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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. |
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| == Overview ==
| | 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: | |
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| {| 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. |
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| !edo
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| !ed12
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| !NTSC*n
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| !PAL-M*n
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| |-
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| |1
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| |3.5849625 | |
| |3.579545 MHz | |
| |3.575611 MHz
| |
| |-
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| |2
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| |7.169925
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| |7.158909
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| |7.151222
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| |-
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| |3
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| |10.7548875
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| |10.7383635
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| |10.726833
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| |-
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| |4
| |
| |14.33985
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| |14.317818
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| |14.302444
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| |-
| |
| |5
| |
| |17.9248125
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| |17.8972725
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| |17.878055
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| |-
| |
| |6
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| |21.509775
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| |21.476727
| |
| |21.453666
| |
| |-
| |
| |7
| |
| |25.0947375
| |
| |25.0561815
| |
| |25.029277
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| |-
| |
| |8
| |
| |28.6797
| |
| |28.635636
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| |28.604888
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| |-
| |
| |9
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| |32.2646625
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| |32.2150905
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| |32.180299
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| |-
| |
| |10
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| |35.849625
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| |35.79545
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| |35.75611
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| |-
| |
| |11
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| |39.4345875
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| |39.374
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| |39.331521
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| |-
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| |12
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| |43.01955
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| |42.953454
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| |42.907332
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| |-
| |
| |13
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| |46.6045125
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| |46.5329085
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| |46.482743
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| |-
| |
| |14
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| |50.189475
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| |50.112363
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| |50.058554
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| |-
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| |15
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| |53.7744375
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| |53.6918175
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| |53.634265
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| |-
| |
| |16
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| |57.3594
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| |57.271272
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| |57.209776
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| |-
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| |17
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| |60.9443625
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| |60.8507265
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| |60.785487
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| |-
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| |18
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| |64.529325
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| |64.430181
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| |64.360598
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| |-
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| |19
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| |68.1142875
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| |68.0096355
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| |67.936709
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| |-
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| |20
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| |71.69925
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| |71.58909
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| |71.51222
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| |-
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| |21
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| |75.2842125
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| |75.1685445
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| |75.087931
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| |-
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| |22
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| |78.869175
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| |78.747999
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| |78.663442
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| |-
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| |23
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| |82.4541375
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| |82.3274535
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| |82.239153
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| |-
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| |24
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| |86.0391
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| |85.906908
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| |85.814664
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| |-
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| |25
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| |89.6240625
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| |89.4863625
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| |89.390375
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| |-
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| |26
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| |93.209025
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| |93.065817
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| |92.965886
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| |-
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| |27
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| |96.7939875
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| |96.6452715
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| |96.541597
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| |-
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| |28
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| |100.37895
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| |100.224726
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| |100.117108
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| |-
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| |29
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| |103.9639125
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| |103.8041805
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| |103.692819
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| |-
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| |30
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| |107.548875
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| |107.38365
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| |107.28633
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| |-
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| |31
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| |111.1338375
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| |110.9630895
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| |110.894041
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| |-
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| |32
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| |114.7188
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| |114.542544
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| |114.437552
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| |-
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| |33
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| |118.3037625
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| |118.1219985
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| |118.045263
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| |-
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| |34
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| |121.888725
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| |121.701453
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| |121.588774
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| |-
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| |35
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| |125.4736875
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| |125.2809075
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| |125.096485
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| |-
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| |36
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| |129.05865
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| |128.860362
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| |128.739296
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| |-
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| |37
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| |132.6436125
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| |132.4398165
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| |132.247707
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| |-
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| |38
