21edf: Difference between revisions

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Lookalikes: [[36edo]], [[57edt]]
Lookalikes: [[36edo]], [[57edt]]


=Approximations=
==3-limit (Pythagorean) approximations (same as 7edf):==
2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.
4/3 = 498.045... cents; 15 degrees of 21edf = 501.9634... cents.
9/8 = 203.910... cents; 6 degrees of 21edf = 200.5585... cents.
16/9 = 996.090... cents; 30 degrees of 21edf = 1002.7928... cents.
27/16 = 905.865... cents; 27 degrees of 21edf = 902.5135... cents.
32/27 = 294.135... cents; 9 degrees of 21edf = 300.8379... cents.
81/64 = 407.820... cents; 12 degrees of 21edf = 401.1171... cents.
128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.
==7-limit approximations:==
===7 only:===
7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.
8/7 = 231.174... cents; 7 degrees of 21edf = 233.985... cents.
49/32 = 737.652... cents; 22 degrees of 21edf = 733.333... cents.
64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.
===3 and 7:===
7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.
12/7 = 933.129... cents; 28 degrees of 21edf = 934.94... cents.
9/7 = 435.084... cents; 13 degrees of 21edf = 434.5435... cents.
14/9 = 764.916... cents; 23 degrees of 21edf = 768.8078... cents.
28/27 = 62.961... cents; 2 degrees of 21edf = 66.8528... cents.
27/14 = 1137.039... cents; 34 degrees of 21edf = 1136.4985... cents.
21/16 = 470.781... cents; 14 degrees of 21edf = 467.97... cents.
32/21 = 729.219... cents; 22 degrees of 21edf = 735.3814... cents.
49/48 = 35.697... cents; 1 degree of 21edf = 33.4264... cents.
96/49 = 1164.303... cents; 35 degrees of 21edf = 1169.925... cents.
49/36 = 533.742... cents; 16 degrees of 21edf = 534.8228... cents.
72/49 = 666.258... cents; 20 degrees of 21edf = 668.5285... cents.
64/63 = 27.264... cents; 1 degree of 21edf = 33.4264... cents.
63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.
The following table gives an overview of all degrees of 36edo.
{| class="wikitable"
|-
! |Degree
! |Size
in [[Cent|cents]]
! |Approximate
ratios of 2.3.7
! |Additional ratios
of 2.3.7.13.17
|-
| style="text-align:center;" |0
| style="text-align:right;" |0.00
| style="text-align:center;" |1/1
| style="text-align:center;" |
|-
| style="text-align:center;" |1
| style="text-align:right;" |33.4264
| style="text-align:center;" |64/63, [[49/48]]
| style="text-align:center;" |
|-
| style="text-align:center;" |2
| style="text-align:right;" |66.8529
| style="text-align:center;" |[[28/27]]
| style="text-align:center;" |
|-
| style="text-align:center;" |3
| style="text-align:right;" |100.2793
| style="text-align:center;" |256/243
| style="text-align:center;" |[[17/16]], [[18/17]]
|-
| style="text-align:center;" |4
| style="text-align:right;" |133.7057
| style="text-align:center;" |243/224
| style="text-align:center;" |[[14/13]], [[13/12]]
|-
| style="text-align:center;" |5
| style="text-align:right;" |167.1321
| style="text-align:center;" |[[54/49]]
| style="text-align:center;" |
|-
| style="text-align:center;" |6
| style="text-align:right;" |200.5586
| style="text-align:center;" |[[9/8]]
| style="text-align:center;" |
|-
| style="text-align:center;" |7
| style="text-align:right;" |233.985
| style="text-align:center;" |[[8/7]]
| style="text-align:center;" |
|-
| style="text-align:center;" |8
| style="text-align:right;" |267.4114
| style="text-align:center;" |[[7/6]]
| style="text-align:center;" |
|-
| style="text-align:center;" |9
| style="text-align:right;" |300.8379
| style="text-align:center;" |[[32/27]]
| style="text-align:center;" |
|-
| style="text-align:center;" |10
| style="text-align:right;" |334.2643
| style="text-align:center;" |98/81
| style="text-align:center;" |[[17/14]]
|-
| style="text-align:center;" |11
| style="text-align:right;" |367.6907
| style="text-align:center;" |243/196
| style="text-align:center;" |[[16/13]], [[26/21]], [[21/17]]
|-
| style="text-align:center;" |12
| style="text-align:right;" |401.1171
