47edo

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47edo

48edo →

Prime factorization

47 (prime)

Step size

25.5319¢

Fifth

27\47 (689.362¢)

Semitones (A1:m2)

1:6 (25.53¢ : 153.2¢)

Dual sharp fifth

28\47 (714.894¢)

Dual flat fifth

27\47 (689.362¢)

Dual major 2nd

8\47 (204.255¢)
(convergent)

Consistency limit

5

Distinct consistency limit

5

47 equal divisions of the octave (abbreviated 47edo or 47ed2), also called 47-tone equal temperament (47tet) or 47 equal temperament (47et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 47 equal parts of about 25.5 ¢ each. Each step represents a frequency ratio of 2^1/47, or the 47th root of 2.

Theory

47edo has a fifth which is 12.593-cent flat, unless you use the alternative fifth which is 12.939-cent sharp, similar to 35edo. The soft diatonic scale generated from its flat fifth is so soft, with L/s = 7/6, that it stops sounding like meantone or even a flattone system like 26edo or 40edo, but just sounds like a circulating temperament of 7edo. It has therefore not aroused much interest, but its best approximation to 9/8 is actually quite good, one-third-of-a-cent sharp.

47edo is a diatonic edo because its 5th falls between 4\7 = 686 ¢ and 3\5 = 720 ¢, as does its alternate 5th as well. 47edo is one of the most difficult diatonic edos to notate, because no other diatonic edos 5th is as flat (see 42edo for the opposite extreme).

Odd harmonics

47edo does a good job of approximating the 2.9.5.7.33.13.17.57.69 23-limit 2*47 subgroup of the 23-limit, on which it tempers out the same commas as 94edo. It provides a good tuning for baldy and silver and their relatives. It also provides a good tuning for the baseball temperament.

47edo can be treated as a dual-fifth system in the 2.3+.3-.5.7.13 subgroup, or the 3+.3-.5.7.11+.11-.13 subgroup for those who aren’t intimidated by lots of basis elements. As a dual-fifth system, it really shines, as both of its fifths have low enough harmonic entropy to sound consonant to many listeners, giving two consonant intervals for the price of one.

Approximation of odd harmonics in 47edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-12.6	-3.3	+1.4	+0.3	+10.4	+2.0	+9.6	-2.8	+8.9	-11.2	+10.0
Error	Relative (%)	-49.3	-13.1	+5.4	+1.4	+40.7	+7.9	+37.6	-11.1	+34.7	-43.9	+39.3
Steps (reduced)		74 (27)	109 (15)	132 (38)	149 (8)	163 (22)	174 (33)	184 (43)	192 (4)	200 (12)	206 (18)	213 (25)

Subsets and supersets

47edo is the 15th prime edo, following 43edo and preceding 53edo.

Intervals

#	Cents	Relative notation		Absolute notation
0	0.0000	perfect unison	P1	D
1	25.5319	aug 1sn	A1	D#
2	51.0638	double-aug 1sn	AA1	Dx
3	76.5957	triple-aug 1sn, triple-dim 2nd	A³1, d³2	D#³, Eb⁴
4	102.1277	double-dim 2nd	dd2	Eb³
5	127.6596	dim 2nd	d2	Ebb
6	153.1915	minor 2nd	m2	Eb
7	178.7234	major 2nd	M2	E
8	204.2553	aug 2nd	A2	E#
9	229.7872	double-aug 2nd	AA2	Ex
10	255.3191	triple-aug 2nd, triple-dim 3rd	A³2, d³3	E#³, Fb³
11	280.8511	double-dim 3rd	dd3	Fbb
12	306.3830	dim 3rd	d3	Fb
13	331.9149	minor 3rd	m3	F
14	357.4468	major 3rd	M3	F#
15	382.9787	aug 3rd	A3	Fx
16	408.5106	double-aug 3rd	AA3	F#³
17	434.0426	triple-aug 3rd, triple-dim 4th	A³3, d³4	F#⁴, Gb³
18	459.5745	double-dim 4th	dd4	Gbb
19	485.1064	dim 4th	d4	Gb
20	510.6383	perfect 4th	P4	G
21	536.1702	aug 4th	A4	G#
22	561.7021	double-aug 4th	AA4	Gx
23	587.2340	triple-aug 4th	A³4	G#³
24	612.7660	triple-dim 5th	d³5	Ab³
25	638.2979	double-dim 5th	dd5	Abb
26	663.8298	dim 5th	d5	Ab
27	689.3617	perfect 5th	P5	A
28	714.8936	aug 5th	A5	A#
29	740.4255	double-aug 5th	AA5	Ax
30	765.9574	triple-aug 5th, triple-dim 6th	A³5, d³6	A#³, Bb⁴
31	791.4894	double-dim 6th	dd6	Bb³
32	817.0213	dim 6th	d6	Bbb
33	842.5532	minor 6th	m6	Bb
34	868.0851	major 6th	M6	B
35	893.6170	aug 6th	A6	B#
36	919.1489	double-aug 6th	AA6	Bx
37	944.6809	triple-aug 6th, triple-dim 7th	A³6, d³7	B#³, Cb³
38	970.2128	double-dim 7th	dd7	Cbb
39	995.7447	dim 7th	d7	Cb
40	1021.2766	minor 7th	m7	C
41	1046.8085	major 7th	M7	C#
42	1072.3404	aug 7th	A7	Cx
43	1097.8723	double-aug 7th	AA7	C#³
44	1123.4043	triple-aug 7th, triple-dim 8ve	A³7, d³8	C#⁴, Db³
45	1148.9362	double-dim 8ve	dd8	Dbb
46	1174.4681	dim 8ve	d8	Db
47	1200.0000	perfect 8ve	P8	D

Notation

A notation using the best 5th has major and minor 2nds of 7 and 6 edosteps respectively, with the naturals creating a 7edo-like scale:

D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D

D# is next to D. This notation requires triple, quadruple and in some keys, quintuple or more sharps and flats. For example, a 0-15-27-38 chord (an approximate 4:5:6:7) on the note three edosteps above D would be spelled either as D#³ - F#⁵ - A#³ - C# or as Eb⁴ - Gbb - Ab⁴ - Db⁶. This is an aug-three double-dim-seven chord, written D#³(A3)dd7 or Eb⁴(A3)dd7. It could also be called a sharp-three triple-flat-seven chord, written D#³(#3)b³7 or Eb⁴(#3)b³7.

Using the 2nd best 5th is even more awkward. The major 2nd is 9 edosteps and the minor is only one. The naturals create a 5edo-like scale, with two of the notes inflected by a comma-sized edostep:

D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D

D# is next to E. This notation requires quadruple, quintuple, and even sextuple ups and downs, as well as single sharps and flats.