 1080
| augmented 7th
| A7
| Cx
| style="text-align: center;" | 28:15
| <small>''1080.5571''</small>
|  -0.55719
|-
! 37
! 1110
| double-aug 7th
| AA7
| C#x, Dbbb
| style="text-align: center;" | 19:10
| <small>''1111.1993''</small>
|  -1.19930
|-
! 38
! 1140
| double-dim 8ve
| dd8
| Dbb
| style="text-align: center;" | 56:29
| <small>''1139.2487''</small>
| 0.75128
|-
! 39
! 1170
| diminished 8ve
| d8
| Db
| style="text-align: center;" | 116:59
| <small>''1170.4055''</small>
|  -0.40553
|-
! 40
! 1200
| perfect octave
| P8
| D
| style="text-align: center;" | 2:1
| <small>''1200''</small>
| 0
|}
 
== Notation ==
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[30edo#Second-best fifth notation|30b]] and [[35edo#Sagittal notation|35]].
 
<imagemap>
File:40-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 647 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:40-EDO_Sagittal.svg]]
</imagemap>
 
== Octave stretch or compression ==
'''127ed9''' optimises 40edo for [[dual-fifth]] usage by distributing error evenly between its two fifths. 127ed9 is just 40edo with [[octave shrinking|octaves compressed]] by 1.9{{c}}.
{{harmonics in equal|127|9|1|intervals=integer|collapsed=yes}}
{{harmonics in equal|40|2|1|intervals=integer|collapsed=yes}}
 
== Scales ==
* Amulet{{idiosyncratic}}, (approximated from [[magic]] in [[25edo]]): 3 2 3 3 2 3 5 3 3 2 3 5 3
* [[Equipentatonic]]: 8 8 8 8 8
* Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]]): 4 13 6 12 5
* Approximations of [[gamelan]] scales:
** 5-tone pelog: 4 5 14 3 14
** 7-tone pelog: 4 5 8 6 3 10 4
** 5-tone slendro: 8 8 8 8 8
* 12-tone 4&10edo scale: 4 4 2 2 4 4 4 4 2 2 4 4
* 12-tone 5&8edo scale: 5 3 2 5 1 4 4 1 5 2 3 5
 
{| class="wikitable mw-collapsible mw-collapsed"
|+Approximated from [[96edo]]
|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
* Flattened major: 7 6 4 6 7 7 3
* Sharpened minor: 7 3 7 6 4 7 6
* Sharpened harmonic minor: 7 3 7 7 3 11 2
* Flattened major pentatonic: 6 7 10 7 10
* Sharpened minor pentatonic: 10 7 6 11 6
* Evened minor hexatonic: 6 4 7 6 10 7
* Roughened augmented: 11 3 10 3 10 3
* Evened dominant pentatonic: 7 7 9 10 7
* Sharpened Dorian: 7 3 7 7 7 3 6
* Flattened Ionian pentatonic: 13 3 7 13 4
* Sharpened Dorian harmonic: 7 3 10 4 7 3 6
* Evened Mixolydian pentatonic: 13 4 6 10 7
* Evened Phrygian dominant: 4 9 4 6 4 6 7
* Evened Phrygian dominant hexatonic: 3 10 4 6 10 7
* Sharpened Phrygian pentatonic: 4 7 13 3 13
* Sharpened minor harmonic pentatonic I: 7 3 14 13 3
* Evened hirajoshi: 7 4 12 4 13
* Sharpened hirajoshi: 7 4 13 4 12
* Extra-sharp hirajoshi: 8 3 13 4 12
* Evened akebono I: 6 5 12 6 11
* Sharpened akebono I: 7 3 14 6 10
* Extra-sharp akebono I: 7 4 13 7 9
* Evened Javanese pentachordal: 4 7 9 4 16
* Moonbeam: 7 4 12 14 3
* Palace (type of [[equiheptatonic]]): 5 6 6 6 6 6 5
* Underpass: 11 12 8 3 6
|}
 
== Instruments ==
* [[Lumatone mapping for 40edo]]
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/x5cnT4Bw1ZQ ''Balance Beam''] (2026)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=EMZu6ZE6A3g ''Happy Birthday Canon'', 6-in-1 Canon in 40edo]
* [https://www.youtube.com/watch?v=eu854Ld_uLE ''Canon on Twinkle Twinkle Little Star'', for Baroque Oboe & Viola] (2023) – ([https://www.youtube.com/watch?v=l7vDHwsboLE for Organ])
 
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=tLmaQK10aYM ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!''] (2025; mostly in 42edo, but also some in 40edo)
** [https://www.youtube.com/watch?v=NE77rwCsGHw live performance of the above in Munich, Germany] (2026)
 
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/40edo"