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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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| == Properties ==
| | 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. |
| === Improving primes ===
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| Ed12 is like [[ed6]] in that it is a compromise between [[prime]]s 2 and 3 — but ed12 leans more towards prime 2 than ed6 does. An ed12 sometimes gives you the right amount of stretch for equal temperaments whose 3 is more inaccurate than its higher [[prime]]s.
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| Here for example, you can choose how much you wish to stretch [[31edo]] depending on your harmonic style: [[80ed6]] vs [[111ed12]].
| | 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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| ''(See also [[Stretched and compressed tuning]].)''
| | 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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| === Width of its period === | | == Individual pages for ed12's == |
| The twelfth harmonic is particularly wide as far as [[period]]s go. There are (at absolute most) ~3.1 dodecataves within the human hearing range; imagine if that were the case with octaves.
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| This width means that the listener probably won't hear the period (12/1) 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 don't.
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| === NTSC and PAL devices ===
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| The ed12-[[edo]] correspondences fall particularly close to the harmonic series of the NTSC or PAL-M color subcarrier. Perhaps a multimedia artist could find some way to put this to use.
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| {| class="wikitable mw-collapsible mw-collapsed"
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| |+ Ed12-edo correspondences
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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
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| |3.579545 MHz
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| |3.575611 MHz
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| |-
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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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| |-
| |
| |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
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| |14.33985
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| |14.317818
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| |14.302444
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| |-
| |
| |5
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| |17.9248125
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| |17.8972725
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| |17.878055
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| |-
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| |6
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| |21.509775
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| |21.476727
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| |21.453666
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| |-
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| |7
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| |25.0947375
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| |25.0561815
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| |25.029277
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| |-
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| |8
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| |28.6797
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| |28.635636
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| |28.604888
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| |-
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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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| |-
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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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| |-
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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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| |-
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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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| |-
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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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| |-
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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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| |}
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| == Table of Ed12s ==
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| === 0…299 ===
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| {| class="wikitable center-all" | | {| class="wikitable center-all" |
| |+ style=white-space:nowrap | 0…99 | | |+ style=white-space:nowrap | 0…99 |
| Line 532: |
Line 232: |
| | [[199ed12|199]] | | | [[199ed12|199]] |
| |} | | |} |
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| | ; 200 and beyond |
| | * [[258ed12|258]] |
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| | <!-- 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 644: |
Line 349: |
| | [[299ed12|299]] | | | [[299ed12|299]] |
| |} | | |} |
| | --> |
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| [[Category:Equal-step tuning]] | | [[Category:Ed12's| ]] |
| [[Category:Edonoi]]
| | <!-- main article --> |
| [[Category:Ed12]] | | [[Category:List of scales]] |
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| {{todo|inline=1|explain edonoi|text=Most people do not think 5/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is.}}
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