5edo

5edo compared to other equipentatonic scales. Relative to just intonation, it narrows 7/4, 7/6, and 4/3, and widens 12/7, 8/7, and 3/2.

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

Theory

Edostep interpretations

5edo's edostep has the following interpretations in the 2.3.7 subgroup:

• 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.

Approximation of prime harmonics in 5edo

Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	0.0	+18.0	+93.7	-8.8	-71.3	+119.5	-105.0	-57.5	+91.7
	Relative (%)	0.0	+7.5	+39.0	-3.7	-29.7	+49.8	-43.7	-24.0	+38.2
Steps (reduced)		5 (0)	8 (3)	12 (2)	14 (4)	17 (2)	19 (4)	20 (0)	21 (1)	23 (3)

Derivation

5edo is derived by equalizing an equipentatonic scale.

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.

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 temperament, and is supported by, most notably, 15edo. Blackwood tempers out the diatonic semitone, meaning that the 3-limit diatonic MOS scale is degenerate, collapsing to a 5-note scale. However, intervals of 5 are distinct, and may be used to generate the Zarlino diatonic.