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| |136.228575
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| |136.019271
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| |135.860518
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| |-
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| |39
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| |139.8135375
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| |139.5987255
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| |135.398929
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| |-
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| |40
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| |143.3985
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| |143.17818
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| |143.02444
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| |-
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| |41
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| |146.41815
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| |146.7576345
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| |146.600151
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| |-
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| |42
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| |150.568425
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| |150.337089
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| |150.175862
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| |-
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| |43
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| |154.0533875
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| |153.9165435
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| |153.751373
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| |-
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| |44
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| |157.73835
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| |157.495998
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| |157.326884
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| |-
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| |45
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| |161.3233125
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| |161.0754525
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| |160.902595
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| |-
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| |46
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| |164.908275
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| |164.654907
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| |164.478306
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| |-
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| |47
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| |168.4932375
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| |168.2343615
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| |168.053817
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| |-
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| |48
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| |172.0782
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| |171.813816
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| |171.629328
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| |-
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| |49
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| |175.6631625
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| |175.3932705
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| |175.205039
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| |-
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| |50
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| |179.248125
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| |178.972725
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| |178.78075
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| |-
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| |51
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| |182.8330875
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| |182.5521795
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| |182.356261
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| |-
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| |52
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| |186.41805
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| |186.131634
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| |185.931772
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| |-
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| |53
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| |190.003125
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| |189.7110885
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| |189.507483
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| |-
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| |54
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| |193.597975
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| |193.290543
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| |193.083194
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| |-
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| |55
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| |197.1729375
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| |196.869975
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| |196.658705
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| |-
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| |56
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| |200.7579
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| |200.449452
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| |200.234216
| |
| |}
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| == Table of Ed12s ==
| | 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]]. |
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| === 0…499 === | | == Individual pages for ed12's == |
| {| class="wikitable center-all" | | {| class="wikitable center-all" |
| |+ style=white-space:nowrap | 0…99 | | |+ style=white-space:nowrap | 0…99 |
| Line 519: |
Line 232: |
| | [[199ed12|199]] | | | [[199ed12|199]] |
| |} | | |} |
| | |
| | ; 200 and beyond |
| | * [[258ed12|258]] |
| | |
| | <!-- Uncomment this when there are more pages |
| {| class="wikitable center-all mw-collapsible mw-collapsed" | | {| class="wikitable center-all mw-collapsible mw-collapsed" |
| |+ style=white-space:nowrap | 200…299 | | |+ style=white-space:nowrap | 200…299 |
| Line 631: |
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| | [[299ed12|299]] | | | [[299ed12|299]] |
| |} | | |} |
| {| class="wikitable center-all mw-collapsible mw-collapsed"
| | --> |
| |+ style=white-space:nowrap | 300…399
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| | [[300ed12|300]]
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| | [[301ed12|301]]
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| | [[302ed12|302]]
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| | [[303ed12|303]]
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| | [[304ed12|304]]
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| | [[305ed12|305]]
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| | [[306ed12|306]]
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| | [[307ed12|307]]
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| | [[308ed12|308]]
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| | [[309ed12|309]]
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| |-
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| | [[310ed12|310]]
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| | [[311ed12|311]]
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| | [[312ed12|312]]
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| | [[313ed12|313]]
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| | [[314ed12|314]]
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| | [[315ed12|315]]
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| | [[316ed12|316]]
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| | [[317ed12|317]]
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| | [[318ed12|318]]
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| | [[319ed12|319]]
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| |-
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| | [[320ed12|320]]
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| | [[321ed12|321]]
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| | [[322ed12|322]]
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| | [[323ed12|323]]
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| | [[324ed12|324]]
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| | [[325ed12|325]]
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| | [[326ed12|326]]
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| | [[327ed12|327]]
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| | [[328ed12|328]]
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| | [[329ed12|329]]
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| |-
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| | [[330ed12|330]]
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| | [[331ed12|331]]
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| | [[332ed12|332]]
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| | [[333ed12|333]]
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| | [[334ed12|334]]
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| | [[335ed12|335]]
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| | [[336ed12|336]]
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| | [[337ed12|337]]
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| | [[338ed12|338]]
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| | [[339ed12|339]]
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| |-
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| | [[340ed12|340]]
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| | [[341ed12|341]]
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| | [[342ed12|342]]
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| | [[343ed12|343]]
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| | [[344ed12|344]]
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| | [[345ed12|345]]
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| | [[346ed12|346]]
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| | [[347ed12|347]]
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| | [[348ed12|348]]
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| | [[349ed12|349]]
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| |-
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| | [[350ed12|350]]
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| | [[351ed12|351]]
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| | [[352ed12|352]]
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| | [[353ed12|353]]
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| | [[354ed12|354]]