| style="text-align:center;" |[[81/64]]
| style="text-align:center;" |
|-
| style="text-align:center;" |13
| style="text-align:right;" |434.5436
| style="text-align:center;" |[[9/7]]
| style="text-align:center;" |
|-
| style="text-align:center;" |14
| style="text-align:right;" |467.97
| style="text-align:center;" |[[64/49]], [[21/16]]
| style="text-align:center;" |[[17/13]]
|-
| style="text-align:center;" |15
| style="text-align:right;" |501.3964
| style="text-align:center;" |[[4/3]]
| style="text-align:center;" |
|-
| style="text-align:center;" |16
| style="text-align:right;" |534.8229
| style="text-align:center;" |[[49/36]]
| style="text-align:center;" |
|-
| style="text-align:center;" |17
| style="text-align:right;" |568.2493
| style="text-align:center;" |
| style="text-align:center;" |[[18/13]]
|-
| style="text-align:center;" |18
| style="text-align:right;" |601.6757
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" |19
| style="text-align:right;" |635.1021
| style="text-align:center;" |
| style="text-align:center;" |[[13/9]]
|-
| style="text-align:center;" |20
| style="text-align:right;" |668.5286
| style="text-align:center;" |72/49
| style="text-align:center;" |
|-
| style="text-align:center;" |21
| style="text-align:right;" |701.955
| style="text-align:center;" |[[3/2]]
| style="text-align:center;" |
|-
| style="text-align:center;" |22
| style="text-align:right;" |735.3814
| style="text-align:center;" |[[49/32]], [[32/21]]
| style="text-align:center;" |[[26/17]]
|-
| style="text-align:center;" |23
| style="text-align:right;" |768.8079
| style="text-align:center;" |[[14/9]]
| style="text-align:center;" |
|-
| style="text-align:center;" |24
| style="text-align:right;" |802.2343
| style="text-align:center;" |[[128/81]]
| style="text-align:center;" |
|-
| style="text-align:center;" |25
| style="text-align:right;" |835.6607
| style="text-align:center;" |392/243
| style="text-align:center;" |[[13/8]], [[21/13]], [[34/21]]
|-
| style="text-align:center;" |26
| style="text-align:right;" |869.0871
| style="text-align:center;" |81/49
| style="text-align:center;" |[[28/17]]
|-
| style="text-align:center;" |27
| style="text-align:right;" |902.5136
| style="text-align:center;" |[[27/16]]
| style="text-align:center;" |
|-
| style="text-align:center;" |28
| style="text-align:right;" |935.94
| style="text-align:center;" |[[12/7]]
| style="text-align:center;" |
|-
| style="text-align:center;" |29
| style="text-align:right;" |969.3664
| style="text-align:center;" |[[7/4]]
| style="text-align:center;" |
|-
| style="text-align:center;" |30
| style="text-align:right;" |1002.7929
| style="text-align:center;" |[[16/9]]
| style="text-align:center;" |
|-
| style="text-align:center;" |31
| style="text-align:right;" |1036.2193
| style="text-align:center;" |49/27
| style="text-align:center;" |
|-
| style="text-align:center;" |32
| style="text-align:right;" |1069.6457
| style="text-align:center;" |448/243
| style="text-align:center;" |[[13/7]], [[24/13]]
|-
| style="text-align:center;" |33
| style="text-align:right;" |1103.0721
| style="text-align:center;" |[[243/128]]
| style="text-align:center;" |[[32/17]], [[17/9]]
|-
| style="text-align:center;" |34
| style="text-align:right;" |1136.4986
| style="text-align:center;" |[[27/14]]
| style="text-align:center;" |
|-
| style="text-align:center;" |35
| style="text-align:right;" |1169.925
| style="text-align:center;" |63/32, 96/49
| style="text-align:center;" |
|-
| style="text-align:center;" |36
| style="text-align:right;" |1203.3514
| style="text-align:center;" |2/1
| style="text-align:center;" |
|}
[[Category:Edf]]
[[Category:Edf]]
[[Category:Edonoi]]
[[Category:Edonoi]]

Revision as of 22:56, 20 February 2019

Division of the just perfect fifth into 21 equal parts (21EDF) is related to 36 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 33.4264 cents. Unlike 36edo, it is only consistent up to the 4-integer-limit, with discrepancy for the 5th harmonic.

