730edo: Difference between revisions

Line 1:

The ''730 equal temperament'' divides the octave into 730 equal parts of 1.644 cents each. It is a very strong five-limit system, but is also distinctly consistent up to the 15-limit. It tempers out the minortone comma, |-16 35 -17>, the kwasy comma, |-53 10 16>, the whoosh comma, |37 25 -33> and the pirate comma, |-90 -15 49>. In the 7-limit it tempers out 4375/4374 and | -21 0 3 5 >, so that it supports [[Ragismic_microtemperaments#Mitonic|mitonic temperament]]. In the 11-limit, 3025/3024 and | 4 -3 -6 4 1 > , so that it supports [[Ragismic_microtemperaments#Deca|deca temperament]]. In the 13-limit, 1001/1000 and 4225/4224, supporting 13-limit deca.

W. S. B. Woolhouse proposed 730edo as a logarithmic measure of [[Interval_size_measure|interval size]], sometimes called the Woolhouse unit. While 730 is divisible by 2, 5, 10, 73, 146 and 365, it is not divisible by 12, which can be regarded as either a good thing or a bad one.

W. S. B. Woolhouse proposed 730edo<ref name="summary">[http://www.webcitation.org/5zxZzQ3eS A summary of W. S. B. Woolhouse's Essay on musical intervals], 1999 by Joe Monzo</ref> as a logarithmic measure of [[Interval_size_measure|interval size]], sometimes called the Woolhouse unit. While 730 is divisible by 2, 5, 10, 73, 146 and 365, it is not divisible by 12, which can be regarded as either a good thing or a bad one.

[~~http~~://~~sonic~~-~~arts~~.~~org~~/~~monzo~~/~~woolhouse~~/essay.~~htm A summary~~ of W. S. B. Woolhouse's ~~Essay on musical intervals~~] [~~http:~~//~~www.webcitation.org~~/~~5zxZzQ3eS Permalink~~]

==Intervals==

W. S. B. Woolhouse, in his 1835 essay<ref name="essay">[https://archive.org/details/essayonmusicali00woolgoog/page/n34/mode/2up Essay on musical intervals, harmonics, and the temperament of the musical scale, &c], 1835 by Wesley Stoker Barker Woolhouse</ref>, proposed:

... dividing the octave into 730 equal intervals, which we shall call ''degrees'', the elemental intervals will be:

<pre>

Major-tone, t = 124

Minor-tone, tˌ= 111

Limma, θ = 68

Comma, c = 13

</pre>

...

These numbers present a more accurate measurement of the musical scale than any other, unless we go to very high numbers. The greatest error which can arise from their natural or melodious combinations is that of the fifth, and does not amount to one half of the error of the major-tone above mentioned.

The concordant intervals are

<pre>

Minor-third ...... 192

Major-third ...... 235

Fourth ........... 303

Fifth ............ 427

Minor-sixth ...... 495

Major-sixth ...... 538

Octave ........... 730

</pre>

</blockquote>

==Woolhouse diatonic scale==

Woolhouse defined the following diatonic/heptonic scale for 730EDO<ref name="essay" />.

According to this division of the octave into 730 degrees, which we shall here-after adopt, the diatonic scale will be —

<pre>

Key ... 0

... 124 ... t ... Major-tone.

2d ... 124

... 111 ... tˌ... Minor-tone.

3d ... 235

... 68 ... θ ... Limma.

4th ... 303

... 124 ... t ... Major-tone.

5th ... 427

... 111 ... tˌ... Minor-tone.

6th ... 538

... 124 ... t ... Major-tone.

7th ... 662

... 68 ... θ ... Limma.

8th ... 730

</pre>

</blockquote>

Woolhouse's diatonic scale in Ls notation is

*'''LMsLMLs''' - L: 124, M: 111, s: 68

Inferred modes are shown in the following table.

{| class="wikitable"

|+Woolhouse 730EDO diatonic scale

|-

!Sequence!!Mode (suggested name)!!I1!!I2!!I3!!I4!!I5!!I6!!I7

|-

|LMsLMLs||Woolhouse Ionian||P1||[[9/8|M2]]||[[5/4|M35]]||[[4/3|P4]]||[[3/2|P5]]||[[5/3|M65]]||[[15/8|M75]]

