7edo: Difference between revisions

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{{Infobox ET

| Prime factorization = 7 (prime)

| Step size = 171.~~42857¢~~

| Step size = 171.429¢

| P5 = 4\7 (686¢)

| M2 = 1\7 (171¢)

| Semitones = 0 : 1

| Semitones = 0:1 (0¢ : 171¢)

| Consistency = 5

~~| Monotonicity = 5~~

}}

'''7 equal divisions of the octave''' ('''7edo''') is the [[tuning system]] derived by dividing the [[octave]] into 7 equal steps of 171.4 [[cent]]s each, or the seventh root of 2. It is the fourth [[prime ~~EDO~~]], after [[2edo]], [[3edo]] and [[5edo]]. It is the third [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral ~~EDO~~]].

'''7 equal divisions of the octave''' ('''7edo''') is the [[tuning system]] derived by dividing the [[octave]] into 7 equal steps of 171.4 [[cent]]s each, or the seventh root of 2. It is the fourth [[prime edo]], after [[2edo]], [[3edo]] and [[5edo]]. It is the third [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]].

== Theory ==

Equal-heptatonic scales are used in non-western music in African cultures as well as an integral part of early Thai and early Chinese music<ref>[https://www.britannica.com/art/African-music Donald Keith Robotham and Gerhard Kubik, ''African music'', Encyclopedia Britannica]</ref>. Also Georgian music seems to be based on near-equal 7-step scales. It has been speculated in "Indian music:history and structure", that the Indian three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones).

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[7]" just as the whole tone scale of [[12edo]] is known as "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[7]" just as the whole tone scale of [[12edo]] is known as "Hexe[6]".

Typically, 7edo exists as the tuning for pentatonic scales in traditional Thai music with the other two pitches acting as auxiliary tones. However, it can be used as an interesting diatonic scale choice as well in tunings such as [[14edo]] or [[21edo]].

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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~~]] ~~scales~~, 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 ~~MOS's~~ 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-~~1Lns~~ 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 ==

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*[https://www.youtube.com/watch?v=rZ8yX6RUDzY crystal glass wishes you happy new year] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]

== Ear ~~Training~~ ==

== Ear training ==

7edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].

@@ Line 7: / Line 7: @@
 {{Infobox ET
 | Prime factorization = 7 (prime)
-| Step size = 171.42857¢
+| Step size = 171.429¢
 | P5 = 4\7 (686¢)
 | M2 = 1\7 (171¢)
-| Semitones = 0 : 1
+| Semitones = 0:1 (0¢ : 171¢)
 | Consistency = 5
-| Monotonicity = 5
 }}
-'''7 equal divisions of the octave''' ('''7edo''') is the [[tuning system]] derived by dividing the [[octave]] into 7 equal steps of 171.4 [[cent]]s each, or the seventh root of 2. It is the fourth [[prime EDO]], after [[2edo]], [[3edo]] and [[5edo]]. It is the third [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].
+'''7 equal divisions of the octave''' ('''7edo''') is the [[tuning system]] derived by dividing the [[octave]] into 7 equal steps of 171.4 [[cent]]s each, or the seventh root of 2. It is the fourth [[prime edo]], after [[2edo]], [[3edo]] and [[5edo]]. It is the third [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]].
 == Theory ==
-{{Harmonics in equal|7|intervals=odd}}
+{{Harmonics in equal|7}}
 Equal-heptatonic scales are used in non-western music in African cultures as well as an integral part of early Thai and early Chinese music<ref>[https://www.britannica.com/art/African-music Donald Keith Robotham and Gerhard Kubik, ''African music'', Encyclopedia Britannica]</ref>. Also Georgian music seems to be based on near-equal 7-step scales. It has been speculated in "Indian music:history and structure", that the Indian three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones).
-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[7]" just as the whole tone scale of [[12edo]] is known as "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[7]" just as the whole tone scale of [[12edo]] is known as "Hexe[6]".
 Typically, 7edo exists as the tuning for pentatonic scales in traditional Thai music with the other two pitches acting as auxiliary tones. However, it can be used as an interesting diatonic scale choice as well in tunings such as [[14edo]] or [[21edo]].
@@ Line 296: / Line 295: @@
 == Temperaments ==
-edo is the first EDO in which regular temperament theory starts to make sense as a way of subdividing the steps into [[MOS]] scales, 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 MOS's 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-1Lns 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. 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 317: / Line 316: @@
 *[https://www.youtube.com/watch?v=rZ8yX6RUDzY crystal glass wishes you happy new year] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
-== Ear Training ==
+== Ear training ==
 edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].