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), and a minor 2nd equals three ups.

41-equal notes from C to D
C ^C ^^C = vvC# vC# C# ^C#
vDb Db ^Db ^^Db = vvD vD D
41-equal intervals from P1 to M2
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 vA4/d5 A4/^d5 ~5 v5 P5 ^5
  • vm6 m6 ^m6 ~6 vM6 M6 ^M6
  • vm7 m7 ^m7 ~7 vM7 M7 ^M7
  • v8 P8

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

step cents Ups and Downs Notation
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).

Scroll to Top