7edo

7edo is the basic equiheptatonic, where all the steps are tuned to be precisely equal. It features steps of (1200/7) ~= 171.4 cents.

Theory

Edostep interpretations

7edo's edostep has the following interpretations in the 2.3.5 subgroup:

• 9/8 (the diatonic major second)
• 10/9 (the interval separating 9/8 and 5/4)
• 16/15 (the interval separating 5/4 and 4/3)

JI approximation

7edo is, very crudely, a 2.3.5 system, and strength in 2.3.5 is generally what carries into other equiheptatonic scales. It can also be viewed in various other subgroups, most notably 2.3.13 and 2.3.11, and equiheptatonic temperaments can be found that represent those subgroups as well. The diatonic scale in 7edo is equivalent to every note in the tuning system; sharps and flats are not meaningful and all intervals are perfect.

Approximation of prime harmonics in 7edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-16.2	-43.5	+59.7	-37.0	+16.6	+66.5	+45.3	+57.4	-1.0	+55.0
	Relative (%)	0.0	-9.5	-25.3	+34.9	-21.6	+9.7	+38.8	+26.5	+33.5	-0.6	+32.1
Steps (reduced)		7 (0)	11 (4)	16 (2)	20 (6)	24 (3)	26 (5)	29 (1)	30 (2)	32 (4)	34 (6)	35 (7)

Thirds in 7edo

Quality	Neutral
Cents	343
Just interpretation	11/9

Diatonic thirds are bolded.

Chords

7edo features, for tertian triadic harmony, only a neutral chord [0 2 4] and the (rather discordant) sus chords [0 1 4] and [0 3 4]. Regardless, due to its triads and due to representing all seven degrees of the diatonic scale, it is the smallest edo where Western functional harmony works.

Scales

7edo is the first edo to distinguish the modes of the pentic scale. However, it is still small enough that it is well-temperable into scales (specifically, those of the 7-form discussed elsewhere in this article). In real world musical cultures which use near-equal 7-note scales, perfect 7edo is almost never used.

Derivation

7edo is derived by equalizing an equiheptatonic scale.

Notation

In 7edo, pretty much all reasonable notation schemes collapse to ABCDEFG on A=440Hz. Accidentals are not used.

Polysomatic tuning

Polysomatic tuning, coined by Cole Parker, refers to an octave stretch of 7edo (close to 11edt, about 6.929edo, step size 173.19c, octave 1212.33 cents) that has the property of approximating the first six harmonics of a standard harmonic instrument and of an unsupported bar (more context), such as a glockenspiel, within 25% relative error (and in fact approximates them within 25 cents absolute error, except for the 6th frequency of an unsupported bar).

Frequency #	Unsupported bar					Harmonic instrument
	Decimal	Cents	Polysomatic tuning	Deviation	Steps	Decimal	Cents	Polysomatic tuning	Deviation	Steps
1	1	0	0	0	0	1	0	0	0	0
2	2.75625	1755.25	1731.91	-23.34	10	2	1200.00	1212.33	12.33	7
3	5.40225	2920.27	2944.24	23.97	17	3	1901.96	1905.10	3.14	11
4	8.93025	3790.44	3810.19	19.75	22	4	2400.00	2424.67	24.67	14
5	13.34025	4485.26	4502.95	17.70	26	5	2786.31	2771.05	-15.26	16
6	18.63225	5063.68	5022.53	-41.15	29	6	3101.96	3117.43	15.48	18

Within the octave, polysomatic tuning also slightly improves the intervals 5/4 and 3/2, but makes 4/3 less accurate.

Whitewood temperament

7edo may be interpreted as Whitewood temperament, which tempers out the Pythagorean chromatic semitone. The most obvious rank-2 extension is to add a free generator corresponding to 7/4, resulting in a system containing multiple copies of 7edo separated by the interval 7/4. This extension is supported by 21edo, which, along with 14edo, supports the omnidiatonic ternary diatonic scale.