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 is the first edo that has two diatonic perfect fifths, as both fall between 4\7 = 686 ¢ and 3\5 = 720 ¢. The fifth closest to 3/2 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. The hard diatonic scale generated from its sharp fifth is extremely hard, with L:s = 9:1. 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 one of the most difficult diatonic edos to notate in native fifths, because no other diatonic edo's fifth is as extreme.

Odd harmonics

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
	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)

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.

Subsets and supersets

47edo is the 15th prime edo, following 43edo and preceding 53edo, so it does not contain any nontrivial subset edos, though it contains 47ed4. 94edo, which doubles it, corrects its approximations of harmonics 3 and 11 to near-just qualities.

Intervals

#	Cents	Relative notation		Absolute notation
0	0.0	perfect unison	P1	D
1	25.5	aug 1sn	A1	D#
2	51.1	double-aug 1sn	AA1	Dx
3	76.6	triple-aug 1sn, triple-dim 2nd	A31, d32	D#3, Eb4
4	102.1	double-dim 2nd	dd2	Eb3
5	127.7	dim 2nd	d2	Ebb
6	153.2	minor 2nd	m2	Eb
7	178.7	major 2nd	M2	E
8	204.3	aug 2nd	A2	E#
9	229.8	double-aug 2nd	AA2	Ex
10	255.3	triple-aug 2nd, triple-dim 3rd	A32, d33	E#3, Fb3
11	280.9	double-dim 3rd	dd3	Fbb
12	306.4	dim 3rd	d3	Fb
13	331.9	minor 3rd	m3	F
14	357.4	major 3rd	M3	F#
15	383.0	aug 3rd	A3	Fx
16	408.5	double-aug 3rd	AA3	F#3
17	434.0	triple-aug 3rd, triple-dim 4th	A33, d34	F#4, Gb3
18	459.6	double-dim 4th	dd4	Gbb
19	485.1	dim 4th	d4	Gb
20	510.6	perfect 4th	P4	G
21	536.2	aug 4th	A4	G#
22	561.7	double-aug 4th	AA4	Gx
23	587.2	triple-aug 4th	A34	G#3
24	612.8	triple-dim 5th	d35	Ab3
25	638.3	double-dim 5th	dd5	Abb
26	663.8	dim 5th	d5	Ab
27	689.4	perfect 5th	P5	A
28	714.9	aug 5th	A5	A#
29	740.4	double-aug 5th	AA5	Ax
30	766.0	triple-aug 5th, triple-dim 6th	A35, d36	A#3, Bb4
31	791.5	double-dim 6th	dd6	Bb3
32	817.0	dim 6th	d6	Bbb
33	842.6	minor 6th	m6	Bb
34	868.1	major 6th	M6	B
35	893.6	aug 6th	A6	B#
36	919.1	double-aug 6th	AA6	Bx
37	944.7	triple-aug 6th, triple-dim 7th	A36, d37	B#3, Cb3
38	970.2	double-dim 7th	dd7	Cbb
39	995.7	dim 7th	d7	Cb
40	1021.3	minor 7th	m7	C
41	1046.8	major 7th	M7	C#
42	1072.3	aug 7th	A7	Cx
43	1097.9	double-aug 7th	AA7	C#3
44	1123.4	triple-aug 7th, triple-dim 8ve	A37, d38	C#4, Db3
45	1148.9	double-dim 8ve	dd8	Dbb
46	1174.5	dim 8ve	d8	Db
47	1200.0	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.

Ups and downs notation

Using Helmholtz–Ellis accidentals and the sharp fifth, 47edo can be notated using ups and downs:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Sharp symbol
Flat symbol

With the flat fifth, notation is identical to standard notation:

Step offset	−2	−1	0	+1	+2
Symbol

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as 42b.

Second-best fifth notation

Evo and Revo flavors

Alternative Evo flavor

Evo-SZ flavor

Scales

• Negri in zeta-stretched 47edo
• Quasi-equal equiheptatonic (Dorian): 7 6 7 7 7 6 7
  • Quasi-equiheptatonic minor pentatonic: 13 7 7 13 7
• Quasi-equal equiheptatonic (Mixolydian): 7 7 6 7 7 6 7
• Quasi-equal equipentatonic: 9 10 9 10 9
• Sabertooth hexatonic: 3 9 3 13 12 7 (this is the original/default tuning; designed for the "gold" and "platinum" timbres in Scale Workshop)
  • Sabertooth pentatonic: 3 9 3 13 19 (this is the original/default tuning)
  • Sabertooth neutral: 3 11 14 11 8 (this is the original/default tuning)

Instruments

Lumatone

• Lumatone mapping for 47edo

Skip fretting

Skip fretting system 47 3 11 is a skip-fretting system for 47edo where strings are 11\47 and frets are 3\47. This is effectively 15.6666...-edo. All examples of this system on this page are for 5-string bass.

Chords

Neutral-dominant 7th: 1 0 1 2 2

Music

• Improvisation in 47edo (octave-compressed tuning, 7-note subset of Negri[9]) by Budjarn Lambeth, Jan 2024