|-

|MsLMLsL||Woolhouse Dorian||P1||m25||m35||P4||P5||m65||m75

|-

|sLMLsLM||Woolhouse Phrygian||P1||M25||M35||P4||P5||M65||m7

|-

|LMLsLMs||Woolhouse Lydian||P1||m25||m35||P4||d517||m65||m7

|-

|MLsLMsL||Woolhouse Mixolydian||P1||M25||m319||P4||-||M65||m7

|-

|LsLMsLM||Woolhouse Aeolian||P1||M2||M35||A45||P5||d7175||M75

|-

|sLMsLML||Woolhouse Locrian||P1||M2||m35||P4197||P5||m65||m75

|-

|}

==References==

@@ Line 1: / Line 1: @@
 The ''730 equal temperament'' divides the octave into 730 equal parts of 1.644 cents each. It is a very strong five-limit system, but is also distinctly consistent up to the 15-limit. It tempers out the minortone comma, |-16 35 -17&gt;, the kwasy comma, |-53 10 16&gt;, the whoosh comma, |37 25 -33&gt; and the pirate comma, |-90 -15 49&gt;. In the 7-limit it tempers out 4375/4374 and | -21 0 3 5 &gt;, so that it supports [[Ragismic_microtemperaments#Mitonic|mitonic temperament]]. In the 11-limit, 3025/3024 and | 4 -3 -6 4 1 &gt; , so that it supports [[Ragismic_microtemperaments#Deca|deca temperament]]. In the 13-limit, 1001/1000 and 4225/4224, supporting 13-limit deca.
-W. S. B. Woolhouse proposed 730edo as a logarithmic measure of [[Interval_size_measure|interval size]], sometimes called the Woolhouse unit. While 730 is divisible by 2, 5, 10, 73, 146 and 365, it is not divisible by 12, which can be regarded as either a good thing or a bad one.
+W. S. B. Woolhouse proposed 730edo<ref name="summary">[http://www.webcitation.org/5zxZzQ3eS A summary of W. S. B. Woolhouse's Essay on musical intervals], 1999 by Joe Monzo</ref> as a logarithmic measure of [[Interval_size_measure|interval size]], sometimes called the Woolhouse unit. While 730 is divisible by 2, 5, 10, 73, 146 and 365, it is not divisible by 12, which can be regarded as either a good thing or a bad one.
-[http://sonic-arts.org/monzo/woolhouse/essay.htm A summary of W. S. B. Woolhouse's Essay on musical intervals] [http://www.webcitation.org/5zxZzQ3eS Permalink]
+==Intervals==
+W. S. B. Woolhouse, in his 1835 essay<ref name="essay">[https://archive.org/details/essayonmusicali00woolgoog/page/n34/mode/2up Essay on musical intervals, harmonics, and the temperament of the musical scale, &c], 1835 by Wesley Stoker Barker Woolhouse</ref>, proposed:
+<blockquote>
+... dividing the octave into 730 equal intervals, which we shall call ''degrees'', the elemental intervals will be:
+<pre>
+      Major-tone, t = 124
+      Minor-tone, tˌ= 111
+      Limma,      θ =  68
+      Comma,      c =  13
+</pre>
+...
+These numbers present a more accurate measurement of the musical scale than any other, unless we go to very high numbers. The greatest error which can arise from their natural or melodious combinations is that of the fifth, and does not amount to one half of the error of the major-tone above mentioned.
+The concordant intervals are
+<pre>
+      Minor-third ...... 192
+      Major-third ...... 235
+      Fourth ........... 303
+      Fifth ............ 427
+      Minor-sixth ...... 495
+      Major-sixth ...... 538
+      Octave ........... 730
+</pre>
+</blockquote>
+==Woolhouse diatonic scale==
+Woolhouse defined the following diatonic/heptonic scale for 730EDO<ref name="essay" />.
+<blockquote>
+According to this division of the octave into 730 degrees, which we shall here-after adopt, the diatonic scale will be —
+<pre>
+Key ...   0
+            ... 124 ... t ... Major-tone.
+d  ... 124
+            ... 111 ... tˌ... Minor-tone.
+d  ... 235
+            ...  68 ... θ ... Limma.
+th ... 303
+            ... 124 ... t ... Major-tone.
+th ... 427
+            ... 111 ... tˌ... Minor-tone.
+th ... 538
+            ... 124 ... t ... Major-tone.
+th ... 662
+            ...  68 ... θ ... Limma.
+th ... 730
+</pre>
+</blockquote>
+Woolhouse's diatonic scale in Ls notation is
+*'''LMsLMLs''' - L: 124, M: 111, s: 68
+Inferred modes are shown in the following table.
+{| class="wikitable"
+|+Woolhouse 730EDO diatonic scale
+|-
+!Sequence!!Mode (suggested name)!!I1!!I2!!I3!!I4!!I5!!I6!!I7
+|-
+|LMsLMLs||Woolhouse Ionian||P1||[[9/8|M2]]||[[5/4|M3<sup>5</sup>]]||[[4/3|P4]]||[[3/2|P5]]||[[5/3|M6<sup>5</sup>]]||[[15/8|M7<sup>5</sup>]]
+|-
+|MsLMLsL||Woolhouse Dorian||P1||m2<sub>5</sub>||m3<sub>5</sub>||P4||P5||m6<sub>5</sub>||m7<sub>5</sub>
+|-
+|sLMLsLM||Woolhouse Phrygian||P1||M2<sup>5</sup>||M3<sup>5</sup>||P4||P5||M6<sup>5</sup>||m7
+|-
+|LMLsLMs||Woolhouse Lydian||P1||m2<sub>5</sub>||m3<sub>5</sub>||P4||d5<sup>17</sup>||m6<sub>5</sub>||m7
+|-
+|MLsLMsL||Woolhouse Mixolydian||P1||M2<sup>5</sup>||m3<sup>19</sup>||P4||-||M6<sup>5</sup>||m7
+|-
+|LsLMsLM||Woolhouse Aeolian||P1||M2||M3<sup>5</sup>||A4<sup>5</sup>||P5||d7<sup>17</sup><sub>5</sub>||M7<sup>5</sup>
+|-
+|sLMsLML||Woolhouse Locrian||P1||M2||m3<sub>5</sub>||P4<sup>19</sup><sub>7</sub>||P5||m6<sub>5</sub>||m7<sub>5</sub>
+|-
+|}
+==References==
+<references />