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'''Ed12''' means '''Division of the Twelfth Harmonic ([[12/1]]) into n equal parts'''.
'''Equal divisions of the 12th harmonic''' ('''ed12''') are [[tuning system|tunings]] obtained by dividing the [[12/1|12th harmonic]] in a certain number of [[equal]] steps.  


= Division of the twelfth harmonic into n equal parts =
The twelfth harmonic, duodecuple, or dodecatave, is particularly wide as far as [[equivalence]]s go, as there are at absolute most about 3.1 instances of the 12th harmonic within the [[human hearing range]]. This width means that the listener probably will not hear the interval as an equivalence, but instead will hear the [[pseudo-octave]] or pseudo-tritave or similar as one – this disconnect between period versus equivalence could be used by a composer to surprise their listener, in a similar way that [[13edo]] can be used to make melodies that sound like [[12edo]], until they suddenly do not.
The twelfth harmonic is particularly wide as far as equivalences go. There are (at absolute most) ~3.1 dodecataves within the human hearing range; imagine if that were the case with octaves. If one does indeed deal with dodecatave equivalence, this fact shapes one's musical approach dramatically. Also, the ed12-[[edo]] correspondences fall particularly close to the harmonic series of the NTSC or PAL-M color subcarrier:


{| class="wikitable"
However, using ed12's does not necessarily imply using the 12th harmonic as an interval of equivalence. The quintessential reason for using a 12th-harmonic based tuning is that it is a compromise between [[2/1|octave]] and [[3/1|twelfth]] based tunings, like an [[ed6]] – but ed12 leans more towards octaves than ed6 does. In fact, ed12's optimize for the 1:2:3:4:6:12 chord, with equal magnitudes and opposite signs of [[error]] on 3 and 4 and on 2 and 6.
|+
 
!edo
As such, an ed12 sometimes gives you the right amount of [[stretched and compressed tuning|stretch]] for equal temperaments whose 3 is more inaccurate than its higher [[prime interval|primes]]. Here for example, you can choose how much you wish to stretch [[31edo]] depending on your harmonic style: [[80ed6]] vs [[111ed12]].
!ed12
 
