Ed8/3: Difference between revisions

(30 intermediate revisions by 8 users not shown)

Line 1:

'''~~EdXI~~''' ~~means~~ '''~~Division of an eleventh interval into n equal parts~~'''.

The '''equal division of 8/3''' ('''ed8/3''') is a [[tuning]] obtained by dividing the [[8/3|Pythagorean perfect eleventh (8/3)]] in a certain number of [[equal]] steps.

~~<font style~~=~~"font-size: 19.5px;">~~Division of ~~an eleventh (e. g.~~ 8/3) into n equal parts</~~font>~~

== Equivalence ==

Division of 8/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed8/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.

~~Division of e. g.~~ the ~~8:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of~~ [[~~equivalence~~]] ~~has not even been posed yet. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy. The eleventh is also the highest equivalence~~ where composers do not need to go beyond the ~~pseudo (~~false) octave just to have a reasonably complete chordal harmony. ~~However, the~~ utility of 8:3 or another eleventh as a ~~base is complicated~~ by the fact that 8:3 is the avoid note in a major modality ~~although this~~ matters less in Mixolydian than it does in Ionian given that the former is the natural dominant scale anyway.

The eleventh is the highest [[period]] where composers do not need to go beyond the false octave just to have a reasonably complete chordal harmony. The structural utility of 8/3 or another eleventh as a period may be undermined, though, by the fact that 8/3 is the {{w|avoid note}} in a major modality. This matters less in Mixolydian than it does in Ionian given that the former is the natural dominant scale anyway.

~~Incidentally, one way~~ to ~~treat 8~~/3 ~~or 11/4 as an equivalence~~ is the use of the 3:4:5:6:(8~~)/4:5:6:8:(11~~) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes twelve octaves to get to 134217718/98415 (tempering out the schisma) or 1071794405/262144 (tempering out the comma |-29 0 8 0 8>). So, doing this yields 7, 10 and 17 or 13, 16 or 19,note MOS. While the notes are rather farther apart, the scheme is uncannily similar to the [[~~Mohajira]] (within 8/3) and [[Magic~~]] ~~(within 11/4) temperaments~~. The terms for it are the "Macromohajira"/"Macromagic" Bolivarian mode (get your mind out of the ''other'' gutter, they don't have to do with ''[[wikipedia:Venezuela|that country that is there right now]]'', at least not particularly directly). Rather, they refer to a place that is slightly cleaner:

One approach to ed8/3 tunings is the use of the 3:4:5:6:(8) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]].

== Joseph Ruhf's approach ==

Whereas in meantone it takes four [[3/2]] to get to [[5/1]], here it takes twelve octaves to get to [[134217718/98415]] (tempering out the schisma). So, doing this yields 7-, 10- and 17- or 13-, 16- or 19-note [[mos scale]]s. While the notes are rather farther apart, the scheme is uncannily similar to the [[mohajira]] (within 8/3) temperaments. [[Joseph Ruhf]] calls this the ''Macromohajira Bolivarian mode''.

=== Temperament areas ===

Galveston Bay Temperament Area

* 2L 8s and 8L 2s, 5L 5s - Galveston Symmetric, Pentachordal Major, Macro-Blackwood

2L 8s and 8L 2s, 5L 5s - Galveston Symmetric, Pentachordal Major, Macro-Blackwood

* 4L 6s and 6L 4s - Baytown

* '''3L 7s and 7L 3s - Bolivar'''

4L 6s and 6L 4s - Baytown

'''3L 7s and 7L 3s - Bolivar'''

The similar decatonic scales in edIXs and edXs belong to the Chesapeake Bay Temperament Area:

* Double Neapolitan[10i]: Scala Mariae/Notre Dame

* Neapolitan/Middletown Valley Dorian[10i]: Annapolis

* Middletown Valley Mixolydian[10i]: Oriole

* Other similar decatonic ± 1 scales have the following names:

* Locrian and Pluperfect/Abundant Phrygian[10i]/Lydian and Perfect Ionian[11i]: Scala Francisci

* Perfect Ionian through Pluperfect/Abundant Phrygian[9i]: Montréal

~~Double Neapolitan~~[~~10i~~]~~: Scala Mariae~~

[[Category:Ed8/3's| ]]

~~Neapolitan/Middletown Valley Dorian~~[~~10i~~]~~: Annapolis~~

[[Category:Lists of scales]]

