# About 41-equal

41-equal approximates just intonation very closely. Prime 3 is extremely accurate, and primes 5 and 7 are both flat, which means their errors partially cancel out in ratios such as 7/5. Unfortunately prime 11 is sharp, so the errors add up, and 11/10 is nearly 11¢ sharp.

prime | 2/1 | 3/2 | 5/4 | 7/4 | 11/8 | 13/8 | 17/16 | 19/16 |
---|---|---|---|---|---|---|---|---|

error | 0.0¢ | +0.48¢ | -5.8¢ | -3.0¢ | +4.8¢ | +8.3¢ | +12.1¢ | -4.8¢ |

The 41 notes can be named with ups and downs. A sharp equals four ups (two frets), thus an up is a quarter-sharp. A minor 2nd equals three ups. Both 81/80 and 64/63 are equal to one up.

C | ^C | ^^C = vvC# | vC# | C# | ^C# | ||

vDb | Db | ^Db | ^^Db = vvD | vD | D |

P1 | ^1 | ^^1 = vvA1 | vA1 | A1 | ^A1 | ||

vm2 | m2 | ^m2 | ~2 | vM2 | M2 |

All 41 intervals:

- P1 ^1
- vm2 m2 ^m2 ~2 vM2 M2 ^M2
- vm3 m3 ^m3 ~3 vM3 M3 ^M3
- v4 P4 ^4
- ~4/vd5 vA4/d5 A4/^d5 ^A4/~5
- v5 P5 ^5
- vm6 m6 ^m6 ~6 vM6 M6 ^M6
- vm7 m7 ^m7 ~7 vM7 M7 ^M7
- v8 P8

Solfege (-u = **u**p, -o = d**o**wn, -a = pl**a**in, -i = m**i**d, Fr- = **f**lat **R**e, etc.)

- Da Du
- Fro Fra Fru Ri Ro Ra Ru
- No Na Nu Mi Mo Ma Mu
- Fo Fa Fu
- Fi/Sho Po/Sha Pa/Shu Pu/Si
- So Sa Su
- Flo Fla Flu Li Lo La Lu
- Tho Tha Thu Ti To Ta Tu
- Do Da

The most dissonant intervals are the off-perfect ones ^1, v4, ^4, v5, ^5 and v8.

step | cents | interval | ||
---|---|---|---|---|

0 | 0¢ | perfect unison | P1 | D |

1 | 29 | up-unison | ^1 | ^D |

2 | 59 | double-up 1sn, downminor 2nd | ^^1, vm2 | ^^D, vEb |

3 | 88 | down-aug 1sn, minor 2nd | vA1, m2 | vD#, Eb |

4 | 117 | augmented 1sn, upminor 2nd | A1, ^m2 | D#, ^Eb |

5 | 146 | mid 2nd | ~2 | ^D#, vvE |

6 | 176 | downmajor 2nd | vM2 | vE |

7 | 205 | major 2nd | M2 | E |

8 | 234 | upmajor 2nd | ^M2 | ^E |

9 | 263 | downminor 3rd | vm3 | vF |

10 | 293 | minor 3rd | m3 | F |

11 | 322 | upminor 3rd | ^m3 | ^F |

12 | 351 | mid 3rd | ~3 | ^^F, vGb |

13 | 380 | downmajor 3rd | vM3 | vF#, Gb |

14 | 410 | major 3rd | M3 | F#, ^Gb |

15 | 439 | upmajor 3rd | ^M3 | ^F#, vvG |

16 | 468 | down-4th | v4 | vG |

17 | 498 | perfect 4th | P4 | G |

18 | 527 | up-4th | ^4 | ^G |

19 | 556 | mid-4th | ~4 | ^^G, vAb |

20 | 585 | downaug 4th, dim 5th | vA4, d5 | vG#, Ab |

21 | 615 | aug 4th, updim 5th | A4, ^d5 | G#, ^Ab |

22 | 644 | mid-5th | ~5 | vvA |

23 | 673 | down-5th | v5 | vA |

24 | 702 | perfect 5th | P5 | A |

25 | 732 | up-5th | ^5 | ^A |

26 | 761 | downminor 6th | vm6 | ^^A, vBb |

27 | 790 | minor 6th | m6 | vA#, Bb |

28 | 820 | upminor 6th | ^m6 | A#, ^Bb |

29 | 849 | mid 6th | ~6 | ^A#, vvB |

30 | 878 | downmajor 6th | vM6 | vB |

31 | 907 | major 6th | M6 | B |

32 | 937 | upmajor 6th | ^M6 | ^B |

33 | 966 | downminor 7th | vm7 | vC |

34 | 995 | minor 7th | m7 | C |

35 | 1024 | upminor 7th | ^m7 | ^C |

36 | 1054 | mid 7th | ~7 | ^^C, vDb |

37 | 1083 | downmajor 7th | vM7 | vC#, Db |

38 | 1112 | major 7th | M7 | C#, ^Db |

39 | 1141 | upmajor 7th | ^M7 | C#^, vvD |

40 | 1171 | dim 8ve | v8 | vD |

41 | 1200 | perfect 8ve | P8 | D |

This chart shows 41-equal in terms of 12-equal. "-ish" means ±1 step of 41. The 12 categories circled in red correspond to the notes of 12-equal. The two innermost and two outermost intervals are duplicates.

These charts show the 7-limit and 11-limit lattices with 41-equal names and steps of 41 in place of ratios. Each color represents a separate plane of the lattice. In the 2nd lattice, the ~3 note in the center next to the P1 is simultaneously 7-limit (49/40 and 60/49), 11-limit (11/9 and 27/22) and 13-limit (16/13 and 39/32).