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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 hendecatave 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]]
|39.3739995
| [[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
|100.2274726
|100.117108
|-
|-
|29
| [[210ed12|210]]
|103.9639125
| [[211ed12|211]]
|103.8041805
| [[212ed12|212]]
|103.692819
| [[213ed12|213]]
| [[214ed12|214]]
| [[215ed12|215]]
| [[216ed12|216]]
| [[217ed12|217]]
| [[218ed12|218]]
| [[219ed12|219]]
|-
|-
|30
| [[220ed12|220]]
|107.548875
| [[221ed12|221]]
|107.38365
| [[222ed12|222]]
|107.28633
| [[223ed12|223]]
| [[224ed12|224]]
| [[225ed12|225]]
| [[226ed12|226]]
| [[227ed12|227]]
| [[228ed12|228]]
| [[229ed12|229]]
|-
|-
|31
| [[230ed12|230]]
|111.1338375
| [[231ed12|231]]
|110.9630895
| [[232ed12|232]]
|110.894041
| [[233ed12|233]]
| [[234ed12|234]]
| [[235ed12|235]]
| [[236ed12|236]]
| [[237ed12|237]]
| [[238ed12|238]]
| [[239ed12|239]]
|-
|-
|32
| [[240ed12|240]]
|114.7188
| [[241ed12|241]]
|114.542544
| [[242ed12|242]]
|114.437552
| [[243ed12|243]]
| [[244ed12|244]]
| [[245ed12|245]]
| [[246ed12|246]]
| [[247ed12|247]]
| [[248ed12|248]]
| [[249ed12|249]]
|-
|-
|33
| [[250ed12|250]]
|118.3037625
| [[251ed12|251]]
|118.1219985
| [[252ed12|252]]
|118.045263
| [[253ed12|253]]
| [[254ed12|254]]
| [[255ed12|255]]
| [[256ed12|256]]
| [[257ed12|257]]
| [[258ed12|258]]
| [[259ed12|259]]
|-
|-
|34
| [[260ed12|260]]
|121.888725
| [[261ed12|261]]
|121.701453
| [[262ed12|262]]
|121.588774
| [[263ed12|263]]
| [[264ed12|264]]
| [[265ed12|265]]
| [[266ed12|266]]
| [[267ed12|267]]
| [[268ed12|268]]
| [[269ed12|269]]
|-
|-
|35
| [[270ed12|270]]
|125.4736875
| [[271ed12|271]]
|125.2809075
| [[272ed12|272]]
|125.096485
| [[273ed12|273]]
| [[274ed12|274]]
| [[275ed12|275]]
| [[276ed12|276]]
| [[277ed12|277]]
| [[278ed12|278]]
| [[279ed12|279]]
|-
|-
|36
| [[280ed12|280]]
|129.05865
| [[281ed12|281]]
|128.860362
| [[282ed12|282]]
|128.739296
| [[283ed12|283]]
| [[284ed12|284]]
| [[285ed12|285]]
| [[286ed12|286]]
| [[287ed12|287]]
| [[288ed12|288]]
| [[289ed12|289]]
|-
|-
|37
| [[290ed12|290]]
|132.6436125
| [[291ed12|291]]
|132.4398165
| [[292ed12|292]]
|132.247707
| [[293ed12|293]]
|-
| [[294ed12|294]]
|38
| [[295ed12|295]]
|136.228575
| [[296ed12|296]]
|136.019271
| [[297ed12|297]]
|135.860518
| [[298ed12|298]]
|-
| [[299ed12|299]]
|39
|139.8135375
|139.5987255
|135.398929
|-
|40
|143.3985
|143.17818
|143.02444
|}
|}
-->
[[Category:Ed12's| ]]
<!-- main article -->
[[Category:List of scales]]
Retrieved from "https://en.xen.wiki/w/Ed12"