5edo: Difference between revisions

{{ED intro}}

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

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.

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.

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

 

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]].

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

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

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

! rowspan="2" | [[Interval region]]

! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])

! rowspan="2" | Audio

| [[1/1]] (just)

| [[File:0-0 unison.mp3|frameless]]

| Second-inter-third

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

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

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

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

| Fourth

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

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

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

| Fifth

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

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

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

| Sixth-inter-seventh

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

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

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

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

| [[File:0-1200 octave.mp3|frameless]]

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

! colspan="2" | [[Chain-of-fifths notation]]

! [[5L 2s|Diatonic]] interval names

! Note names (on D)

| '''Perfect unison (P1)'''<br>Minor second (m2)<br>Diminished third (d3)

| Augmented unison (A1)<br>'''Major second (M2)'''<br>'''Minor third (m3)'''<br>Diminished fourth (d4)

| D#<br>'''E'''<br>'''F'''<br>Gb

| Augmented second (A2)<br>Major third (M3)<br>'''Perfect fourth (P4)'''<br>Diminished fifth (d5)

| E#<br>F#<br>'''G'''<br>Ab

| Augmented fourth (A4)<br>'''Perfect fifth (P5)'''<br>Minor sixth (m6)<br>Diminished seventh (d7)

| G#<br>'''A'''<br>Bb<br>Cb

| Augmented fifth (A5)<br>'''Major sixth (M6)'''<br>'''Minor seventh (m7)'''<br>Diminished octave (d8)

| A#<br>'''B'''<br>'''C'''<br>Db

| Augmented sixth (A6)<br>Major seventh (M7)<br>'''Perfect octave (P8)'''

| B#<br>C#<br>'''D'''

* [[ups and downs notation]] is identical to circle-of-fifths notation;

* mixed [[sagittal notation]] is identical to circle-of-fifths notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

 

 

 

Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation.

Intervals can be named penta-2nd, penta-3rd, penta-4th, penta-5th and hexave. The circle of fifths: 1sn -- penta-4th -- penta-2nd -- penta-5th -- penta-3rd -- 1sn.

[[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 ⺘(unicode 2E98 or 624C) or ꘙ (unicode A619) or 𐐆 (unicode 10406). Eef can also be used to notate [[15edo]].

== Solfege ==

 

== Approximation to JI ==

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

5edo does not have any strong consonance nor dissonance. It could be considered [[omniconsonant scale|omniconsonant]]. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.

{{Uniform map|edo=5}}

5et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 5 8 12 14 17 19 }}.  

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

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

| Blackwood comma, Pythagorean limma

| Bug comma, large limma

| Father comma, classic diatonic semitone

| Syntonic comma, Didymus' comma, meantone comma

| [[Vulture comma]]

| Mint comma, septimal quartertone

| Semaphoresma, slendro diesis

| Sensamagic comma

| Orwellisma

| Cataharry comma

| Hemifamity comma

| Pentacircle comma

| Superleap comma, biome comma

 

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

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

<references group="note" />