47edo: Difference between revisions
m Cleanup table, sort key |
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 47 | | Prime factorization = 47 (prime) | ||
| Step size = 25. | | Step size = 25.532¢ | ||
| Fifth = 27\47 (689¢) | | Fifth = 27\47 (689¢) | ||
| Major 2nd = 7\47 (179¢) | | Major 2nd = 7\47 (179¢) | ||
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== Theory == | == Theory == | ||
''' | '''47edo''' divides the octave into 47 equal parts of 25.532{{cent}} each. It has a fifth which is 12.593{{cent}} flat, unless you use the alternative fifth which is 12.939{{cent}} 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 [[k*N_subgroups|2*47 subgroup]] of the [[23-limit]], on which it tempers out the same commas as [[94edo]]. It provides a good tuning for [[Chromatic_pairs#Baldy|baldy]] and [[Chromatic_pairs#Silver|silver]] temperaments and relatives. | ||
47edo is the 15th [[ | 47edo is the 15th [[prime EDO]], following [[43edo]] and preceding [[53edo]]. | ||
47edo is a diatonic edo because its 5th falls between 4\7 = | 47edo is a diatonic edo because its 5th falls between 4\7 = 686{{cent}} and 3\5 = 720{{cent}}, 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: | 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: | ||
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D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D | 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#< | 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#<sup>3</sup> - F#<sup>5</sup> - A#<sup>3</sup> - C# or as Eb<sup>4</sup> - Gbb - Ab<sup>4</sup> - Db<sup>6</sup>. This is an aug-three double-dim-seven chord, written D#<sup>3</sup>(A3)dd7 or Eb<sup>4</sup>(A3)dd7. It could also be called a sharp-three triple-flat-seven chord, written D#<sup>3</sup>(#3)b<sup>3</sup>7 or Eb<sup>4</sup>(#3)b<sup>3</sup>7. | ||
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: | 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: | ||
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D# is next to E. This notation requires quadruple, quintuple, and even sextuple ups and downs, as well as single sharps and flats. | D# is next to E. This notation requires quadruple, quintuple, and even sextuple ups and downs, as well as single sharps and flats. | ||
{{ | |||
{{Harmonics in equal|47}} | |||
== Intervals == | == Intervals == | ||
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|- | |- | ||
| 10 | | 10 | ||
| 255. | | 255.3191 | ||
| triple-aug 2nd, triple-dim 3rd | | triple-aug 2nd, triple-dim 3rd | ||
| A<sup>3</sup>2, d<sup>3</sup>3 | | A<sup>3</sup>2, d<sup>3</sup>3 | ||
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|- | |- | ||
| 12 | | 12 | ||
| 306. | | 306.3830 | ||
| dim 3rd | | dim 3rd | ||
| d3 | | d3 | ||
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|- | |- | ||
| 17 | | 17 | ||
| 434. | | 434.0426 | ||
| triple-aug 3rd, triple-dim 4th | | triple-aug 3rd, triple-dim 4th | ||
| A<sup>3</sup>3, d<sup>3</sup>4 | | A<sup>3</sup>3, d<sup>3</sup>4 | ||
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|- | |- | ||
| 30 | | 30 | ||
| 765. | | 765.9574 | ||
| triple-aug 5th, triple-dim 6th | | triple-aug 5th, triple-dim 6th | ||
| A<sup>3</sup>5, d<sup>3</sup>6 | | A<sup>3</sup>5, d<sup>3</sup>6 | ||
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|- | |- | ||
| 35 | | 35 | ||
| 893. | | 893.6170 | ||
| aug 6th | | aug 6th | ||
| A6 | | A6 | ||
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|- | |- | ||
| 37 | | 37 | ||
| 944. | | 944.6809 | ||
| triple-aug 6th, triple-dim 7th | | triple-aug 6th, triple-dim 7th | ||
| A<sup>3</sup>6, d<sup>3</sup>7 | | A<sup>3</sup>6, d<sup>3</sup>7 | ||
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|- | |- | ||
| 47 | | 47 | ||
| 1200 | | 1200.0000 | ||
| perfect 8ve | | perfect 8ve | ||
| P8 | | P8 | ||