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An '''equipentatonic scale''' is a scale with 5 approximately equally spaced notes within the octave. A tuning system that generates an equipentatonic scale may be conceptualized with the '''5-form'''. Below are several examples of equipentatonic scales.
[[File:Qetqbnx.png|thumb|The derivation of equipentatonic and [[equiheptatonic]] scales by adding melodic steps to [1/1 4/3 3/2 2/1].]]
An '''equipentatonic scale''' is a scale with 5 approximately equally spaced notes within the octave. A tuning system that generates an equipentatonic scale may be conceptualized with the '''5-form'''.  


== 5edo ==
The basic equipentatonic is [[5edo]], where all the steps are tuned to be precisely equal. It features steps of (1200/5) = 240 cents. Below are several other examples of equipentatonic scales.
5edo is the basic equipentatonic, where all the steps are tuned to be precisely equal. It features steps of (1200/5) = 240 cents.


=== Theory ===
== Equal trichordal scale ==
The equal trichordal scale is the scale of the form and mode LLsLL, where s represents [[9/8]] and L represents sqrt([[4/3]]). It is so named by Vector because it is constructed from the equally divided [[Polychordal scale|trichord]]. It is tuned close to just in [[24edo]].


===== Edostep interpretations =====
=== Semaphore ===
5edo's edostep has the following interpretations in the 2.3.7 subgroup:
Semaphore is a possible temperament interpretation of the equal trichordal scale, also supported by 24edo. Semaphore equalizes 6:7:8 (that is, tempers 7/6 and 8/7 together, or tempers out 49/48), turning the just equipentatonic scale into the equal trichordal one. It is sometimes considered an exotemperament due to the fact that doing so requires tuning 7/4 to 950 cents assuming a just fourth. The fourth can be flattened (bringing the generator closer to 8/7 at the cost of 7/6, at which point the natural next step is [[blackwood]] temperament), or sharpened (bringing the generator closer to 7/6 at the cost of 8/7).
 
* 7/6  
* 8/7
* 9/8
 
===== JI approximation =====
5edo is most obviously a 2.3.7 system (and this property carries to 5-form systems as a whole). although its 3 is audibly sharper than a just 3/2. There is no [[diatonic]] scale in 5edo; there are too few notes. However, 5edo interestingly is simple enough, and at least vaguely accurate enough to just intonation, that essentially any combination of notes taken from it sounds consonant to some extent.{{Harmonics in ED|5|7|edo=5|EDO=5|ET=5}}
===== Chords =====
Essentially any combination of notes in 5edo is a chord of some kind. Up to inversion and modulation, there are a total of 5 chords in 5edo, not counting a single note or the whole scale. Predictably, triadic harmony is rare in 5edo, and most harmonic systems in this tuning rely on drones or modulation.
 
===== Scales =====
5edo is simple enough that the entire edo is usually used as a scale (although composer Hideya Amano avoids this by using subsets). The construction of scales from 5edo is usually not by taking subsets but by detuning the notes to reach one of the equipentatonic scales discussed elsewhere on the page. In fact, in real world musical cultures that use equipentatonic scales, they are almost never perfect 5edo.
 
Notably, any subset of 5edo is a MOS, and 5edo is the last edo for which this is true.
 
=== Notation ===
5edo may be notated with any diatonic notation, as long as you keep in mind that E and F are the same note, and B and C are also the same note; the note sequence is usually C-D-E-G-A like a major pentatonic scale, and it is preferable to avoid accidentals.
 
=== Blackwood temperament ===
Owing to 5edo's accuracy in 2.3.7 but failure to represent 5, a perfectly tuned 5edo can be taken as the 2.3.7 structure in a rank-2 temperament, with 5 (or any other prime) added as a [[generator]]. This is called blackwood (5edo.5) temperament, and is supported by, most notably, [[15edo]]. Blackwood tempers out the [[diatonic semitone]], meaning that the 3-limit diatonic is degenerate, collapsing to a 5-note scale. However, intervals of 5 are distinct, and may be used to generate the Zarlino [[diatonic]].
 
