5edo: Difference between revisions

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5edo is notable for being the smallest [[edo]] containing xenharmonic ~~intervals — 1edo~~, 2edo, 3edo, and 4edo are all subsets of [[12edo]].

5edo is notable for being the smallest [[edo]] containing xenharmonic intervals—1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]].

== Theory ==

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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.

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 [[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]].

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

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

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== Intervals ==

{| class="wikitable center-all"

|+ Intervals of 5edo

|+ style="font-size: 105%;" | Intervals of 5edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

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{| class="wikitable center-all"

|+ Notation of 5edo

|+ style="font-size: 105%;" | Notation of 5edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

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== Solfege ==

{| class="wikitable center-all"

|+ Solfege of 5edo

|+ style="font-size: 105%;" | Solfege of 5edo

|-

! [[Degree]]

! [[Cents]]

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|-

! [[Harmonic limit|Prime<br>limit]]

! [[Ratio]]<ref group="note">~~Ratios longer than 10 digits are presented by placeholders with informative hints~~</ref>

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

! [[Cent]]s

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| Island comma, parizeksma

|}

~~<references group="note"/>~~

== Ear training ==

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There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]".

== Notes ==

[[Category:5-tone scales]]

@@ Line 9: / Line 9: @@
 {{EDO intro|5}}
-edo is notable for being the smallest [[edo]] containing xenharmonic intervals — 1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]].
+edo is notable for being the smallest [[edo]] containing xenharmonic intervals—1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]].
 == Theory ==
@@ Line 16: / Line 16: @@
 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.
+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 [[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]].
-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 the optimal patent val for the no-5s [[Trienstonic clan|trienstonic]] (or [[Color notation/Temperament Names|Zo]]) temperament, although this is a very inaccurate 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]]). 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.
 edo is an example of an [[equipentatonic]] scale.
@@ Line 32: / Line 32: @@
 == Intervals ==
 {| class="wikitable center-all"
-|+ Intervals of 5edo
+|+ style="font-size: 105%;" | Intervals of 5edo
+|-
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
@@ Line 103: / Line 104: @@
 {| class="wikitable center-all"
-|+ Notation of 5edo
+|+ style="font-size: 105%;" | Notation of 5edo
+|-
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
@@ Line 158: / Line 160: @@
 == Solfege ==
 {| class="wikitable center-all"
-|+ Solfege of 5edo
+|+ style="font-size: 105%;" | Solfege of 5edo
+|-
 ! [[Degree]]
 ! [[Cents]]
@@ Line 243: / Line 246: @@
 |-
 ! [[Harmonic limit|Prime<br>limit]]
-! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
+! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]
 ! [[Cent]]s
@@ Line 417: / Line 420: @@
 | Island comma, parizeksma
 |}
-<references group="note"/>
 == Ear training ==
@@ Line 432: / Line 434: @@
 There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]".
+== Notes ==
+<references group="note" />
 [[Category:5-tone scales]]