47edo
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) | 7mus | relative notation | absolute notation | |
|---|---|---|---|---|---|
| 0 | 0.0000 | 0 | perfect unison | P1 | D |
| 1 | 25.5319 | 32.68085 (20.AE4C16) | aug 1sn | A1 | D# |
| 2 | 51.0638 | 65.3617 (41.5C98816) | double-aug 1sn | AA1 | Dx |
| 3 | 76.5957 | 98.04255 (62.0AE516) | triple-aug 1sn, triple-dim 2nd | A31, d32 | D#3, Eb4 |
| 4 | 102.1277 | 130.7234 (82.B93116) | double-dim 2nd | dd2 | Eb3 |
| 5 | 127.6596 | 163.4043 (A3.677D16) | dim 2nd | d2 | Ebb |
| 6 | 153.1915 | 196.0851 (C4.15CA16) | minor 2nd | m2 | Eb |
| 7 | 178.7234 | 228.766 (E4.C41616) | major 2nd | M2 | E |
| 8 | 204.2553 | 261.4468 (105.726216) | aug 2nd | A2 | E# |
| 9 | 229.7872 | 294.1277 (126.20AE16) | double-aug 2nd | AA2 | Ex |
| 10 | 255.31915 | 326.8085 (146.CEFB16) | triple-aug 2nd, triple-dim 3rd | A32, d33 | E#3, Fb3 |
| 11 | 280.8511 | 359.4894 (167.7D4716) | double-dim 3rd | dd3 | Fbb |
| 12 | 306.383 | 392.1702 (188.2B9316) | dim 3rd | d3 | Fb |
| 13 | 331.9149 | 424.8511 (1A8.D9DF16) | minor 3rd | m3 | F |
| 14 | 357.4468 | 457.5319 (1C9.882C16) | major 3rd | M3 | F# |
| 15 | 382.9787 | 490.2128 (1EA.367816) | aug 3rd | A3 | Fx |
| 16 | 408.5106 | 522.8936 (20A.E4C416) | double-aug 3rd | AA3 | F#3 |
| 17 | 434.04255 | 555.5745 (22B.93116) | triple-aug 3rd, triple-dim 4th | A33, d34 | F#4, Gb3 |
| 18 | 459.5745 | 588.2553 (24C.415D16) | double-dim 4th | dd4 | Gbb |
| 19 | 485.1064 | 620.9362 (26C.EFA916) | dim 4th | d4 | Gb |
| 20 | 510.6383 | 653.617 (28D.9DF516) | perfect 4th | P4 | G |
| 21 | 536.1702 | 686.2979 (2AE.4C4116) | aug 4th | A4 | G# |
| 22 | 561.7021 | 718.9787 (2CE.FA8E16) | double-aug 4th | AA4 | Gx |
| 23 | 587.2340 | 751.6595 (2EF.A8DA16) | triple-aug 4th | A34 | G#3 |
| 24 | 612.7660 | 784.3404 (310.572616) | triple-dim 5th | d35 | Ab3 |
| 25 | 638.2979 | 817.0213 (331.057216) | double-dim 5th | dd5 | Abb |
| 26 | 663.8298 | 849.7021 (351.B3BF16) | dim 5th | d5 | Ab |
| 27 | 689.3617 | 882.383 (372.620B16) | perfect 5th | P5 | A |
| 28 | 714.8936 | 915.0638 (393.105716) | aug 5th | A5 | A# |
| 29 | 740.4255 | 947.7447 (3B3.BEA316) | double-aug 5th | AA5 | Ax |
| 30 | 765.95745 | 980.4255 (3D4.6CF16) | triple-aug 5th, triple-dim 6th | A35, d36 | A#3, Bb4 |
| 31 | 791.4894 | 1013.1064 (3F5.1B3C16) | double-dim 6th | dd6 | Bb3 |
| 32 | 817.0213 | 1045.7872 (415.C98816) | dim 6th | d6 | Bbb |
| 33 | 842.5532 | 1078.4681 (436.77D416) | minor 6th | m6 | Bb |
| 34 | 868.0851 | 1111.1489 (457.262116) | major 6th | M6 | B |
| 35 | 893.617 | 1143.8298 (477.D46D16) | aug 6th | A6 | B# |
| 36 | 919.1489 | 1176.5106 (498.82B916) | double-aug 6th | AA6 | Bx |
| 37 | 944.68085 | 1209.1915 (4B9.310516) | triple-aug 6th, triple-dim 7th | A36, d37 | B#3, Cb3 |
| 38 | 970.2128 | 1241.8723 (4D9.DF5216) | double-dim 7th | dd7 | Cbb |
| 39 | 995.7447 | 1274.5532 (4FA.8D9E16) | dim 7th | d7 | Cb |
| 40 | 1021.2766 | 1307.234 (51B.3BEA16) | minor 7th | m7 | C |
| 41 | 1046.8085 | 1339.9149 (53B.EA3616) | major 7th | M7 | C# |
| 42 | 1072.3404 | 1372.5957 (55C.988316) | aug 7th | A7 | Cx |
| 43 | 1097.8723 | 1405.2766 (57D.46CF16) | double-aug 7th | AA7 | C#3 |
| 44 | 1123.4043 | 1437.95745 (59D.F51B16) | triple-aug 7th, triple-dim 8ve | A37, d38 | C#4, Db3 |
| 45 | 1148.9362 | 1470.6383 (5BE.A367816) | double-dim 8ve | dd8 | Dbb |
| 46 | 1174.4681 | 1503.31915 (5DF.51B416) | dim 8ve | d8 | Db |
| 47 | 1200 | 1536 (60016) | perfect 8ve | P8 | D |