4172edo: Difference between revisions

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4172edo

4173edo →

Prime factorization

2² × 7 × 149

Step size

0.287632 ¢

Fifth

2440\4172 (701.822 ¢) (→ 610\1043)

Semitones (A1:m2)

392:316 (112.8 ¢ : 90.89 ¢)

Dual sharp fifth

2441\4172 (702.109 ¢)

Dual flat fifth

2440\4172 (701.822 ¢) (→ 610\1043)

Dual major 2nd

709\4172 (203.931 ¢)

Consistency limit

7

Distinct consistency limit

7

4172 equal divisions of the octave (abbreviated 4172edo or 4172ed2), also called 4172-tone equal temperament (4172tet) or 4172 equal temperament (4172et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4172 equal parts of about 0.288 ¢ each. Each step represents a frequency ratio of 2^1/4172, or the 4172nd root of 2.

Theory

The first 8 prime harmonics below 25% in 4172edo are 2, 5, 13, 17, 31, 37, 53, 61. Therefore, 4172edo can be thought of as a 2.5.13.17.31.37.53.61 subgroup temperament.

Subsets and supersets

4172's divisors are 1, 2, 4, 7, 14, 28, 149, 298, 596, 1043, 2086. Notable member of the group is 149edo, which is the smallest edo uniquely consistent in the 17-odd limit, although its approximations have long been diluted by edo of this size.

Odd harmonics

Approximation of odd harmonics in 4172edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-0.133	-0.024	-0.082	+0.021	+0.072	-0.067	+0.130	+0.030	-0.102	+0.072	-0.086
Error	Relative (%)	-46.4	-8.4	-28.5	+7.3	+25.1	-23.5	+45.2	+10.5	-35.4	+25.2	-30.0
Steps (reduced)		6612 (2440)	9687 (1343)	11712 (3368)	13225 (709)	14433 (1917)	15438 (2922)	16300 (3784)	17053 (365)	17722 (1034)	18325 (1637)	18872 (2184)

Todo: explain its xenharmonic value

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|4172}}
+{{ED intro}}
-==Theory==
+== Theory ==
-{{Harmonics in equal|4172}}
+The first 8 prime harmonics below 25% in 4172edo are 2, 5, 13, 17, 31, 37, 53, 61. Therefore, 4172edo can be thought of as a 2.5.13.17.31.37.53.61 subgroup temperament.
-The first 8 prime harmonics below 25% in 4172edo are 2, 5, 13, 17, 31, 37, 53, 61. Therefore, 4172edo can be thought of as a 2.5.13.17.31.37.53.61 subgroup temperament, on which it is consistent. Other than that, it offers satisfactory representation of the 13-odd-limit (<28% error).
+=== Subsets and supersets ===
+'s divisors are {{EDOs|1, 2, 4, 7, 14, 28, 149, 298, 596, 1043, 2086}}. Notable member of the group is 149edo, which is the smallest edo uniquely consistent in the 17-odd limit, although its approximations have long been diluted by edo of this size.
-'s divisors are {{EDOs|1, 2, 4, 7, 14, 28, 149, 298, 596, 1043, 2086}}. Notable member of the group is 149edo, which is the smallest edo uniquely consistent in the 17-odd limit. Therefore from a logarithmic pitch or highly composite EDO theory perspective, 4172edo can be thought of as a compound of 28 149edos interlocked together.
+=== Odd harmonics ===
+{{harmonics in equal|4172}}
 <!-- 4-digit number -->
 [[Category:Equal divisions of the octave|####]]
+{{todo|inline=1|explain its xenharmonic value}}