# 145edo

 ← 144edo 145edo 146edo →
Prime factorization 5 × 29
Step size 8.27586¢
Fifth 85\145 (703.448¢) (→17\29)
Semitones (A1:m2) 15:10 (124.1¢ : 82.76¢)
Consistency limit 11
Distinct consistency limit 11

145 equal divisions of the octave (abbreviated 145edo or 145ed2), also called 145-tone equal temperament (145tet) or 145 equal temperament (145et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 145 equal parts of about 8.28 ¢ each. Each step represents a frequency ratio of 21/145, or the 145th root of 2.

## Theory

145et tempers out 1600000/1594323 in the 5-limit; 4375/4374 and 5120/5103 in the 7-limit; 441/440 and 896/891 in the 11-limit; 196/195, 352/351 and 364/363 in the 13-limit; 595/594 in the 17-limit; 343/342 and 476/475 in the 19-limit.

It is the optimal patent val for the 11-limit mystery temperament and the 11-limit rank-3 pele temperament. It also supports and provides a good tuning for 13-limit mystery, and because it tempers out 441/440 it allows werckismic chords, because it tempers out 196/195 it allows mynucumic chords, because it tempers out 352/351 it allows minthmic chords, because it tempers out 364/363 it allows gentle chords, and because it tempers out 847/845 it allows the cuthbert chords, making it a very flexible harmonic system. The same is true of 232edo, the optimal patent val for 13-limit mystery.

The 145c val provides a tuning for magic which is nearly identical to the POTE tuning.

### Prime harmonics

Approximation of odd harmonics in 145edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.49 +2.65 -0.55 +2.99 +3.16 +3.61 -4.13 +2.63 +0.42 +0.94 +0.69
Relative (%) +18.0 +32.0 -6.6 +36.1 +38.2 +43.6 -49.9 +31.8 +5.1 +11.4 +8.4
Steps
(reduced)
230
(85)
337
(47)
407
(117)
460
(25)
502
(67)
537
(102)
566
(131)
593
(13)
616
(36)
637
(57)
656
(76)

### Subsets and supersets

145 = 5 × 29, and 145edo shares the same excellent fifth with 29edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 1600000/1594323, [28 -3 -10 [145 230 337]] -0.695 0.498 6.02
2.3.5.7 4375/4374, 5120/5103, 50421/50000 [145 230 337 407]] -0.472 0.578 6.99
2.3.5.7.11 441/440, 896/891, 3388/3375, 4375/4374 [145 230 337 407 502]] -0.561 0.547 6.61
2.3.5.7.11.13 196/195, 352/351, 364/363, 676/675, 4375/4374 [145 230 337 407 502 537]] -0.630 0.522 6.32
2.3.5.7.11.13.17 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155 [145 230 337 407 502 537 593]] -0.632 0.484 5.85

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 2\145 16.55 100/99 Quincy
1 12\145 99.31 18/17 Quinticosiennic
1 14\145 115.86 77/72 Countermiracle
1 39\145 322.76 3087/2560 Seniority / senator
1 41\145 339.31 128/105 Amity / catamite
5 67\145
(9\145)
554.48
(74.48)
11/8
(25/24)
Trisedodge / countdown
29 60\145
(2\145)
496.55
(16.55)
4/3
(100/99)
Mystery

## Music

Chris Vaisvil (site)