!NTSC*n
== Individual pages for ed12's ==
!PAL-M*n
{| class="wikitable center-all"
|+ style=white-space:nowrap | 0…99
| [[0ed12|0]]
| [[1ed12|1]]
| [[2ed12|2]]
| [[3ed12|3]]
| [[4ed12|4]]
| [[5ed12|5]]
| [[6ed12|6]]
| [[7ed12|7]]
| [[8ed12|8]]
| [[9ed12|9]]
|-
|-
|1
| [[10ed12|10]]
|3.5849625
| [[11ed12|11]]
|3.579545 MHz
| [[12ed12|12]]
|3.575611 MHz
| [[13ed12|13]]
| [[14ed12|14]]
| [[15ed12|15]]
| [[16ed12|16]]
| [[17ed12|17]]
| [[18ed12|18]]
| [[19ed12|19]]
|-
|-
|2
| [[20ed12|20]]
|7.169925
| [[21ed12|21]]
|7.158909
| [[22ed12|22]]
|7.151222
| [[23ed12|23]]
| [[24ed12|24]]
| [[25ed12|25]]
| [[26ed12|26]]
| [[27ed12|27]]
| [[28ed12|28]]
| [[29ed12|29]]
|-
|-
|3
| [[30ed12|30]]
|10.7548875
| [[31ed12|31]]
|10.7383635
| [[32ed12|32]]
|10.726833
| [[33ed12|33]]
| [[34ed12|34]]
| [[35ed12|35]]
| [[36ed12|36]]
| [[37ed12|37]]
| [[38ed12|38]]
| [[39ed12|39]]
|-
|-
|4
| [[40ed12|40]]
|14.33985
| [[41ed12|41]]
|14.317818
| [[42ed12|42]]
|14.302444
| [[43ed12|43]]
| [[44ed12|44]]
| [[45ed12|45]]
| [[46ed12|46]]
| [[47ed12|47]]
| [[48ed12|48]]
| [[49ed12|49]]
|-
|-
|5
| [[50ed12|50]]
|17.9248125
| [[51ed12|51]]
|17.8972725
| [[52ed12|52]]
|17.878055
| [[53ed12|53]]
| [[54ed12|54]]
| [[55ed12|55]]
| [[56ed12|56]]
| [[57ed12|57]]
| [[58ed12|58]]
| [[59ed12|59]]
|-
|-
|6
| [[60ed12|60]]
|21.509775
| [[61ed12|61]]
|21.476727
| [[62ed12|62]]
|21.453666
| [[63ed12|63]]
| [[64ed12|64]]
| [[65ed12|65]]
| [[66ed12|66]]
| [[67ed12|67]]
| [[68ed12|68]]
| [[69ed12|69]]
|-
|-
|7
| [[70ed12|70]]
|25.0947375
| [[71ed12|71]]
|25.0561815
| [[72ed12|72]]
|25.029277
| [[73ed12|73]]
| [[74ed12|74]]
| [[75ed12|75]]
| [[76ed12|76]]
| [[77ed12|77]]
| [[78ed12|78]]
| [[79ed12|79]]
|-
|-
|8
| [[80ed12|80]]
|28.6797
| [[81ed12|81]]
|28.635636
| [[82ed12|82]]
|28.604888
| [[83ed12|83]]
| [[84ed12|84]]
| [[85ed12|85]]
| [[86ed12|86]]
| [[87ed12|87]]
| [[88ed12|88]]
| [[89ed12|89]]
|-
|-
|9
| [[90ed12|90]]
|32.2646625
| [[91ed12|91]]
|32.2150905
| [[92ed12|92]]
|32.180299
| [[93ed12|93]]
| [[94ed12|94]]
| [[95ed12|95]]
| [[96ed12|96]]
| [[97ed12|97]]
| [[98ed12|98]]
| [[99ed12|99]]
|}
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | 100…199
| [[100ed12|100]]
| [[101ed12|101]]
| [[102ed12|102]]
| [[103ed12|103]]
| [[104ed12|104]]
| [[105ed12|105]]
| [[106ed12|106]]
| [[107ed12|107]]
| [[108ed12|108]]
| [[109ed12|109]]
|-
|-
|10
| [[110ed12|110]]
|35.849625
| [[111ed12|111]]
|35.79545
| [[112ed12|112]]
|35.75611
| [[113ed12|113]]
| [[114ed12|114]]
| [[115ed12|115]]
| [[116ed12|116]]
| [[117ed12|117]]
| [[118ed12|118]]
| [[119ed12|119]]
|-
|-
|11
| [[120ed12|120]]
|39.4345875
| [[121ed12|121]]
|39.374
| [[122ed12|122]]
|39.331521
| [[123ed12|123]]
| [[124ed12|124]]
| [[125ed12|125]]
| [[126ed12|126]]
| [[127ed12|127]]
| [[128ed12|128]]
| [[129ed12|129]]
|-
|-
|12
| [[130ed12|130]]
|43.01955
| [[131ed12|131]]
|42.953454
| [[132ed12|132]]
|42.907332
| [[133ed12|133]]
| [[134ed12|134]]
| [[135ed12|135]]
| [[136ed12|136]]
| [[137ed12|137]]
| [[138ed12|138]]
| [[139ed12|139]]
|-
|-
|13
| [[140ed12|140]]
|46.6045125
| [[141ed12|141]]
|46.5329085
| [[142ed12|142]]
|46.482743
| [[143ed12|143]]
| [[144ed12|144]]
| [[145ed12|145]]
| [[146ed12|146]]
| [[147ed12|147]]
| [[148ed12|148]]
| [[149ed12|149]]
|-
|-
|14
| [[150ed12|150]]
|50.189475
| [[151ed12|151]]
|50.112363
| [[152ed12|152]]
|50.058554
| [[153ed12|153]]
| [[154ed12|154]]
| [[155ed12|155]]
| [[156ed12|156]]
| [[157ed12|157]]
| [[158ed12|158]]
| [[159ed12|159]]
|-
|-
|15
| [[160ed12|160]]
|53.7744375
| [[161ed12|161]]
|53.6918175
| [[162ed12|162]]
|53.634265
| [[163ed12|163]]
| [[164ed12|164]]
| [[165ed12|165]]
| [[166ed12|166]]
| [[167ed12|167]]
| [[168ed12|168]]
| [[169ed12|169]]
|-
|-
|16
| [[170ed12|170]]
|57.3594
| [[171ed12|171]]
|57.271272
| [[172ed12|172]]
|57.209776
| [[173ed12|173]]
| [[174ed12|174]]
| [[175ed12|175]]
| [[176ed12|176]]
| [[177ed12|177]]
| [[178ed12|178]]
| [[179ed12|179]]
|-
|-
|17
| [[180ed12|180]]
|60.9443625
| [[181ed12|181]]
|60.8507265
| [[182ed12|182]]
|60.785487
| [[183ed12|183]]
| [[184ed12|184]]
| [[185ed12|185]]
| [[186ed12|186]]
| [[187ed12|187]]
| [[188ed12|188]]
| [[189ed12|189]]
|-
|-
|18
| [[190ed12|190]]
|64.529325
| [[191ed12|191]]
|64.430181
| [[192ed12|192]]
|64.360598
| [[193ed12|193]]
|-
| [[194ed12|194]]
|19
| [[195ed12|195]]
|68.1142875
| [[196ed12|196]]
|68.0096355
| [[197ed12|197]]
|67.936709
| [[198ed12|198]]
|-
| [[199ed12|199]]
|20
|}
|71.69925
 
