5edo: Difference between revisions

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[[File:5edo scale.mp3|thumb|A chromatic 5edo scale on C.]]

If 5edo is regarded as a temperament, which is to say as 5tet, then the most salient fact is that [[16/15]] is tempered out. This means in 5tet the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[father]] temperament.

If 5edo is regarded as a temperament, which is to say as 5tet, then the most salient fact is that [[16/15]] is [[tempering out|tempered out]]. This means in 5tet the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[father]] temperament.

Also tempered out is 27/25. If we temper this out in preference to 16/15, we obtain [[bug]], which equates 10/9 with 6/5, making it a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, ~~I-IV-V-I~~ is the same as ~~I-III-V-I~~ and involves triads with common intervals because of fourth-thirds equivalence.

Also tempered out is [[27/25]]. If we temper this out in preference to 16/15, we obtain [[bug]], which equates [[10/9]] with [[6/5]], making it a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I–IV–V–I is the same as I–III–V–I and involves triads with common intervals because of fourth-thirds equivalence.

Despite its lack of accuracy, 5edo is the second [[~~The Riemann Zeta Function and Tuning #Zeta edo lists|~~zeta integral edo]], after [[2edo]]. It also is the smallest ~~equal division~~ representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct value modulo ~~five~~ to 1, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[~~The_Seven_Limit_Symmetrical_Lattices~~|lattice]] of tetrads/pentads together with the number of scale steps in 5edo. However, while 2edo represents the [[3-odd-limit]] consistently, 3edo the [[5-odd-limit]], 4edo the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]]. Nevertheless, because the comma tempered out for this edo's circle of fifths is [[256/243]], and since this interval is smaller than half a step, 5edo is the second edo to demonstrate 3-to-2 [[telicity]]—that is, when not counting the comparatively trivial [[1edo]].

Despite its lack of accuracy, 5edo is the second [[zeta integral edo]], after [[2edo]]. It also is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct value modulo 5 to 1, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5edo. However, while 2edo represents the [[3-odd-limit]] consistently, 3edo the [[5-odd-limit]], 4edo the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]]. Nevertheless, because the comma tempered out for this edo's [[circle of fifths]] is [[256/243]], and since this interval is smaller than half a step, 5edo is the second edo to demonstrate 3-to-2 [[telicity]]—that is, when not counting the comparatively trivial [[1edo]].

In addition, considering 5edo as a no-5's temperament improves its standing significantly. It is especially prominent as a simple 2.3.7 temperament with high relative accuracy (the next edo doing it better being [[17edo|17]]). It is also the optimal patent val for the no-5s [[~~Trienstonic~~ clan|trienstonic]] (or [[Color notation/Temperament ~~Names~~|Zo]]) temperament, although this temperament is very inaccurate.

In addition, considering 5edo as a no-5's temperament improves its standing significantly. It is especially prominent as a simple [[2.3.7 subgroup|2.3.7-subgroup]] temperament with high relative accuracy (the next edo doing it better being [[17edo|17]]). It is also the optimal patent val for the no-5's [[trienstonic clan|trienstonic]] (or [[Color notation/Temperament names|Zo]]) temperament, although this temperament is very inaccurate.

5edo is an example of an [[equipentatonic]] scale.

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=== Subsets and supersets ===

5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. Multiples such as [[10edo]], [[15edo]], … up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.

5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It contains [[5ed4]]. Multiples such as [[10edo]], [[15edo]], … up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.

== Intervals ==

@@ Line 14: / Line 14: @@
 [[File:5edo scale.mp3|thumb|A chromatic 5edo scale on C.]]
-If 5edo is regarded as a temperament, which is to say as 5tet, then the most salient fact is that [[16/15]] is tempered out. This means in 5tet the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[father]] temperament.
+If 5edo is regarded as a temperament, which is to say as 5tet, then the most salient fact is that [[16/15]] is [[tempering out|tempered out]]. This means in 5tet the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[father]] temperament.
-Also tempered out is 27/25. If we temper this out in preference to 16/15, we obtain [[bug]], which equates 10/9 with 6/5, making it a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.
+Also tempered out is [[27/25]]. If we temper this out in preference to 16/15, we obtain [[bug]], which equates [[10/9]] with [[6/5]], making it a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I–IV–V–I is the same as I–III–V–I and involves triads with common intervals because of fourth-thirds equivalence.
-Despite its lack of accuracy, 5edo is the second [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]], after [[2edo]]. It also is the smallest equal division representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct value modulo five to 1, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The_Seven_Limit_Symmetrical_Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5edo. However, while 2edo represents the [[3-odd-limit]] consistently, 3edo the [[5-odd-limit]], 4edo the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]]. Nevertheless, because the comma tempered out for this edo's circle of fifths is [[256/243]], and since this interval is smaller than half a step, 5edo is the second edo to demonstrate 3-to-2 [[telicity]]—that is, when not counting the comparatively trivial [[1edo]].
+Despite its lack of accuracy, 5edo is the second [[zeta integral edo]], after [[2edo]]. It also is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct value modulo 5 to 1, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5edo. However, while 2edo represents the [[3-odd-limit]] consistently, 3edo the [[5-odd-limit]], 4edo the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]]. Nevertheless, because the comma tempered out for this edo's [[circle of fifths]] is [[256/243]], and since this interval is smaller than half a step, 5edo is the second edo to demonstrate 3-to-2 [[telicity]]—that is, when not counting the comparatively trivial [[1edo]].
-In addition, considering 5edo as a no-5's temperament improves its standing significantly. It is especially prominent as a simple 2.3.7 temperament with high relative accuracy (the next edo doing it better being [[17edo|17]]). It is also the optimal patent val for the no-5s [[Trienstonic clan|trienstonic]] (or [[Color notation/Temperament Names|Zo]]) temperament, although this temperament is very inaccurate.
+In addition, considering 5edo as a no-5's temperament improves its standing significantly. It is especially prominent as a simple [[2.3.7 subgroup|2.3.7-subgroup]] temperament with high relative accuracy (the next edo doing it better being [[17edo|17]]). It is also the optimal patent val for the no-5's [[trienstonic clan|trienstonic]] (or [[Color notation/Temperament names|Zo]]) temperament, although this temperament is very inaccurate.
 edo is an example of an [[equipentatonic]] scale.
@@ Line 28: / Line 28: @@
 === Subsets and supersets ===
-edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. Multiples such as [[10edo]], [[15edo]], … up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.
+edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It contains [[5ed4]]. Multiples such as [[10edo]], [[15edo]], … up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.
 == Intervals ==