28ed5: Difference between revisions
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28ed5 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 1216/1215, 1445/1444, and 6144/6137, which is a [[cluster temperament]] with 12 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 1088/1083 ~ 256/255 ~ 289/288 ~ 324/323 ~ 361/360 all tempered together. This temperament is supported by [[12edo]], [[205edo]], and [[217edo]] among others. | 28ed5 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 1216/1215, 1445/1444, and 6144/6137, which is a [[cluster temperament]] with 12 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 1088/1083 ~ 256/255 ~ 289/288 ~ 324/323 ~ 361/360 all tempered together. This temperament is supported by [[12edo]], [[205edo]], and [[217edo]] among others. | ||
'''<font style="font-size: 1.2em">5-limit 12&193</font>''' | '''<font style="font-size: 1.2em">5-limit 12&193 (quinsa-quingu)</font>''' | ||
Comma: |56 -28 -5> | Comma: |56 -28 -5> | ||
Line 217: | Line 217: | ||
Badness: 0.093971<br> | Badness: 0.093971<br> | ||
'''<font style="font-size: 1.2em">2.3.5.17.19 subgroup 12&193</font>''' | '''<font style="font-size: 1.2em">7-limit 12&422</font>''' | ||
Commas: 1216/1215, 1445/1444, 6144/6137 | |||
POTE generator: ~18/17 = 99.524 | Commas: 102760448/102515625, 1220703125/1219784832 | ||
Map: [<1 2 0 5 4|, <0 -5 28 -11 3|] | |||
EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603 | POTE generator: ~1323/1250 = 99.521 | ||
Map: [<2 4 0 -5|, <0 -5 28 64|] | |||
EDOs: 12, 398, 410, 422, 808, 832, 1242 | |||
Badness: 0.233140<br> | |||
'''<font style="font-size: 1.2em">11-limit 12&422</font>''' | |||
Commas: 5632/5625, 9801/9800, 85937500/85766121 | |||
POTE generator: ~1323/1250 = 99.525 | |||
Map: [<2 4 0 -5 -10|, <0 -5 28 64 102|] | |||
EDOs: 12, 410, 422, 832 | |||
Badness: 0.093926<br> | |||
'''<font style="font-size: 1.2em">2.3.5.17.19 subgroup 12&193</font>''' | |||
Commas: 1216/1215, 1445/1444, 6144/6137 | |||
POTE generator: ~18/17 = 99.524 | |||
Map: [<1 2 0 5 4|, <0 -5 28 -11 3|] | |||
EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603<br> | |||
==See also== | ==See also== |
Revision as of 11:12, 13 April 2019
Division of the 5th harmonic into 28 equal parts (28ed5) is related to 12 edo, but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 99.5112 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12edo, as well as 18/17, 19/16, and 24/17.
degree | cents value | corresponding JI intervals |
comments |
---|---|---|---|
0 | 0.0000 | exact 1/1 | |
1 | 99.5112 | 18/17 | |
2 | 199.0224 | 55/49 | |
3 | 298.5336 | 19/16 | |
4 | 398.0448 | 34/27 | pseudo-5/4 |
5 | 497.5560 | 4/3 | |
6 | 597.0672 | 24/17 | |
7 | 696.5784 | meantone fifth (pseudo-3/2) | |
8 | 796.0896 | 19/12 | |
9 | 895.6008 | 57/34 | pseudo-5/3 |
10 | 995.1120 | 16/9 | |
11 | 1094.6232 | 32/17 | |
12 | 1194.1344 | 255/128 | pseudo-octave |
13 | 1293.6457 | 19/9 | |
14 | 1393.1569 | 38/17, 85/38 | meantone major second plus an octave |
15 | 1492.6681 | 45/19 | |
16 | 1592.1793 | 128/51 | pseudo-5/2 |
17 | 1691.6905 | 85/32 | |
18 | 1791.2017 | 45/16 | |
19 | 1890.7129 | 170/57 | pseudo-3/1 |
20 | 1990.2241 | 60/19 | |
21 | 2089.7353 | meantone major sixth plus an octave (pseudo-10/3) | |
22 | 2189.2465 | 85/24 | |
23 | 2288.7577 | 15/4 | |
24 | 2388.2689 | 135/34 | pseudo-4/1 |
25 | 2487.7801 | 80/19 | |
26 | 2587.2913 | 49/11 | |
27 | 2686.8025 | 85/18 | |
28 | 2786.3137 | exact 5/1 | just major third plus two octaves |
28ed5 as a generator
28ed5 can also be thought of as a generator of the 2.3.5.17.19 subgroup temperament which tempers out 1216/1215, 1445/1444, and 6144/6137, which is a cluster temperament with 12 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 1088/1083 ~ 256/255 ~ 289/288 ~ 324/323 ~ 361/360 all tempered together. This temperament is supported by 12edo, 205edo, and 217edo among others.
5-limit 12&193 (quinsa-quingu)
Comma: |56 -28 -5>
POTE generator: ~4428675/4194304 = 99.526
Map: [<1 2 0|, <0 -5 28|]
EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603, 627, 639, 784, 808, 832, 856, 989, 1001, 1013, 1037, 1049, 1061, 1242
Badness: 0.399849
7-limit 12&193
Commas: 5120/5103, 9765625/9680832
POTE generator: ~625/588 = 99.483
Map: [<1 2 0 -2|, <0 -5 28 58|]
EDOs: 12, 169, 181, 193, 205, 374
Badness: 0.155536
11-limit 12&193
Commas: 1375/1372, 4375/4356, 5120/5103
POTE generator: ~35/33 = 99.472
Map: [<1 2 0 -2 -4|, <0 -5 28 58 90|]
EDOs: 12, 169e, 181, 193, 205e, 374
Badness: 0.073158
7-limit 12&229
Commas: 3136/3125, 33554432/33480783
POTE generator: ~200/189 = 99.555
Map: [<1 2 0 -3|, <0 -5 28 70|]
EDOs: 12, 217, 229, 241, 446
Badness: 0.142897
11-limit 12&229
Commas: 3136/3125, 8019/8000, 15488/15435
POTE generator: ~200/189 = 99.570
Map: [<1 2 0 -3 -6|, <0 -5 28 70 114|]
EDOs: 12, 217e, 229, 241
Badness: 0.093971
7-limit 12&422
Commas: 102760448/102515625, 1220703125/1219784832
POTE generator: ~1323/1250 = 99.521
Map: [<2 4 0 -5|, <0 -5 28 64|]
EDOs: 12, 398, 410, 422, 808, 832, 1242
Badness: 0.233140
11-limit 12&422
Commas: 5632/5625, 9801/9800, 85937500/85766121
POTE generator: ~1323/1250 = 99.525
Map: [<2 4 0 -5 -10|, <0 -5 28 64 102|]
EDOs: 12, 410, 422, 832
Badness: 0.093926
2.3.5.17.19 subgroup 12&193
Commas: 1216/1215, 1445/1444, 6144/6137
POTE generator: ~18/17 = 99.524
Map: [<1 2 0 5 4|, <0 -5 28 -11 3|]
EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603