7edo: Difference between revisions

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== Theory ==

[[File:7edo scale.mp3|thumb|A chromatic 7edo scale on C.]]

7edo can be thought of as the result of stacking seven [[11/9]]'s on top of each other, and then tempering to remove the comma {{monzo| -2 -14 0 0 7 }}. As a temperament, [[William Lynch]] gives it the name "[[Neutron|Neutron[7]]]" just as the whole tone scale of [[12edo]] is known as "[[Hexe|Hexe[6]]]".

7edo can be thought of as the result of stacking seven [[11/9]]'s on top of each other, and then tempering to remove the [[comma]] {{monzo| -2 -14 0 0 7 }}. As a temperament, [[William Lynch]] gives it the name "[[Neutron|Neutron[7]]]" just as the whole tone scale of [[12edo]] is known as "[[Hexe|Hexe[6]]]".

=== Prime harmonics ===

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It has been speculated in ''Indian music: history and structure''<ref>Nambiyathiri, Tarjani. ''[https://archive.org/details/indianmusichistoryandstructureemmietenijenhuisbrill Indian Music History And Structure Emmie Te Nijenhuis Brill]''</ref> that the [[Indian]] three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones), one step of 7edo.

In equiheptatonic systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. (Similar to [[adaptive just intonation]] but with equal tuning instead).

In [[equiheptatonic]] systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. (Similar to [[adaptive just intonation]] but with equal tuning instead).

One region of Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern [[Angolan]] area. This music is heptatonic and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds ([[5/4]], 386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.

One region of Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern [[Angolan]] area. This music is [[heptatonic]] and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds ([[5/4]], 386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.

A [[Ugandan]], [[Chopi]] xylophone measured by Haddon (1952) was also tuned something close to this.

It has often been stated that 7edo approximates tunings used in Thai classical music, though this is a myth unsupported by empirical studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>

It has often been stated that 7edo approximates tunings used in [[Thai]] classical music, though this is a myth unsupported by [[empirical]] studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>

=== Observations ===

The seventh of 7edo is almost exactly the 29th harmonic ([[29/16]]), which can have a very agreeable sound with harmonic ~~timbres~~. However it also finds itself nested between ratios such as 20/11 and 9/5, which gives it considerably higher [[harmonic entropy]] than [[7/4]], a much simpler overtone seventh.

The seventh of 7edo is almost exactly the 29th harmonic ([[29/16]]), which can have a very agreeable sound with harmonic [[timbre]]s. However it also finds itself nested between ratios such as [[20/11]] and [[9/5]], which gives it considerably higher [[harmonic entropy]] than [[7/4]], a much simpler [[overtone]] seventh.

7edo is the unique intersection of the temperaments of [[meantone]] (specifically [[3/4-comma meantone]]) and [[porcupine]].

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The [[Octave stretch|stretched-octaves]] tuning [[Ed257/128#7ed257/128|7ed257/128]] greatly improves 7edo’s approximation of 3/1, 5/1 and 11/1, at the cost of slightly worsening 2/1 and 7/1, and greatly worsening 13/1. If one is hoping to use 7edo for [[11-limit]] harmonies, then 7ed257/128 is a good choice to make that easier.

The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo’s approximation of JI in ~~the same~~ way.

The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo’s approximation of JI in a similar way.

=== Subsets and supersets ===

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== Temperaments ==

7edo is the first edo in which regular temperament theory starts to make sense as a way of subdividing the steps into [[mos scale]]s, with three different ways of dividing it, although there is still quite a lot of ambiguity as each step can be considered as the sharp extreme of one temperament or the flat end of another. 1/7 can be considered the intersection of sharp [[porcupine]] and flat [[tetracot]] temperaments, as three steps makes a 4th and four a 5th. 2/7 can be interpreted as critically flat [[Mohajira]] or critically sharp [[amity]], and creates mosses of 322 and 2221. 3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s pentatonic mos. This is in part because 7edo is a [[The Riemann zeta function and tuning #Zeta edo lists|strict zeta edo]] (close to low-complexity JI for its size), and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.

7edo is the first edo in which [[regular temperament theory]] starts to make sense as a way of subdividing the steps into [[mos scale]]s, with three different ways of dividing it, although there is still quite a lot of ambiguity as each step can be considered as the sharp extreme of one temperament or the flat end of another.

1/7 can be considered the intersection of sharp [[porcupine]] and flat [[tetracot]] temperaments, as three steps makes a 4th and four a 5th. 2/7 can be interpreted as critically flat [[Mohajira]] or critically sharp [[amity]], and creates mosses of 322 and 2221.

3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is a [[The Riemann zeta function and tuning #Zeta edo lists|strict zeta edo]] (close to low-complexity JI for its size), and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.

