# 47 tone Equal Temperament

47-EDO divides the octave into 47 equal parts of 25.5319 cents each. It has a fifth which is 12.5933 cents flat, unless you use the alternative fifth which is 12.9386 cents sharp, similar to 35edo. It has therefore not aroused much interest, but its best approximation to 9/8 is actually quite good, one-third of a cent sharp. It 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 temperaments and relatives.

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

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

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

## Intervals of 47edo

Degree Size (Cents) pions 7mus relative notation absolute notation
0 perfect unison P1 D
1 25.5319 27.0638 32.68085 (20.AE4C16) aug 1sn A1 D#
2 51.0638 54.1277 65.3617 (41.5C98816) double-aug 1sn AA1 Dx
3 76.5957 81.1915 98.04255 (62.0AE516) triple-aug 1sn, triple-dim 2nd A31, d32 D#3, Eb4
4 102.1277 108.2553 130.7234 (82.B93116) double-dim 2nd dd2 Eb3
5 127.6596 135.31915 163.4043 (A3.677D16) dim 2nd d2 Ebb
6 153.1915 162.383 196.0851 (C4.15CA16) minor 2nd m2 Eb
7 178.7234 189.4468 228.766 (E4.C41616) major 2nd M2 E
8 204.2553 216.5106 261.4468 (105.726216) aug 2nd A2 E#
9 229.7872 243.5745 294.1277 (126.20AE16) double-aug 2nd AA2 Ex
10 255.31915 270.6383 326.8085 (146.CEFB16) triple-aug 2nd, triple-dim 3rd A32, d33 E#3, Fb3
11 280.8511 297.7021 359.4894 (167.7D4716) double-dim 3rd dd3 Fbb
12 306.383 324.766 392.1702 (188.2B9316) dim 3rd d3 Fb
13 331.9149 351.8298 424.8511 (1A8.D9DF16) minor 3rd m3 F
14 357.4468 378.8936 457.5319 (1C9.882C16) major 3rd M3 F#
15 382.9787 405.95745 490.2128 (1EA.367816) aug 3rd A3 Fx
16 408.5106 433.0213 522.8936 (20A.E4C416) double-aug 3rd AA3 F#3
17 434.04255 460.0851 555.5745 (22B.93116) triple-aug 3rd, triple-dim 4th A33, d34 F#4, Gb3
18 459.5745 487.1489 588.2553 (24C.415D16) double-dim 4th dd4 Gbb
19 485.1064 514.2128 620.9362 (26C.EFA916) dim 4th d4 Gb
20 510.6383 541.2766 653.617 (28D.9DF516) perfect 4th P4 G
21 536.1702 568.3404 686.2979 (2AE.4C4116) aug 4th A4 G#
22 561.7021 595.4043 718.9787 (2CE.FA8E16) double-aug 4th AA4 Gx
23 587.2340 622.4681 751.6595 (2EF.A8DA16) triple-aug 4th A34 G#3
24 612.7660 649.5319 784.3404 (310.572616) triple-dim 5th d35 Ab3
25 638.2979 676.5957 817.0213 (331.057216) double-dim 5th dd5 Abb
26 663.8298 703.6596 849.7021 (351.B3BF16) dim 5th d5 Ab
27 689.3617 730.7234 882.383 (372.620B16) perfect 5th P5 A
28 714.8936 757.7872 915.0638 (393.105716) aug 5th A5 A#
29 740.4255 784.8511 947.7447 (3B3.BEA316) double-aug 5th AA5 Ax
30 765.95745 811.9149 980.4255 (3D4.6CF16) triple-aug 5th, triple-dim 6th A35, d36 A#3, Bb4
31 791.4894 838.9787 1013.1064 (3F5.1B3C16) double-dim 6th dd6 Bb3
32 817.0213 866.04255 1045.7872 (415.C98816) dim 6th d6 Bbb
33 842.5532 893.1064 1078.4681 (436.77D416) minor 6th m6 Bb
34 868.0851 920.1702 1111.1489 (457.262116) major 6th M6 B
35 893.617 947.234 1143.8298 (477.D46D16) aug 6th A6 B#
36 919.1489 974.2979 1176.5106 (498.82B916) double-aug 6th AA6 Bx
37 944.68085 1001.3617 1209.1915 (4B9.310516) triple-aug 6th, triple-dim 7th A36, d37 B#3, Cb3
38 970.2128 1028.4255 1241.8723 (4D9.DF5216) double-dim 7th dd7 Cbb
39 995.7447 1055.4894 1274.5532 (4FA.8D9E16) dim 7th d7 Cb
40 1021.2766 1082.5532 1307.234 (51B.3BEA16) minor 7th m7 C
41 1046.8085 1109.617 1339.9149 (53B.EA3616) major 7th M7 C#
42 1072.3404 1136.68085 1372.5957 (55C.988316) aug 7th A7 Cx
43 1097.8723 1163.7447 1405.2766 (57D.46CF16) double-aug 7th AA7 C#3
44 1123.4043 1190.8085 1437.95745 (59D.F51B16) triple-aug 7th, triple-dim 8ve A37, d38 C#4, Db3
45 1148.9362 1217.8723 1470.6383 (5BE.A367816) double-dim 8ve dd8 Dbb
46 1174.4681 1244.9362 1503.31915 (5DF.51B416) dim 8ve d8 Db
47 1200 1272 1536 (60016) perfect 8ve P8 D