# 21edf

← 20edf | 21edf | 22edf → |

**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.

# 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 = 935.94... 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 | 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 | |

37 | 1236.7779 | 128/63, 49/24 | |

38 | 1270.2043 | 56/27 | |

39 | 1303.6307 | 512/243 | 17/8, 36/17 |

40 | 1337.05715 | 243/112 | 28/13, 13/6 |

41 | 1370.4836 | 108/49 | |

42 | 1403.91 | 9/4 |

## Scale tree

If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger EDOs pop up in our continuum.

Generator range: 32.65306 cents (4\7/21 = 4\147) to 34.28571 cents (3\5/21 = 1\35)

Fifth | Cents | Comments | ||||||
---|---|---|---|---|---|---|---|---|

4\7 | 32.6531 | |||||||

27\47 | 32.82675 | |||||||

23\40 | 32.8571 | |||||||

42\73 | 32.8767 | |||||||

19\33 | 32.9004 | |||||||

53\92 | 32.91925 | |||||||

34\59 | 32.9298 | |||||||

49\85 | 32.9412 | |||||||

15\26 | 33.9670 | |||||||

56\97 | 32.9897 | |||||||

41\71 | 32.9980 | |||||||

67\116 | 33.0049 | |||||||

26\45 | 33.0159 | Flattone is in this region | ||||||

63\109 | 33.0275 | |||||||

37\64 | 33.0357 | |||||||

48\83 | 33.0465 | |||||||

11\19 | 33.0827 | |||||||

51\88 | 33.1167 | |||||||

40\69 | 33.1263 | |||||||

69\119 | 33.13325 | |||||||

29\50 | 33.1429 | |||||||

76\131 | 33.1516 | Golden meantone (696.2145¢) | ||||||

47\81 | 33.1570 | |||||||

65\112 | 33.1633 | |||||||

18\31 | 33.1797 | Meantone is in this region | ||||||

61\105 | 33.1973 | |||||||

43\74 | 33.2046 | |||||||

68\117 | 33.2112 | |||||||

25\43 | 33.2226 | |||||||

57\98 | 33.23615 | |||||||

32\55 | 33.24675 | |||||||

39\67 | 33.2623 | |||||||

7\12 | 33.3 | |||||||

38\65 | 33.4066 | |||||||

31\53 | 33.4232 | The fifth closest to a just 3/2 for EDOs less than 200 | ||||||

55\94 | 33.43465 | Garibaldi / Cassandra | ||||||

24\41 | 33.4495 | |||||||

65\111 | 33.4620 | |||||||

41\70 | 33.4694 | |||||||

58\99 | 33.4776 | |||||||

17\29 | 33.4975 | |||||||

61\104 | 33.5165 | |||||||

44\75 | 33.5328 | |||||||

71\121 | 33.5301 | Golden neogothic (704.0956¢) | ||||||

27\46 | 33.5404 | Neogothic is in this region | ||||||

64\109 | 33.5518 | |||||||

37\63 | 33.5601 | |||||||

47\80 | 33.5714 | |||||||

10\17 | 33.61345 | The generator closest to a just 17/14 for EDOs less than 4200 | ||||||

43\73 | 33.6595 | |||||||

33\56 | 33.6735 | |||||||

56\95 | 33.6842 | |||||||

23\39 | 33.6996 | |||||||

59\100 | 33.7143 | |||||||

36\61 | 33.72365 | |||||||

49\83 | 33.7349 | |||||||

13\22 | 33.7662 | Archy is in this region | ||||||

42\71 | 33.8028 | |||||||

29\49 | 33.8192 | |||||||

45\76 | 33.8346 | |||||||

16\27 | 33.8624 | |||||||

35\59 | 33.8983 | |||||||

19\32 | 33.9286 | |||||||

22\37 | 33.9768 | |||||||

3\5 | 34.2857 |

Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.