41edo: Difference between revisions
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For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | ||
== | == Notations == | ||
=== Red-Blue Notation === | |||
A red-note/blue-note system, similar to the one proposed for [[36edo|36edo]], is one option for notating 41edo. (This is separate from and not compatible with Kite's [[Kite's_color_notation|color notation]].) We have the "white key" albitonic notes A-G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are: | A red-note/blue-note system, similar to the one proposed for [[36edo|36edo]], is one option for notating 41edo. (This is separate from and not compatible with Kite's [[Kite's_color_notation|color notation]].) We have the "white key" albitonic notes A-G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are: | ||
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If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[Ups_and_Downs_Notation|ups and downs notation]]. The only difference is the use of minor tritone and major tritone. | If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[Ups_and_Downs_Notation|ups and downs notation]]. The only difference is the use of minor tritone and major tritone. | ||
=== Selected just intervals | == Just approximation == | ||
=== Selected just intervals === | |||
{| class="wikitable" style="text-align:center;" | {| class="wikitable" style="text-align:center;" | ||
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|- | |- | ||
! rowspan="2" |Error | ! rowspan="2" |Error | ||
!absolute (¢) | ! absolute (¢) | ||
| 0.0 | | 0.0 | ||
| +0.48 | | +0.48 | ||
| Line 516: | Line 518: | ||
| -4.8 | | -4.8 | ||
|- | |- | ||
!relative (%) | ! relative (%) | ||
|0.0 | | 0.0 | ||
| +1.7 | | +1.7 | ||
| -19.9 | | -19.9 | ||
| Line 615: | Line 617: | ||
| 14.079 | | 14.079 | ||
|} | |} | ||
=== Temperament measures === | |||
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 41et. | |||
{| class="wikitable center-all" | |||
! colspan="2" | | |||
! 3-limit | |||
! 5-limit | |||
! 7-limit | |||
! 11-limit | |||
! 13-limit | |||
! 2.3.5.7.11.19 | |||
! 2.3.5.7.11.13.19 | |||
|- | |||
! colspan="2" |Octave stretch (¢) | |||
| -0.153 | |||
| +0.734 | |||
| +0.815 | |||
| +0.375 | |||
| -0.060 | |||
| +0.502 | |||
| +0.111 | |||
|- | |||
! rowspan="2" |Error | |||
![[TE error|absolute]] (¢) | |||
| 0.153 | |||
| 1.26 | |||
| 1.10 | |||
| 1.32 | |||
| 1.55 | |||
| 1.24 | |||
| 1.49 | |||
|- | |||
! [[TE simple badness|relative]] (%) | |||
| 0.521 | |||
| 4.31 | |||
| 3.76 | |||
| 4.51 | |||
| 5.29 | |||
| 4.23 | |||
| 5.10 | |||
|} | |||
* 41et has a lower relative error than any previous ETs in the 3-, 13- and 19-limit. The next ET that does better in these subgroups is 53, 53, and 46, respectively. | |||
* 41et is prominent in the 2.3.5.7.11.19 and 2.3.5.7.11.13.19 subgroup. The next ET that does better in these subgroups is 72 and 53, respectively. | |||
== Relationship to 12-edo == | == Relationship to 12-edo == | ||