47edo: Difference between revisions
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= | ==Theory== | ||
'''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 [[k*N_subgroups|2*47 subgroup]] of the [[23-limit|23-limit]], on which it tempers out the same commas as [[94edo|94edo]]. It provides a good tuning for [[Chromatic_pairs#Baldy|baldy]] and [[Chromatic_pairs#Silver|silver]] temperaments and relatives. | '''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 [[k*N_subgroups|2*47 subgroup]] of the [[23-limit|23-limit]], on which it tempers out the same commas as [[94edo|94edo]]. It provides a good tuning for [[Chromatic_pairs#Baldy|baldy]] and [[Chromatic_pairs#Silver|silver]] temperaments and relatives. | ||
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47edo is the 15th [[prime_numbers|prime]] edo, following [[43edo|43edo]] and preceding [[53edo|53edo]]. | 47edo is the 15th [[prime_numbers|prime]] edo, following [[43edo|43edo]] and preceding [[53edo|53edo]]. | ||
47edo is a | 47edo is a diatonic 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 diatonic edos to notate, because no other diatonic 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 | 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 * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D | ||
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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#<span style="font-size: 90%; vertical-align: super;">3</span> - F#<span style="font-size: 90%; vertical-align: super;">5</span> - A#<span style="font-size: 90%; vertical-align: super;">3</span> - C# or as Eb<span style="font-size: 90%; vertical-align: super;">4</span> - Gbb - Ab<span style="font-size: 90%; vertical-align: super;">4</span> - Db<span style="font-size: 90%; vertical-align: super;">6</span>. This is an aug-three double-dim-seven chord, written D#<span style="font-size: 90%; vertical-align: super;">3</span>(A3)dd7 or Eb<span style="font-size: 90%; vertical-align: super;">4</span>(A3)dd7. It could also be called a sharp-three triple-flat-seven chord, written D#<span style="font-size: 90%; vertical-align: super;">3</span>(#3)b<span style="font-size: 90%; vertical-align: super;">3</span>7 or Eb<span style="font-size: 90%; vertical-align: super;">4</span>(#3)b<span style="vertical-align: super;">3</span>7. | 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#<span style="font-size: 90%; vertical-align: super;">3</span> - F#<span style="font-size: 90%; vertical-align: super;">5</span> - A#<span style="font-size: 90%; vertical-align: super;">3</span> - C# or as Eb<span style="font-size: 90%; vertical-align: super;">4</span> - Gbb - Ab<span style="font-size: 90%; vertical-align: super;">4</span> - Db<span style="font-size: 90%; vertical-align: super;">6</span>. This is an aug-three double-dim-seven chord, written D#<span style="font-size: 90%; vertical-align: super;">3</span>(A3)dd7 or Eb<span style="font-size: 90%; vertical-align: super;">4</span>(A3)dd7. It could also be called a sharp-three triple-flat-seven chord, written D#<span style="font-size: 90%; vertical-align: super;">3</span>(#3)b<span style="font-size: 90%; vertical-align: super;">3</span>7 or Eb<span style="font-size: 90%; vertical-align: super;">4</span>(#3)b<span style="vertical-align: super;">3</span>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 | 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 * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D | ||