# 44edo

← 43edo | 44edo | 45edo → |

^{2}× 11**44 equal divisions of the octave** (abbreviated **44edo** or **44ed2**), also called **44-tone equal temperament** (**44tet**) or **44 equal temperament** (**44et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 44 equal parts of about 27.3 ¢ each. Each step represents a frequency ratio of 2^{1/44}, or the 44th root of 2.

## Theory

44edo is a double of 22edo, to which it adds the ratios of 13, 19, and 23. While not the most accurate 2.3.5.7.11 tuning, 22edo is certainly a relatively compact one, and it's natural to extend it this way. The most practically useful of these additions is easily the 13th harmonic with its neutral intervals, but the 17th, 19th, and 23rd are not to be dismissed. In the 13-limit it supplies the optimal patent val for vigin temperament. The 2*44 subgroup of 44edo is 2.9.5.21.11.13.17.19.23, on which 44 tempers out the same commas as the patent val for 88edo.

### Harmonics

Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +7.1 | -4.5 | +13.0 | -13.0 | -5.9 | +4.9 | +2.6 | +4.1 | +2.5 | -7.1 | -1.0 |

Relative (%) | +26.2 | -16.5 | +47.6 | -47.7 | -21.5 | +18.1 | +9.7 | +15.2 | +9.1 | -26.2 | -3.7 | |

Steps (reduced) |
70 (26) |
102 (14) |
124 (36) |
139 (7) |
152 (20) |
163 (31) |
172 (40) |
180 (4) |
187 (11) |
193 (17) |
199 (23) |

### Subsets and supersets

44edo has subsets 2, 4, 11, 22.

One step of 44edo is very close (only 0.0086 cents sharp) to 64/63 (the septimal comma). Ruthenium temperament realizes this proximity through a regular temperament perspective, and it is supported by a large number of edos which are a multiple of 44 - for example 1012edo, 1848edo, and 2684edo.

## Intervals

In 44edo, sharps and flats alter pitch by 6 EDOsteps. This means intervals can be notated with half sharps and half flats equal to 3 EDOsteps, in addition to ups and downs. The table below uses only sharps, flats, and ups and downs. When translating music from 22edo to 44edo, single ups and downs simply become double ups and downs (vEb in 22edo would be vvEb in 44edo).

Degrees | Cents | Ups and Downs Notation | ||
---|---|---|---|---|

0 | 0.000 | Perfect 1sn | P1 | D |

1 | 27.273 | Up 1sn | ^1 | ^D |

2 | 54.545 | Minor 2nd | m2 | Eb |

3 | 81.818 | Upminor 2nd | ^m2 | ^Eb |

4 | 109.091 | Downmid 2nd | v~2 | ^^Eb |

5 | 136.364 | Mid 2nd | ~2 | vvvE, ^^^Eb |

6 | 163.636 | Upmid 2nd | ^~2 | vvE |

7 | 190.909 | Downmajor 2nd | vM2 | vE |

8 | 218.182 | Major 2nd | M2 | E |

9 | 245.455 | Upmajor 2nd, Downminor 3rd | ^M2, vm3 | ^E, vF |

10 | 272.727 | Minor 3rd | m3 | F |

11 | 300.000 | Upminor 3rd | ^m3 | ^F |

12 | 327.273 | Downmid 3rd | v~3 | ^^F |

13 | 354.545 | Mid 3rd | ~3 | ^^^F, vvvF# |

14 | 381.818 | Upmid 3rd | ^~3 | vvF# |

15 | 409.091 | Downmajor 3rd | vM3 | vF# |

16 | 436.364 | Major 3rd | M3 | F# |

17 | 463.636 | Upmajor 3rd, Down 4th | ^M3, v4 | ^F#, vG |

18 | 490.909 | Perfect 4th | P4 | G |

19 | 518.182 | Up 4th | ^4 | ^G |

20 | 545.455 | Dim 5th, Downmid 4th | d5, v~4 | Ab, ^^G |

21 | 572.727 | Mid 4th | ~4 | ^^^G, vvvG# |

22 | 600.000 | Upmid 4th, Downmid 5th | ^~4, v~5 | vvG#, ^^Ab |

23 | 627.273 | Mid 5th | ~5 | vvvA, ^^^Ab |

24 | 654.545 | Aug 4th, Upmid 5th | A4, ^~5 | G#, vvA |

25 | 681.818 | Down 5th | v5 | vA |

26 | 709.091 | Perfect 5th | P5 | A |

27 | 736.364 | Up 5th, Downminor 6th | ^5, vm6 | ^A, vBb |

28 | 763.636 | Minor 6th | m6 | Bb |

29 | 790.909 | Upminor 6th | ^m6 | ^Bb |

30 | 818.182 | Downmid 6th | v~6 | ^^Bb |

31 | 845.455 | Mid 6th | ~6 | ^^^Bb, vvvB |

32 | 872.727 | Upmid 6th | ^~6 | vvB |

33 | 900.000 | Downmajor 6th | vM6 | vB |

34 | 927.273 | Major 6th | M6 | B |

35 | 954.545 | Upmajor 6th, Downminor 7th | ^M6, vm7 | ^B, vC |

36 | 981.818 | Minor 7th | m7 | C |

37 | 1009.091 | Upminor 7th | ^m7 | ^C |

38 | 1036.364 | Downmid 7th | v~7 | ^^C |

39 | 1063.636 | Mid 7th | ~7 | ^^^C, vvvC# |

40 | 1090.909 | Upmid 7th | ^~7 | vvC# |

41 | 1118.182 | Downmajor 7th | vM7 | vC# |

42 | 1145.455 | Major 7th | M7 | C# |

43 | 1172.727 | Upmajor 7th, Down 8ve | ^M7, v8 | ^C#, vD |

44 | 1200.000 | Perfect 8ve | P8 | D |