1547edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 1546edo1547edo1548edo →
Prime factorization 7 × 13 × 17
Step size 0.775695¢
Fifth 905\1547 (702.004¢)
Semitones (A1:m2) 147:116 (114¢ : 89.98¢)
Consistency limit 15
Distinct consistency limit 15

1547 equal divisions of the octave (abbreviated 1547edo), or 1547-tone equal temperament (1547tet), 1547 equal temperament (1547et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1547 equal parts of about 0.776 ¢ each. Each step of 1547edo represents a frequency ratio of 21/1547, or the 1547th root of 2.

Theory

1547edo is consistent to the 15-odd-limit and is excellent in the 7-limit. It tempers out 4375/4374 and it is a member of the optimal ET sequence for the rank-3 temperament associated with this comma.

In the 5-limit, it supports gross, which is a very high-accuracy temperament. The 118-tone maximal evenness scale produced by gross is concoctic, since it uses 118\1547 as the generator. In addition, 1547edo tempers out the septendecima and thus supports the chlorine temperament in 5-limit and also in the 7-limit. 1547edo tempers out the 5-limit comma [236 -61 -60, thus associating a stack of sixty 15/8's with 4/3, and sixty-one of them make 5/4.

In the 7-limit, it provides the optimal patent val for 7-limit brahmagupta, the 441 & 1106 temperament, and supports an alternative 11-limit extension to it. It also supports semidimi, the 171 & 1376 temperament.

In the 11-limit, 1547edo provides the optimal patent val for the aluminium temperament, which maps 135/128 to 1/13th of the occtave. It also tempers out 117649/117612, and is a tuning for the rank-3 temperament heimdall. In higher limits, it supports 91th-octave temperament protactinium.

Prime harmonics

Approximation of prime harmonics in 1547edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.000 +0.049 -0.018 +0.017 +0.201 +0.326 -0.237 +0.354 +0.039 -0.230 -0.110
relative (%) +0 +6 -2 +2 +26 +42 -31 +46 +5 -30 -14
Steps
(reduced)
1547
(0)
2452
(905)
3592
(498)
4343
(1249)
5352
(711)
5725
(1084)
6323
(135)
6572
(384)
6998
(810)
7515
(1327)
7664
(1476)

Subsets and supersets

Since 1547 factors into 7 × 13 × 17, 1547edo has subset edos 7, 13, 17, 91, 119, and 221.

Regular temperament properties

Subgroup Comma List Mapping Optimal 8ve
Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [2452 -1547 [1547 2542]] -0.015 0.015 1.99
2.3.5 [-52 -17 34, [40 -56 21 [1547 2542 3592]] -0.008 0.017 2.14
2.3.5.7 4375/4374, [-1 4 11 -11, [46 -14 -3  -6 [1547 2542 3592 4343]] -0.007 0.014 1.86
2.3.5.7.11 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 [1547 2542 3592 4343 5352]] -0.017 0.024 3.10
2.3.5.7.11.13 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 [1547 2542 3592 4343 5352 5725]] -0.029 0.034 4.42

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 118\1547 91.532 [9 -32 18 Gross
1 579\1547 449.127 35/27 Semidimi
7 670\1547
(7\1547)
519.715
(5.429)
27/20
(325/324)
Brahmagupta (7-limit)
7 11\1547 8.533 1029/1024 Nitrogen
13 642\1547
(47\1547)
497.996
(36.458)
4/3
(?)
Aluminium
17 321\1547
(48\1547)
248.998
(37.233)
[-23 5 9 -2
(100352/98415)
Chlorine
91 642\1547
(13\1547)
497.996
(10.084)
4/3
(176/175)
Protactinium

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Eliora