47edo
| ← 46edo | 47edo | 48edo → |
(convergent)
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 25.532 ¢ each. Each step represents a frequency ratio of 21/47, or the 47th root of 2.
Theory
47edo has a fifth which is 12.593 ¢ flat, unless you use the alternative fifth which is 12.939 ¢ 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. 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. It also provides a good tuning for baseball temperament.
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).
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.
Harmonics
| 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 | -13 | +5 | +1 | +41 | +8 | +38 | -11 | +35 | -44 | +39 | |
| 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 |