5edo: Difference between revisions

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{{Infobox ET

| Prime factorization = 5 (prime)

| Step size = ~~240¢~~

| Step size = 240.000¢

| Fifth = 3\5 = 720¢

| Fifth = 3\5 (720¢)

| Major 2nd = 1\5 = 240¢

| Major 2nd = 1\5 (240¢)

| Semitones = 1 : 0

| Semitones = 1:0 (240¢ : 0¢)

| Consistency = 9

}}

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== Theory ==

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 ~~5-TET~~ 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 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]] temperament, which equates 10/9 with 6/5: it is 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 2, 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-limit]] consistently, 3edo the [[5-limit]], 4edo the [[7-limit]] and 5edo the 9-limit, to represent the [[11-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 [[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 2, 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-limit]] consistently, 3edo the [[5-limit]], 4edo the [[7-limit]] and 5edo the 9-limit, to represent the [[11-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-5s 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]]), and is the optimal patent val for the no-5s [[Trienstonic clan|trienstonic]] (or [[Color notation/Temperament Names|Zo]]) temperament.

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]]), and is the optimal patent val for the no-5s [[Trienstonic clan|trienstonic]] (or [[Color notation/Temperament Names|Zo]]) temperament.

== Intervals ==

@@ Line 7: / Line 7: @@
 {{Infobox ET
 | Prime factorization = 5 (prime)
-| Step size = 240¢
+| Step size = 240.000¢
-| Fifth = 3\5 = 720¢
+| Fifth = 3\5 (720¢)
-| Major 2nd = 1\5 = 240¢
+| Major 2nd = 1\5 (240¢)
-| Semitones = 1 : 0
+| Semitones = 1:0 (240¢ : 0¢)
 | Consistency = 9
 }}
@@ Line 16: / Line 16: @@
 == Theory ==
-{{Harmonics in equal|5|intervals=odd}}
+{{Harmonics in equal|5}}
-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 5-TET 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 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]] temperament, which equates 10/9 with 6/5: it is 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 2, 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-limit]] consistently, 3edo the [[5-limit]], 4edo the [[7-limit]] and 5edo the 9-limit, to represent the [[11-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 [[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 2, 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-limit]] consistently, 3edo the [[5-limit]], 4edo the [[7-limit]] and 5edo the 9-limit, to represent the [[11-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-5s 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]]), and is the optimal patent val for the no-5s [[Trienstonic clan|trienstonic]] (or [[Color notation/Temperament Names|Zo]]) temperament.
+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]]), and is the optimal patent val for the no-5s [[Trienstonic clan|trienstonic]] (or [[Color notation/Temperament Names|Zo]]) temperament.
 == Intervals ==