Lookalikes: 36edo, 57edt

Approximations

3-limit (Pythagorean) approximations (same as 7edf):

2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.

4/3 = 498.045... cents; 15 degrees of 21edf = 501.9634... cents.

9/8 = 203.910... cents; 6 degrees of 21edf = 200.5585... cents.

16/9 = 996.090... cents; 30 degrees of 21edf = 1002.7928... cents.

27/16 = 905.865... cents; 27 degrees of 21edf = 902.5135... cents.

32/27 = 294.135... cents; 9 degrees of 21edf = 300.8379... cents.

81/64 = 407.820... cents; 12 degrees of 21edf = 401.1171... cents.

128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.

7-limit approximations:

7 only:

7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.

8/7 = 231.174... cents; 7 degrees of 21edf = 233.985... cents.

49/32 = 737.652... cents; 22 degrees of 21edf = 733.333... cents.

64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.

3 and 7:

7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.

12/7 = 933.129... cents; 28 degrees of 21edf = 934.94... cents.

9/7 = 435.084... cents; 13 degrees of 21edf = 434.5435... cents.

14/9 = 764.916... cents; 23 degrees of 21edf = 768.8078... cents.

28/27 = 62.961... cents; 2 degrees of 21edf = 66.8528... cents.

27/14 = 1137.039... cents; 34 degrees of 21edf = 1136.4985... cents.

21/16 = 470.781... cents; 14 degrees of 21edf = 467.97... cents.

32/21 = 729.219... cents; 22 degrees of 21edf = 735.3814... cents.

49/48 = 35.697... cents; 1 degree of 21edf = 33.4264... cents.

96/49 = 1164.303... cents; 35 degrees of 21edf = 1169.925... cents.

49/36 = 533.742... cents; 16 degrees of 21edf = 534.8228... cents.

72/49 = 666.258... cents; 20 degrees of 21edf = 668.5285... cents.

64/63 = 27.264... cents; 1 degree of 21edf = 33.4264... cents.

63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.

The following table gives an overview of all degrees of 36edo.

Degree Size

in cents

Approximate

ratios of 2.3.7

Additional ratios

of 2.3.7.13.17

0 0.00 1/1
1 33.4264 64/63, 49/48
2 66.8529 28/27
3 100.2793 256/243 17/16, 18/17
4 133.7057 243/224 14/13, 13/12
5 167.1321 54/49
6 200.5586 9/8
7 233.985 8/7
8 267.4114 7/6
9 300.8379 32/27
10 334.2643 98/81 17/14
11 367.6907 243/196 16/13, 26/21, 21/17
12 401.1171 81/64
13 434.5436 9/7
14 467.97 64/49, 21/16 17/13
15 501.3964 4/3
16 534.8229 49/36
17 568.2493 18/13
18 601.6757
19 635.1021 13/9
20 668.5286 72/49
21 701.955 3/2
22 735.3814 49/32, 32/21 26/17
23 768.8079 14/9
24 802.2343 128/81
25 835.6607 392/243 13/8, 21/13, 34/21
26 869.0871 81/49 28/17
27 902.5136 27/16
28 935.94 12/7
29 969.3664 7/4
30 1002.7929 16/9
31 1036.2193 49/27
32 1069.6457 448/243 13/7, 24/13
33 1103.0721 243/128 32/17, 17/9
34 1136.4986 27/14
35 1169.925 63/32, 96/49
36 1203.3514 2/1
Retrieved from "https://en.xen.wiki/index.php?title=21edf&oldid=40411"