47edo: Difference between revisions

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== Theory ==

47edo has a fifth which is 12.593{{cent}} flat, unless you use the alternative fifth which is 12.939{{cent}} 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 [[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. It also provides a good tuning for [[baseball]] temperament.

47edo has a fifth which is 12.593-cent flat, unless you use the alternative fifth which is 12.939-cent sharp, similar to [[35edo]]. The soft [[5L 2s|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 [[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 [[baldy]] and [[silver]] and their relatives. It also provides a good tuning for the [[baseball]] temperament.

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:~~

=== Odd harmonics ===

~~D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D~~

=== Subsets and supersets ===

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

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 ===~~

~~{{Harmonics in equal|47}}~~

~~=== Subsets and supersets ===~~

~~47edo is the 15th [[prime EDO]], following [[43edo]] and preceding [[53edo]].~~

== Intervals ==

{| class="wikitable center-all right-2"

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| D

|}

== Notation ==

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.

== Scales ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|47}}
 == Theory ==
-edo has a fifth which is 12.593{{cent}} flat, unless you use the alternative fifth which is 12.939{{cent}} 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 [[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. It also provides a good tuning for [[baseball]] temperament.
+edo has a fifth which is 12.593-cent flat, unless you use the alternative fifth which is 12.939-cent sharp, similar to [[35edo]]. The soft [[5L 2s|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 [[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 [[baldy]] and [[silver]] and their relatives. It also provides a good tuning for the [[baseball]] temperament.
 edo 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:
+=== Odd harmonics ===
+{{Harmonics in equal|47}}
-D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D
+=== Subsets and supersets ===
+edo is the 15th [[prime edo]], following [[43edo]] and preceding [[53edo]].
-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:
-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 ===
-{{Harmonics in equal|47}}
-=== Subsets and supersets ===
-edo is the 15th [[prime EDO]], following [[43edo]] and preceding [[53edo]].
 == Intervals ==
 {| class="wikitable center-all right-2"
@@ Line 317: / Line 309: @@
 | D
 |}
+== Notation ==
+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#<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:
+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.
 == Scales ==