Ed7/3: Difference between revisions

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{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 7/3 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.}}
The '''equal division of 7/3''' ('''ed7/3''') is a [[tuning]] obtained by dividing the [[7/3|septimal minor tenth (7/3)]] in a certain number of [[equal]] steps.
The '''equal division of 7/3''' ('''ed7/3''') is a [[tuning]] obtained by dividing the [[7/3|septimal minor tenth (7/3)]] in a certain number of [[equal]] steps.


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== Chords and harmonies ==
== Chords and harmonies ==
{{main|Pseudo-traditional harmonic functions of enneatonic scale degrees}}
[[:Category:9-tone scales|Enneatonic scale]]s, especially those equivalent at 7/3, can sensibly take [[tetrad]]s as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structurally important as it is. Many, though not all, of these scales have a perceptually important [[Pseudo-octave|pseudo (false) octave]], with various degrees of accuracy.
[[:Category:9-tone scales|Enneatonic scale]]s, especially those equivalent at 7/3, can sensibly take [[tetrad]]s as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structurally important as it is. Many, though not all, of these scales have a perceptually important [[Pseudo-octave|pseudo (false) octave]], with various degrees of accuracy.


Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:(7) 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 two [[28/15]] to get to [[7/2]] (tempering out the comma [[225/224]]). So, doing this yields 15-, 19-, and 34-note [[mos]] 2/1 apart. While the notes are rather farther apart, the scheme is uncannily similar to meantone. [[Joseph Ruhf]] named this scheme "macrobichromatic".
Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:(7) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]].
 
== Middletown ==
{{idiosyncratic terms}}
7/3 provides a fairly trivial point to split the difference between the [[octave]] and the [[tritave]], which is why Ruhf has named the region of intervals between 6 and 7 degrees of [[5edo]] the "[[Middletown valley]]".
 
The proper [[Middletown family|Middletown temperament family]] is based on an [[enneatonic]] scale [[generator|generated]] by a third or a fifth optionally with a [[period]] of a [[Wolf interval|wolf]] fourth at most 560 [[cents]] wide) and, as is the twelfth (tritave), an alternative interval where {{w|Inversion (music) #Counterpoint|invertible counterpoint}} has classically occurred.
 
The branches of the Middletown family are named thus:
* 3&6: Tritetrachordal
* 4&5: Montrose (between 5\4edo and 4\3edo in particular, MOS generated by [pseudo] octaves belong to this branch)
* 2&7: Terra Rubra
 
The family of interlaced [[octatonic scale]]-based temperaments in the "Middletown valley" is called Vesuvius (i.e. the volcano east of Naples).
 
The Middlebury temperament falls in the "Middletown valley", but its enneatonic scales are "[[generator-remainder]]".
 
The temperaments neighboring Middletown proper are named thus:
* 5&6: Rosablanca
* 4&7: Saptimpun (10 1/2)
* 5&7: 8bittone (Old Middetown)
 
The [[pyrite]] tuning of [[edX]]s will turn out to divide a barely mistuned [[5/2]] of almost exactly 45\[[34edo]].


== Individual pages for ed7/3's ==
== Individual pages for ed7/3's ==
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<!-- main article -->
<!-- main article -->
[[Category:Lists of scales]]
[[Category:Lists of scales]]
{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 7/3 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.}}

Latest revision as of 21:27, 13 May 2026

Todo: cleanup, explain edonoi

Most people do not think 7/3 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.

The equal division of 7/3 (ed7/3) is a tuning obtained by dividing the septimal minor tenth (7/3) in a certain number of equal steps.

Applications

Division of 7/3 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed7/3 scales have a perceptually important false octave, with various degrees of accuracy.

The structural utility of 7/3 (or another tenth) is apparent by being the absolute widest range most generally used in popular songs[citation needed] (and even the range of a dastgah[citation needed]).

Chords and harmonies

Enneatonic scales, especially those equivalent at 7/3, can sensibly take tetrads as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structurally important as it is. Many, though not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:(7) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone.

Individual pages for ed7/3's

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