== Music ==

@@ Line 10: / Line 10: @@
 == Theory ==
 [[File:7edo scale.mp3|thumb|A chromatic 7edo scale on C.]]
-edo can be thought of as the result of stacking seven [[11/9]]'s on top of each other, and then tempering to remove the comma {{monzo| -2 -14 0 0 7 }}. As a temperament, [[William Lynch]] gives it the name "[[Neutron|Neutron[7]]]" just as the whole tone scale of [[12edo]] is known as "[[Hexe|Hexe[6]]]".
+edo can be thought of as the result of stacking seven [[11/9]]'s on top of each other, and then tempering to remove the [[comma]] {{monzo| -2 -14 0 0 7 }}. As a temperament, [[William Lynch]] gives it the name "[[Neutron|Neutron[7]]]" just as the whole tone scale of [[12edo]] is known as "[[Hexe|Hexe[6]]]".
 === Prime harmonics ===
@@ Line 27: / Line 27: @@
 It has been speculated in ''Indian music: history and structure''<ref>Nambiyathiri, Tarjani. ''[https://archive.org/details/indianmusichistoryandstructureemmietenijenhuisbrill Indian Music History And Structure Emmie Te Nijenhuis Brill]''</ref> that the [[Indian]] three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones), one step of 7edo.
-In equiheptatonic systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. (Similar to [[adaptive just intonation]] but with equal tuning instead).
+In [[equiheptatonic]] systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. (Similar to [[adaptive just intonation]] but with equal tuning instead).
-One region of Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern [[Angolan]] area. This music is heptatonic and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds ([[5/4]], 386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.
+One region of Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern [[Angolan]] area. This music is [[heptatonic]] and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds ([[5/4]], 386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.
 A [[Ugandan]], [[Chopi]] xylophone measured by Haddon (1952) was also tuned something close to this.
-It has often been stated that 7edo approximates tunings used in Thai classical music, though this is a myth unsupported by empirical studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>
+It has often been stated that 7edo approximates tunings used in [[Thai]] classical music, though this is a myth unsupported by [[empirical]] studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>
 === Observations ===
-The seventh of 7edo is almost exactly the 29th harmonic ([[29/16]]), which can have a very agreeable sound with harmonic timbres. However it also finds itself nested between ratios such as 20/11 and 9/5, which gives it considerably higher [[harmonic entropy]] than [[7/4]], a much simpler overtone seventh.
+The seventh of 7edo is almost exactly the 29th harmonic ([[29/16]]), which can have a very agreeable sound with harmonic [[timbre]]s. However it also finds itself nested between ratios such as [[20/11]] and [[9/5]], which gives it considerably higher [[harmonic entropy]] than [[7/4]], a much simpler [[overtone]] seventh.
 edo is the unique intersection of the temperaments of [[meantone]] (specifically [[3/4-comma meantone]]) and [[porcupine]].
@@ Line 42: / Line 42: @@
 The [[Octave stretch|stretched-octaves]] tuning [[Ed257/128#7ed257/128|7ed257/128]] greatly improves 7edo’s approximation of 3/1, 5/1 and 11/1, at the cost of slightly worsening 2/1 and 7/1, and greatly worsening 13/1. If one is hoping to use 7edo for [[11-limit]] harmonies, then 7ed257/128 is a good choice to make that easier.
-The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo’s approximation of JI in the same way.
+The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo’s approximation of JI in a similar way.
 === Subsets and supersets ===
@@ Line 467: / Line 467: @@
 == Temperaments ==
-edo is the first edo in which regular temperament theory starts to make sense as a way of subdividing the steps into [[mos scale]]s, with three different ways of dividing it, although there is still quite a lot of ambiguity as each step can be considered as the sharp extreme of one temperament or the flat end of another. 1/7 can be considered the intersection of sharp [[porcupine]] and flat [[tetracot]] temperaments, as three steps makes a 4th and four a 5th. 2/7 can be interpreted as critically flat [[Mohajira]] or critically sharp [[amity]], and creates mosses of 322 and 2221. 3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s pentatonic mos. This is in part because 7edo is a [[The Riemann zeta function and tuning #Zeta edo lists|strict zeta edo]] (close to low-complexity JI for its size), and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.
+edo is the first edo in which [[regular temperament theory]] starts to make sense as a way of subdividing the steps into [[mos scale]]s, with three different ways of dividing it, although there is still quite a lot of ambiguity as each step can be considered as the sharp extreme of one temperament or the flat end of another.
+/7 can be considered the intersection of sharp [[porcupine]] and flat [[tetracot]] temperaments, as three steps makes a 4th and four a 5th. 2/7 can be interpreted as critically flat [[Mohajira]] or critically sharp [[amity]], and creates mosses of 322 and 2221.
+/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is a [[The Riemann zeta function and tuning #Zeta edo lists|strict zeta edo]] (close to low-complexity JI for its size), and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.
 == Music ==