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 [[~~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~~.

5edo is the smallest edo that contains a usable [[Perfect fifth (interval region)|perfect fifth]] at 720{{Cent}}, being 18{{C}} sharp of a [[just]]ly tuned [[3/2]] ratio at 702{{C}}. As such, it is the smallest edo where elements of traditional music theory begin to make sense.

~~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~~.

The 720{{C}} fifth generates an [[equalized]] tuning of the [[pentic]] (2L 3s) scale, where every step is the same size at 240{{C}}, or one step of 5edo. It also generates a [[collapsed]] tuning of the [[diatonic]] (5L 2s) scale, where the [[diatonic semitone]] or minor second is mapped to 0 steps, meaning that E and F as well as B and C are the same note in 5edo.

~~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]].

5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.

~~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~~.

{{W|Tertian harmony}} is also possible in 5edo, but barely: the only chords available are suspended chords, which may also be seen as inframinor (very flat minor) and ultramajor (very sharp major) chords, also known as [[Extraclassical tonality|arto and tendo]] chords, due to how sharp the fifth is. As a result, many triads will share the same three notes, so rootedness is much more important to explicitly establish.

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

In terms of just intonation, besides the perfect fifth, 5edo also contains a relatively accurate approximation the harmonic seventh [[7/4]] at 4 steps (960{{C}}), being 8.8{{C}} flat of just. 5edo can thus be used as a simplified version of the [[2.3.7 subgroup]], and defines much of its underlying structure. For example, in 5edo, the perfect fifth is 3 steps, meaning it can be divided into 3 equal parts, each representing the supermajor second [[8/7]]. This is known as [[slendric]] temperament, where [[1029/1024]], the gamelisma, is tempered out. Two intervals of [[7/6]] or 8/7 make the perfect fourth [[4/3]], tempering out [[49/48]], known as [[semaphore]] temperament. Finally, the harmonic seventh may be found by going up two perfect fourths, tempering out [[64/63]], which is [[superpyth]] temperament (sometimes known as ''archy'' in the 2.3.7 subgroup).

With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third [[9/7]] is mapped very sharply to 480{{C}}, which is the same interval as the perfect fourth. Thus [[28/27]] is tempered out, leading to the rather inaccurate [[Trienstonic clan|trienstonic]] temperament. However, this interval can still be used as a third, as referenced above.

If we attempt to add prime [[5/1|5]] to the mix and extend 5et to the full [[7-limit]], then the major third [[5/4]] is mapped very sharply to 2 steps (480{{C}}), almost a full semitone sharper than the just 5/4 at 386.3{{C}}. This results in 5edo supporting several [[exotemperament]]s when intervals of 5 are introduced. For example, the best 5/4 is the same interval as 4/3, meaning that the semitone that separates them in JI, [[16/15]], is tempered out, leading to the very inaccurate [[father]] temperament. Exploring more complex intervals, we find that the minor tone [[10/9]] and the minor third [[6/5]] are best mapped to the same step of 240 cents, meaning that the semitone separating them, [[27/25]], is tempered out as well—this is [[bug]] temperament, which is a little more perverse even than father.

Because 5edo's step is so large, such analysis is less significant with 5edo than it becomes with larger and more accurate divisions, but it still plays a role. For example, if we attempt to analyze 5edo as supporting standard [[Diatonic functional harmony|diatonic harmony]], I–IV–V–I is the same as I–III–V–I and involves triads with common intervals because major thirds and fourths are equivalent.

If 5edo is taken as only a tuning of the [[3-limit]], we find that the circle of fifths closes after only 5 steps, rather than 12, meaning [[256/243]] is tempered out. This is called [[blackwood]] temperament, and in 5edo, this is a "good" tuning of a circle of fifths—more formally, since the comma being tempered out, the 256/243 semitone at 90.2{{C}}, is smaller than half a step at 120{{C}}, 5edo demonstrates [[Telicity|3-to-2 telicity]], and is in fact the third edo to do so after [[1edo]] and [[2edo]].

5edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct [[octave-reduced]] step to harmonics 1, 3, 5, 7 and 9—specifically, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). 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]].

Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].

=== Prime harmonics ===

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=== Subsets and supersets ===

5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though 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.

5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though it contains 5 equal divisions of the double octave [[4/1]], or [[5ed4]]. Multiples of 5edo, such as [[10edo]], [[15edo]], …, up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.