~~Middletown Valley Mixolydian[10i]: Oriole~~

~~Pluperfect~~/~~Abundant Phrygian[10i]/Lydian[11i]: Scala Francisci~~

{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 8/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.}}

~~[[Category:Equal-step tuning]]~~

@@ Line 1: / Line 1: @@
-'''EdXI''' means '''Division of an eleventh interval into n equal parts'''.
+The '''equal division of 8/3''' ('''ed8/3''') is a [[tuning]] obtained by dividing the [[8/3|Pythagorean perfect eleventh (8/3)]] in a certain number of [[equal]] steps.
-<font style="font-size: 19.5px;">Division of an eleventh (e. g. 8/3) into n equal parts</font>
+== Equivalence ==
+Division of 8/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed8/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.
-Division of e. g. the 8:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy. The eleventh is also the highest equivalence where composers do not need to go beyond the pseudo (false) octave just to have a reasonably complete chordal harmony. However, the utility of 8:3 or another eleventh as a base is complicated by the fact that 8:3 is the avoid note in a major modality although this matters less in Mixolydian than it does in Ionian given that the former is the natural dominant scale anyway.
+The eleventh is the highest [[period]] where composers do not need to go beyond the false octave just to have a reasonably complete chordal harmony. The structural utility of 8/3 or another eleventh as a period may be undermined, though, by the fact that 8/3 is the {{w|avoid note}} in a major modality. This matters less in Mixolydian than it does in Ionian given that the former is the natural dominant scale anyway.
-Incidentally, one way to treat 8/3 or 11/4 as an equivalence is the use of the 3:4:5:6:(8)/4:5:6:8:(11) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes twelve octaves to get to 134217718/98415 (tempering out the schisma) or 1071794405/262144 (tempering out the comma |-29 0 8 0 8>). So, doing this yields 7, 10 and 17 or 13, 16 or 19,note MOS. While the notes are rather farther apart, the scheme is uncannily similar to the [[Mohajira]] (within 8/3) and [[Magic]] (within 11/4) temperaments. The terms for it are the "Macromohajira"/"Macromagic" Bolivarian mode (get your mind out of the ''other'' gutter, they don't have to do with ''[[wikipedia:Venezuela|that country that is there right now]]'', at least not particularly directly). Rather, they refer to a place that is slightly cleaner:
+One approach to ed8/3 tunings is the use of the 3:4:5:6:(8) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]].
+== Joseph Ruhf's approach ==
+{{idiosyncratic terms}}
+Whereas in meantone it takes four [[3/2]] to get to [[5/1]], here it takes twelve octaves to get to [[134217718/98415]] (tempering out the schisma). So, doing this yields 7-, 10- and 17- or 13-, 16- or 19-note [[mos scale]]s. While the notes are rather farther apart, the scheme is uncannily similar to the [[mohajira]] (within 8/3) temperaments. [[Joseph Ruhf]] calls this the ''Macromohajira Bolivarian mode''.
+=== Temperament areas ===
 Galveston Bay Temperament Area
+* 2L 8s and 8L 2s, 5L 5s - Galveston Symmetric, Pentachordal Major, Macro-Blackwood
-L 8s and 8L 2s, 5L 5s - Galveston Symmetric, Pentachordal Major, Macro-Blackwood
+* 4L 6s and 6L 4s - Baytown
+* '''3L 7s and 7L 3s - Bolivar'''
-L 6s and 6L 4s - Baytown
-'''3L 7s and 7L 3s - Bolivar'''
 The similar decatonic scales in edIXs and edXs belong to the Chesapeake Bay Temperament Area:
+* Double Neapolitan[10i]: Scala Mariae/Notre Dame
+* Neapolitan/Middletown Valley Dorian[10i]: Annapolis
+* Middletown Valley Mixolydian[10i]: Oriole
+* Other similar decatonic ± 1 scales have the following names:
+* Locrian and Pluperfect/Abundant Phrygian[10i]/Lydian and Perfect Ionian[11i]: Scala Francisci
+* Perfect Ionian through Pluperfect/Abundant Phrygian[9i]: Montréal
-Double Neapolitan[10i]: Scala Mariae
+[[Category:Ed8/3's| ]]
+<!-- main article -->
-Neapolitan/Middletown Valley Dorian[10i]: Annapolis
+[[Category:Lists of scales]]
-Middletown Valley Mixolydian[10i]: Oriole
-Pluperfect/Abundant Phrygian[10i]/Lydian[11i]: Scala Francisci
+{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 8/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.}}
-[[Category:Equal-step tuning]]