|71.58909
; 200 and beyond
|71.51222
* [[258ed12|258]]
|-
 
|21
<!-- Uncomment this when there are more pages
|75.2842125
{| class="wikitable center-all mw-collapsible mw-collapsed"
|75.1685445
|+ style=white-space:nowrap | 200…299
|75.087931
| [[200ed12|200]]
|-
| [[201ed12|201]]
|22
| [[202ed12|202]]
|78.869175
| [[203ed12|203]]
|78.747999
| [[204ed12|204]]
|78.663442
| [[205ed12|205]]
|-
| [[206ed12|206]]
|23
| [[207ed12|207]]
|82.4541375
| [[208ed12|208]]
|82.3274535
| [[209ed12|209]]
|82.239153
|-
|24
|86.0391
|85.906908
|85.814664
|-
|25
|89.6240625
|89.4863625
|89.390375
|-
|26
|93.209025
|93.065817
|92.965886
|-
|27
|96.7939875
|96.6452715
|96.541597
|-
|28
|100.37895
|100.224726
|100.117108
|-
|29
|103.9639125
|103.8041805
|103.692819
|-
|30
|107.548875
|107.38365
|107.28633
|-
|31
|111.1338375
|110.9630895
|110.894041
|-
|32
|114.7188
|114.542544
|114.437552
|-
|33
|118.3037625
|118.1219985
|118.045263
|-
|34
|121.888725
|121.701453
|121.588774
|-
|35
|125.4736875
|125.2809075
|125.096485
|-
|36
|129.05865
|128.860362
|128.739296
|-
|37
|132.6436125
|132.4398165
|132.247707
|-
|38
|136.228575
|136.019271
|135.860518
|-
|39
|139.8135375
|139.5987255
|135.398929
|-
|40
|143.3985
|143.17818
|143.02444
|-
|41
|146.41815
|146.7576345
|146.600151
|-
|42
|150.568425
|150.337089
|150.175862
|-
|43
|154.0533875
|153.9165435
|153.751373
|-
|44
|157.73835
|157.495998
|157.326884
|-
|45
|161.3233125
|161.0754525
|160.902595
|-
|46
|164.908275
|164.654907
|164.478306
|-
|47
|168.4932375
|168.2343615
|168.053817
|-
|-
|48
| [[210ed12|210]]
|172.0782
| [[211ed12|211]]
|171.813816
| [[212ed12|212]]
|171.629328
| [[213ed12|213]]
| [[214ed12|214]]
| [[215ed12|215]]
| [[216ed12|216]]
| [[217ed12|217]]
| [[218ed12|218]]
| [[219ed12|219]]
|-
|-
|49
| [[220ed12|220]]
|175.6631625
| [[221ed12|221]]
|175.3932705
| [[222ed12|222]]
|175.205039
| [[223ed12|223]]
| [[224ed12|224]]
| [[225ed12|225]]
| [[226ed12|226]]
| [[227ed12|227]]
| [[228ed12|228]]
| [[229ed12|229]]
|-
|-
|50
| [[230ed12|230]]
|179.248125
| [[231ed12|231]]
|178.972725
| [[232ed12|232]]
|178.78075
| [[233ed12|233]]
| [[234ed12|234]]
| [[235ed12|235]]
| [[236ed12|236]]
| [[237ed12|237]]
| [[238ed12|238]]
| [[239ed12|239]]
|-
|-
|51
| [[240ed12|240]]
|182.8330875
| [[241ed12|241]]
|182.5521795
| [[242ed12|242]]
|182.356261
| [[243ed12|243]]
| [[244ed12|244]]
| [[245ed12|245]]
| [[246ed12|246]]
| [[247ed12|247]]
| [[248ed12|248]]
| [[249ed12|249]]
|-
|-
|52
| [[250ed12|250]]
|186.41805
| [[251ed12|251]]
|186.131634
| [[252ed12|252]]
|185.931772
| [[253ed12|253]]
| [[254ed12|254]]
| [[255ed12|255]]
| [[256ed12|256]]
| [[257ed12|257]]
| [[258ed12|258]]
| [[259ed12|259]]
|-
|-
|53
| [[260ed12|260]]
|190.003125
| [[261ed12|261]]
|189.7110885
| [[262ed12|262]]
|189.507483
| [[263ed12|263]]
| [[264ed12|264]]
| [[265ed12|265]]
| [[266ed12|266]]
| [[267ed12|267]]
| [[268ed12|268]]
| [[269ed12|269]]
|-
|-
|54
| [[270ed12|270]]
|193.597975
| [[271ed12|271]]
|193.290543
| [[272ed12|272]]
|193.083194
| [[273ed12|273]]
| [[274ed12|274]]
| [[275ed12|275]]
| [[276ed12|276]]
| [[277ed12|277]]
| [[278ed12|278]]
| [[279ed12|279]]
|-
|-
|55
| [[280ed12|280]]
|197.1729375
| [[281ed12|281]]
|196.869975
| [[282ed12|282]]
|196.658705
| [[283ed12|283]]
| [[284ed12|284]]
| [[285ed12|285]]
| [[286ed12|286]]
| [[287ed12|287]]
| [[288ed12|288]]
| [[289ed12|289]]
|-
|-
|56
| [[290ed12|290]]
|200.7579
| [[291ed12|291]]
|200.449452
| [[292ed12|292]]
|200.234216
| [[293ed12|293]]
| [[294ed12|294]]
| [[295ed12|295]]
| [[296ed12|296]]
| [[297ed12|297]]
| [[298ed12|298]]
| [[299ed12|299]]
|}
|}
-->