== Intervals ==

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| 1

| 240

| ~~Supermajor second Inframinor~~ third

| Second-inter-third

|

| [[144/125]] (-4.969) [[125/108]] (-13.076)

| [[8/7]] (+8.826) [[7/6]] (-26.871)

| [[224/195]] (-0.030)

| [[23/20]] (-1.960) [[31/27]] (+0.829) [[224/195]] (-0.030)

| [[File:0-240 second, third (5-EDO).mp3|frameless]]

|-

| 2

| 480

| ~~Subfourth~~

| Fourth

| [[4/3]] (-18.045)

|

| [[21/16]] (+9.219)

| [[33/25]] (-0.686)

| [[33/25]] (-0.686) [[120/91]] (-1.085)

| [[File:0-480 fourth (5-EDO).mp3|frameless]]

|-

| 3

| 720

| ~~Superfifth~~

| Fifth

| [[3/2]] (+18.045)

|

| [[32/21]] (-9.219)

| [[50/33]] (+0.686)

| [[50/33]] (+0.686) [[91/60]] (+1.085)

| [[File:0-720 fifth (5-EDO).mp3|frameless]]

|-

| 4

| 960

| ~~Augmented sixth Subminor~~ seventh

| Sixth-inter-seventh

|

| [[216/125]] (+13.076) [[125/72]] (+4.969)

| [[12/7]] (+26.871) [[7/4]] (-8.826)

| [[195/112]] (+0.030)

| [[40/23]] (+1.960) [[54/31]] (-0.829) [[195/112]] (+0.030)

| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]

|-

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[[Kite Giedraitis]] has proposed pentatonic interval names that retain the appearance of heptatonic names, to avoid the confusion caused by one's lifelong association of "fourth" with 4/3, not 3/2. The interval names are unisoid, subthird, fourthoid, fifthoid, subseventh and octoid, or 1d s3 4d 5d s7 8d. The circle of fifths: 1d -- 5d -- s3 -- s7 -- 4d -- 1d. When notating larger edos such as 8 or 13 this way, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid and 16/15 is a unisoid.

For note names, Kite often omits B and merges E and F into a new letter, "eef" (rhymes with leaf). Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A Eef C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ꘙ (unicode ~~A619~~) or ⊧ (unicode ~~22A7~~) or 𐐆 (unicode 10406). Eef can also be used to notate [[15edo]].

For note names, Kite often omits B and merges E and F into a new letter, "eef" (rhymes with leaf). Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A Eef C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ⺘(unicode 2E98 or 624C) or ꘙ (unicode A619) or 𐐆 (unicode 10406). Eef can also be used to notate [[15edo]].

== Solfege ==

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== Approximation to JI ==

=== Selected 7-limit intervals ===

[[File:~~5ed2-001.svg|alt=alt : Your browser has no SVG support.]]~~

[[File:5ed2-001.svg]]

~~[[:File:5ed2-001.svg|~~5ed2-001.svg]]

== Observations ==

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== Regular temperament properties ==

=== Uniform maps ===

=== Commas ===

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| 111.731

| Gubi

| ~~Dicot~~ comma, classic ~~chroma~~

| Father comma, classic diatonic semitone

|-

| 5

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| Island comma, parizeksma

|}

== Octave stretch or compression ==

If one wishes to use 5edo as a 2.3.7 [[subgroup]] tuning, then it benefits from slight [[octave shrinking]] to improve its prime 3. Some compressed-octave 5edo tunings include [[14ed7]] or [[ed12|18ed12]]. [[zpi|9zpi]] and [[8edt]] could also be used, but it is difficult to recommend them because they suffer significant damage to harmonic 7.

== Ear training ==

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* [https://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid]

== Instruments ==

* [[Lumatone mapping for 5edo]]

== Music ==

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

== See also ==

* [[Alpha, beta, and gamma family of equal divisions]]

== Notes ==

[[Category:3-limit record edos|#]]

[[Category:5-tone scales]]

