Ed7/3: Difference between revisions

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== Properties ==

Division of 7/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. The question of equivalence has not even been posed yet.

The utility of 7/3 (or another tenth) as a base though, is apparent by being the absolute widest range most generally used in popular songs{{citation needed}} (and even the range of a {{w|Dastg%C4%81h-e_M%C4%81hur|dastgah}}{{citation needed}}).

It also provides a fairly trivial point to split the difference between the [[octave]] and the [[tritave]], which is why [[Joseph Ruhf]] has named the region of intervals between 6 and 7 degrees of [[5edo]] the "[[Middletown valley]]", the proper [[~~middletown~~]] ~~temperament family~~ being 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.

It also provides a fairly trivial point to split the difference between the [[octave]] and the [[tritave]], which is why [[Joseph Ruhf]] has named the region of intervals between 6 and 7 degrees of [[5edo]] the "[[Middletown valley]]", the proper [[Middletown family|Middletown temperament family]] being 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.

~~Incidentally, [[~~Pseudo-traditional harmonic functions of enneatonic scale_degrees|enneatonic scales]], especially those equivalent at e.g. 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 (false) octave, with various degrees of accuracy.

=== Chords and harmonies ===

{{main|Pseudo-traditional harmonic functions of enneatonic scale_degrees}}

Incidentally, enneatonic scales, especially those equivalent at e.g. 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 (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. "Macrobichromatic" might be a practically perfect term for it if it hasn't been named yet.

The branches of the [[Middletown family]] are named thus:

=== The Middletown family ===

The branches of the Middletown family are named thus:

* 3&6: [[Tritetrachordal]]

* 3&6: Tritetrachordal

* 4&5: [[Montrose]] (between 5\4edo and 4\3edo in particular, MOS generated by [pseudo] octaves belong to this branch)

* 4&5: Montrose (between 5\4edo and 4\3edo in particular, MOS generated by [pseudo] octaves belong to this branch)

* 2&7: ~~[[Terra rubra|~~Terra Rubra]]

* 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 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]]

* 5&6: Rosablanca

* 4&7: [[Saptimpun]] (10 1/2)

* 4&7: Saptimpun (10 1/2)

* 5&7: [[8bittone]] (Old Middetown)

* 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]].

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 == Properties ==
 Division of 7/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. The question of equivalence has not even been posed yet.
 The utility of 7/3 (or another tenth) as a base though, is apparent by being the absolute widest range most generally used in popular songs{{citation needed}} (and even the range of a {{w|Dastg%C4%81h-e_M%C4%81hur|dastgah}}{{citation needed}}).
-It also provides a fairly trivial point to split the difference between the [[octave]] and the [[tritave]], which is why [[Joseph Ruhf]] has named the region of intervals between 6 and 7 degrees of [[5edo]] the "[[Middletown valley]]", the proper [[middletown]] temperament family being 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.
+It also provides a fairly trivial point to split the difference between the [[octave]] and the [[tritave]], which is why [[Joseph Ruhf]] has named the region of intervals between 6 and 7 degrees of [[5edo]] the "[[Middletown valley]]", the proper [[Middletown family|Middletown temperament family]] being 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.
-Incidentally, [[Pseudo-traditional harmonic functions of enneatonic scale_degrees|enneatonic scales]], especially those equivalent at e.g. 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 (false) octave, with various degrees of accuracy.
+=== Chords and harmonies ===
+{{main|Pseudo-traditional harmonic functions of enneatonic scale_degrees}}
+Incidentally, enneatonic scales, especially those equivalent at e.g. 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 (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. "Macrobichromatic" might be a practically perfect term for it if it hasn't been named yet.
-The branches of the [[Middletown family]] are named thus:
+=== The Middletown family ===
+The branches of the Middletown family are named thus:
-* 3&amp;6: [[Tritetrachordal]]
+* 3&amp;6: Tritetrachordal
-* 4&amp;5: [[Montrose]] (between 5\4edo and 4\3edo in particular, MOS generated by [pseudo] octaves belong to this branch)
+* 4&amp;5: Montrose (between 5\4edo and 4\3edo in particular, MOS generated by [pseudo] octaves belong to this branch)
-* 2&amp;7: [[Terra rubra|Terra Rubra]]
+* 2&amp;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 Middlebury temperament falls in the "Middletown valley", but its enneatonic scales are “[[generator-remainder]]".
 The temperaments neighboring Middletown proper are named thus:
-* 5&amp;6: [[Rosablanca]]
+* 5&amp;6: Rosablanca
-* 4&amp;7: [[Saptimpun]] (10 1/2)
+* 4&amp;7: Saptimpun (10 1/2)
-* 5&amp;7: [[8bittone]] (Old Middetown)
+* 5&amp;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]].