[[Category:Edonoi]]
[[Category:Ed12's| ]]
[[Category:Ed12]]
<!-- main article -->
[[Category:List of scales]]

Latest revision as of 19:39, 1 August 2025

Equal divisions of the 12th harmonic (ed12) are tunings obtained by dividing the 12th harmonic in a certain number of equal steps.

The twelfth harmonic, duodecuple, or dodecatave, is particularly wide as far as equivalences go, as there are at absolute most about 3.1 instances of the 12th harmonic within the human hearing range. This width means that the listener probably will not hear the interval as an equivalence, but instead will hear the pseudo-octave or pseudo-tritave or similar as one – this disconnect between period versus equivalence could be used by a composer to surprise their listener, in a similar way that 13edo can be used to make melodies that sound like 12edo, until they suddenly do not.

However, using ed12's does not necessarily imply using the 12th harmonic as an interval of equivalence. The quintessential reason for using a 12th-harmonic based tuning is that it is a compromise between octave and twelfth based tunings, like an ed6 – but ed12 leans more towards octaves than ed6 does. In fact, ed12's optimize for the 1:2:3:4:6:12 chord, with equal magnitudes and opposite signs of error on 3 and 4 and on 2 and 6.

As such, an ed12 sometimes gives you the right amount of stretch for equal temperaments whose 3 is more inaccurate than its higher primes. Here for example, you can choose how much you wish to stretch 31edo depending on your harmonic style: 80ed6 vs 111ed12.

Individual pages for ed12's

0…99
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
100…199
100 101 102 103 104 105 106 107 108 109
110 111 112 113 114 115 116 117 118 119
120 121 122 123 124 125 126 127 128 129
130 131 132 133 134 135 136 137 138 139
140 141 142 143 144 145 146 147 148 149
150 151 152 153 154 155 156 157 158 159
160 161 162 163 164 165 166 167 168 169
170 171 172 173 174 175 176 177 178 179
180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199
200 and beyond
Retrieved from "https://en.xen.wiki/index.php?title=Ed12&oldid=206067"