~~[[Category:7-limit]]~~

~~[[Category:9-odd-limit]]~~

@@ 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 [[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.
+edo is the smallest edo that contains a usable [[Perfect fifth (interval region)|perfect fifth]] at 720{{Cent}}, being 18{{C}} sharp of a [[just]]ly tuned [[3/2]] ratio at 702{{C}}. As such, it is the smallest edo where elements of traditional music theory begin to make sense.
-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.
+The 720{{C}} fifth generates an [[equalized]] tuning of the [[pentic]] (2L 3s) scale, where every step is the same size at 240{{C}}, or one step of 5edo. It also generates a [[collapsed]] tuning of the [[diatonic]] (5L 2s) scale, where the [[diatonic semitone]] or minor second is mapped to 0 steps, meaning that E and F as well as B and C are the same note in 5edo.
-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]].
+edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.
-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.
+{{W|Tertian harmony}} is also possible in 5edo, but barely: the only chords available are suspended chords, which may also be seen as inframinor (very flat minor) and ultramajor (very sharp major) chords, also known as [[Extraclassical tonality|arto and tendo]] chords, due to how sharp the fifth is. As a result, many triads will share the same three notes, so rootedness is much more important to explicitly establish.
-edo is an example of an [[equipentatonic]] scale.
+In terms of just intonation, besides the perfect fifth, 5edo also contains a relatively accurate approximation the harmonic seventh [[7/4]] at 4 steps (960{{C}}), being 8.8{{C}} flat of just. 5edo can thus be used as a simplified version of the [[2.3.7 subgroup]], and defines much of its underlying structure. For example, in 5edo, the perfect fifth is 3 steps, meaning it can be divided into 3 equal parts, each representing the supermajor second [[8/7]]. This is known as [[slendric]] temperament, where [[1029/1024]], the gamelisma, is tempered out. Two intervals of [[7/6]] or 8/7 make the perfect fourth [[4/3]], tempering out [[49/48]], known as [[semaphore]] temperament. Finally, the harmonic seventh may be found by going up two perfect fourths, tempering out [[64/63]], which is [[superpyth]] temperament (sometimes known as ''archy'' in the 2.3.7 subgroup).
+With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third [[9/7]] is mapped very sharply to 480{{C}}, which is the same interval as the perfect fourth. Thus [[28/27]] is tempered out, leading to the rather inaccurate [[Trienstonic clan|trienstonic]] temperament. However, this interval can still be used as a third, as referenced above.
+If we attempt to add prime [[5/1|5]] to the mix and extend 5et to the full [[7-limit]], then the major third [[5/4]] is mapped very sharply to 2 steps (480{{C}}), almost a full semitone sharper than the just 5/4 at 386.3{{C}}. This results in 5edo supporting several [[exotemperament]]s when intervals of 5 are introduced. For example, the best 5/4 is the same interval as 4/3, meaning that the semitone that separates them in JI, [[16/15]], is tempered out, leading to the very inaccurate [[father]] temperament. Exploring more complex intervals, we find that the minor tone [[10/9]] and the minor third [[6/5]] are best mapped to the same step of 240 cents, meaning that the semitone separating them, [[27/25]], is tempered out as well—this is [[bug]] temperament, which is a little more perverse even than father.
+Because 5edo's step is so large, such analysis is less significant with 5edo than it becomes with larger and more accurate divisions, but it still plays a role. For example, if we attempt to analyze 5edo as supporting standard [[Diatonic functional harmony|diatonic harmony]], I–IV–V–I is the same as I–III–V–I and involves triads with common intervals because major thirds and fourths are equivalent.
+If 5edo is taken as only a tuning of the [[3-limit]], we find that the circle of fifths closes after only 5 steps, rather than 12, meaning [[256/243]] is tempered out. This is called [[blackwood]] temperament, and in 5edo, this is a "good" tuning of a circle of fifths—more formally, since the comma being tempered out, the 256/243 semitone at 90.2{{C}}, is smaller than half a step at 120{{C}}, 5edo demonstrates [[Telicity|3-to-2 telicity]], and is in fact the third edo to do so after [[1edo]] and [[2edo]].
+edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct [[octave-reduced]] step to harmonics 1, 3, 5, 7 and 9—specifically, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). 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]].
+Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].
 === Prime harmonics ===
@@ Line 28: / Line 40: @@
 === Subsets and supersets ===
-edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though 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.
+edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though it contains 5 equal divisions of the double octave [[4/1]], or [[5ed4]]. Multiples of 5edo, such as [[10edo]], [[15edo]], …, up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.
 == Intervals ==
@@ Line 56: / Line 68: @@
 | 1
 | 240
-| Supermajor second<br>Inframinor third
+| Second-inter-third
 |
 | [[144/125]] (-4.969)<br>[[125/108]] (-13.076)
 | [[8/7]] (+8.826)<br>[[7/6]] (-26.871)
-| [[224/195]] (-0.030)
+| [[23/20]] (-1.960)<br>[[31/27]] (+0.829)<br>[[224/195]] (-0.030)
 | [[File:0-240 second, third (5-EDO).mp3|frameless]]
 |-
 | 2
 | 480
-| Subfourth
+| Fourth
 | [[4/3]] (-18.045)
 |
 | [[21/16]] (+9.219)
-| [[33/25]] (-0.686)
+| [[33/25]] (-0.686)<br>[[120/91]] (-1.085)
 | [[File:0-480 fourth (5-EDO).mp3|frameless]]
 |-
 | 3
 | 720
-| Superfifth
+| Fifth
 | [[3/2]] (+18.045)
 |
 | [[32/21]] (-9.219)
-| [[50/33]] (+0.686)
+| [[50/33]] (+0.686)<br>[[91/60]] (+1.085)
 | [[File:0-720 fifth (5-EDO).mp3|frameless]]
 |-
 | 4
 | 960
-| Augmented sixth<br>Subminor seventh
+| Sixth-inter-seventh
 |
 | [[216/125]] (+13.076)<br>[[125/72]] (+4.969)
 | [[12/7]] (+26.871)<br>[[7/4]] (-8.826)
-| [[195/112]] (+0.030)
+| [[40/23]] (+1.960)<br>[[54/31]] (-0.829)<br>[[195/112]] (+0.030)
 | [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
 |-
@@ Line 170: / Line 182: @@
 [[Kite Giedraitis]] has proposed pentatonic interval names that retain the appearance of heptatonic names, to avoid the confusion caused by one's lifelong association of "fourth" with 4/3, not 3/2. The interval names are unisoid, subthird, fourthoid, fifthoid, subseventh and octoid, or 1d s3 4d 5d s7 8d. The circle of fifths: 1d -- 5d -- s3 -- s7 -- 4d -- 1d. When notating larger edos such as 8 or 13 this way, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid and 16/15 is a unisoid.
-For note names, Kite often omits B and merges E and F into a new letter, "eef" (rhymes with leaf). Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A Eef C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ꘙ (unicode A619) or ⊧ (unicode 22A7) or 𐐆 (unicode 10406). Eef can also be used to notate [[15edo]].
+For note names, Kite often omits B and merges E and F into a new letter, "eef" (rhymes with leaf). Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A Eef C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ⺘(unicode 2E98 or 624C) or ꘙ (unicode A619) or 𐐆 (unicode 10406). Eef can also be used to notate [[15edo]].
 == Solfege ==
@@ Line 214: / Line 226: @@
 == Approximation to JI ==
 === Selected 7-limit intervals ===
-[[File:5ed2-001.svg|alt=alt : Your browser has no SVG support.]]
+[[File:5ed2-001.svg]]
-[[:File:5ed2-001.svg|5ed2-001.svg]]
 == Observations ==
@@ Line 252: / Line 262: @@
 == Regular temperament properties ==
 === Uniform maps ===
-{{Uniform map|13|4.5|5.5}}
+{{Uniform map|edo=5}}
 === Commas ===
@@ Line 285: / Line 295: @@
 | 111.731
 | Gubi
-| Dicot comma, classic chroma
+| Father comma, classic diatonic semitone
 |-
 | 5
@@ Line 434: / Line 444: @@
 | Island comma, parizeksma
 |}
+== Octave stretch or compression ==
+If one wishes to use 5edo as a 2.3.7 [[subgroup]] tuning, then it benefits from slight [[octave shrinking]] to improve its prime 3. Some compressed-octave 5edo tunings include [[14ed7]] or [[ed12|18ed12]]. [[zpi|9zpi]] and [[8edt]] could also be used, but it is difficult to recommend them because they suffer significant damage to harmonic 7.
 == Ear training ==
@@ Line 442: / Line 455: @@
 * [https://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid]
+== Instruments ==
+* [[Lumatone mapping for 5edo]]
 == Music ==
@@ Line 448: / Line 464: @@
 There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]".
+== See also ==
+* [[Alpha, beta, and gamma family of equal divisions]]
 == Notes ==
 <references group="note" />
+[[Category:3-limit record edos|#]] <!-- 1-digit number -->
 [[Category:5-tone scales]]
-[[Category:7-limit]]
-[[Category:9-odd-limit]]