== Equal trichordal scale ==
The equal trichordal scale is the scale of the form and mode LLsLL, where s represents [[9/8]] and L represents sqrt([[4/3]]). It is so named by Vector because it is constructed from the equally divided [[Polychordal scale|trichord]]. It can represent [[Semaphore]] temperament or simply an equally divided [[Ploidacot#Diploid dicot|hemipythagorean]] subset. It is tuned close to just in [[24edo]].


== Just equipentatonic scale ==
== Just equipentatonic scale ==
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== Slendro ==
== Slendro ==
Slendro is an equipentatonic gamelan tuning used in Indonesian music. It is often approximated as 5edo, however its tuning varies. It is usually tuned with stretched octaves. One tuning in particular more closely resembles an extremely soft [[pentic]] MOS than a perfect equipentatonic.
Slendro is an equipentatonic gamelan tuning used in Indonesian music. It is often approximated as 5edo, however its tuning varies. It is usually tuned with stretched octaves. One tuning in particular more closely resembles an extremely soft pentic (2L 3s) MOS than a perfect equipentatonic.<nowiki>[[citation needed]]</nowiki>


== Other equipentatonic scales ==
== Other equipentatonic scales ==

Latest revision as of 09:03, 27 March 2026

The derivation of equipentatonic and equiheptatonic scales by adding melodic steps to [1/1 4/3 3/2 2/1].

An equipentatonic scale is a scale with 5 approximately equally spaced notes within the octave. A tuning system that generates an equipentatonic scale may be conceptualized with the 5-form.

The basic equipentatonic is 5edo, where all the steps are tuned to be precisely equal. It features steps of (1200/5) = 240 cents. Below are several other examples of equipentatonic scales.

Equal trichordal scale

The equal trichordal scale is the scale of the form and mode LLsLL, where s represents 9/8 and L represents sqrt(4/3). It is so named by Vector because it is constructed from the equally divided trichord. It is tuned close to just in 24edo.

Semaphore

Semaphore is a possible temperament interpretation of the equal trichordal scale, also supported by 24edo. Semaphore equalizes 6:7:8 (that is, tempers 7/6 and 8/7 together, or tempers out 49/48), turning the just equipentatonic scale into the equal trichordal one. It is sometimes considered an exotemperament due to the fact that doing so requires tuning 7/4 to 950 cents assuming a just fourth. The fourth can be flattened (bringing the generator closer to 8/7 at the cost of 7/6, at which point the natural next step is blackwood temperament), or sharpened (bringing the generator closer to 7/6 at the cost of 8/7).

Just equipentatonic scale

The just equipentatonic is the scale 12:14:16:18:21:24, either as an otonal or utonal chain. It has the pattern LMsLM, where L is 7/6, M is 8/7, and s is 9/8. It is a part of the reason why 5edo represents the structure of the 2.3.7 group so well.

Slendro

Slendro is an equipentatonic gamelan tuning used in Indonesian music. It is often approximated as 5edo, however its tuning varies. It is usually tuned with stretched octaves. One tuning in particular more closely resembles an extremely soft pentic (2L 3s) MOS than a perfect equipentatonic.[[citation needed]]

Other equipentatonic scales

Any pentatonic MOS has a range of equipentatonic tunings. The following is a table of equipentatonic MOSes with related temperaments, which all serve as basic temperaments of the 2.3.7 subgroup, though due to being 5-form their extensions to 5/4 are rather complex and sensitive to detuning.

MOS Pattern Temperament EDO
4L 1s LLsLL Semaphore 29edo
3L 2s LLsLs Buzzard 28edo
2L 3s LssLs Archy 27edo
1L 4s Lssss Slendric 26edo

Table of equipentatonic intervals

# Name (ADIN) Tuning range Just intonation Equal-tempered Equitrichordal Just equipentatonic Soft pentic
0 Unison 0c 1/1 0c 0c 0c 0c
1 Major second, minor third 220-270c 9/8, 8/7, 7/6 240c 249c 231c, 267c 222-267c
2 Perfect fourth 480-500c 21/16, 4/3 480c 498c 498c 480-489c
3 Perfect fifth 700-720c 3/2, 32/21 720c 702c 702c 711-720c
4 Major sixth, minor seventh 930-980c 7/4, 12/7, 16/9 960c 951c 933c, 969c 933-978c
5 Octave 1200c 2/1 1200c 1200c 1200c 1200c
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