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| | [[355ed12|355]]
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| | [[356ed12|356]]
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| | [[357ed12|357]]
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| | [[358ed12|358]]
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| | [[359ed12|359]]
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| |-
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| | [[360ed12|360]]
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| | [[361ed12|361]]
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| | [[362ed12|362]]
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| | [[363ed12|363]]
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| | [[364ed12|364]]
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| | [[365ed12|365]]
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| | [[366ed12|366]]
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| | [[367ed12|367]]
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| | [[368ed12|368]]
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| | [[369ed12|369]]
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| |-
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| | [[370ed12|370]]
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| | [[371ed12|371]]
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| | [[372ed12|372]]
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| | [[373ed12|373]]
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| | [[374ed12|374]]
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| | [[375ed12|375]]
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| | [[376ed12|376]]
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| | [[377ed12|377]]
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| | [[378ed12|378]]
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| | [[379ed12|379]]
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| |-
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| | [[380ed12|380]]
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| | [[381ed12|381]]
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| | [[382ed12|382]]
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| | [[383ed12|383]]
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| | [[384ed12|384]]
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| | [[385ed12|385]]
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| | [[386ed12|386]]
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| | [[387ed12|387]]
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| | [[388ed12|388]]
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| | [[389ed12|389]]
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| |-
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| | [[390ed12|390]]
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| | [[391ed12|391]]
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| | [[392ed12|392]]
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| | [[393ed12|393]]
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| | [[394ed12|394]]
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| | [[395ed12|395]]
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| | [[396ed12|396]]
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| | [[397ed12|397]]
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| | [[398ed12|398]]
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| | [[399ed12|399]]
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| |}
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| {| class="wikitable center-all mw-collapsible mw-collapsed"
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| |+ style=white-space:nowrap | 400…499
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| | [[400ed12|400]]
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| | [[401ed12|401]]
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| | [[402ed12|402]]
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| | [[403ed12|403]]
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| | [[404ed12|404]]
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| | [[405ed12|405]]
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| | [[406ed12|406]]
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| | [[407ed12|407]]
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| | [[408ed12|408]]
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| | [[409ed12|409]]
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| |-
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| | [[410ed12|410]]
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| | [[411ed12|411]]
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| | [[412ed12|412]]
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| | [[413ed12|413]]
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| | [[414ed12|414]]
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| | [[415ed12|415]]
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| | [[416ed12|416]]
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| | [[417ed12|417]]
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| | [[418ed12|418]]
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| | [[419ed12|419]]
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| |-
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| | [[420ed12|420]]
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| | [[421ed12|421]]
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| | [[422ed12|422]]
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| | [[423ed12|423]]
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| | [[424ed12|424]]
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| | [[425ed12|425]]
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| | [[426ed12|426]]
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| | [[427ed12|427]]
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| | [[428ed12|428]]
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| | [[429ed12|429]]
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| |-
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| | [[430ed12|430]]
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| | [[431ed12|431]]
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| | [[432ed12|432]]
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| | [[433ed12|433]]
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| | [[434ed12|434]]
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| | [[435ed12|435]]
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| | [[436ed12|436]]
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| | [[437ed12|437]]
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| | [[438ed12|438]]
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| | [[439ed12|439]]
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| |-
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| | [[440ed12|440]]
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| | [[441ed12|441]]
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| | [[442ed12|442]]
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| | [[443ed12|443]]
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| | [[444ed12|444]]
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| | [[445ed12|445]]
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| | [[446ed12|446]]
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| | [[447ed12|447]]
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| | [[448ed12|448]]
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| | [[449ed12|449]]
| |
| |-
| |
| | [[450ed12|450]]
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| | [[451ed12|451]]
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| | [[452ed12|452]]
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| | [[453ed12|453]]
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| | [[454ed12|454]]
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| | [[455ed12|455]]
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| | [[456ed12|456]]
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| | [[457ed12|457]]
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| | [[458ed12|458]]
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| | [[459ed12|459]]
| |
| |-
| |
| | [[460ed12|460]]
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| | [[461ed12|461]]
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| | [[462ed12|462]]
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| | [[463ed12|463]]
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| | [[464ed12|464]]
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| | [[465ed12|465]]
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| | [[466ed12|466]]
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| | [[467ed12|467]]
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| | [[468ed12|468]]
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| | [[469ed12|469]]
| |
| |-
| |
| | [[470ed12|470]]
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| | [[471ed12|471]]
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| | [[472ed12|472]]
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| | [[473ed12|473]]
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| | [[474ed12|474]]
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| | [[475ed12|475]]
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| | [[476ed12|476]]
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| | [[477ed12|477]]
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| | [[478ed12|478]]
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| | [[479ed12|479]]
| |
| |-
| |
| | [[480ed12|480]]
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| | [[481ed12|481]]
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| | [[482ed12|482]]
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| | [[483ed12|483]]
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| | [[484ed12|484]]
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| | [[485ed12|485]]
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| | [[486ed12|486]]
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| | [[487ed12|487]]
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| | [[488ed12|488]]
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| | [[489ed12|489]]
| |
| |-
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| | [[490ed12|490]]
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| | [[491ed12|491]]
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| | [[492ed12|492]]
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| | [[493ed12|493]]
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| | [[494ed12|494]]
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| | [[495ed12|495]]
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| | [[496ed12|496]]
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| | [[497ed12|497]]
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| | [[498ed12|498]]
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| | [[499ed12|499]]
| |
| |}
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| [[Category:Edonoi]] | | [[Category:Ed12's| ]] |
| [[Category:Ed12]] | | <!-- main article --> |
| | [[Category:List of scales]] |