Equiheptatonic

An equiheptatonic scale is a scale with 7 approximately equally spaced notes within the octave. A tuning system that generates an equiheptatonic scale may be conceptualized with the 7-form. Below are several examples of equiheptatonic scales.

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. 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 (which Cole Parker claims are the most common cross-culturally), perfect 7edo is almost never used.

Notation

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

Whitewood temperament

7edo may be interpreted as Whitewood temperament, which tempers out the Pythagorean chromatic semitone. 7edo.7 is the most obvious rank-2 expansion, though 7edo.5 is also viable if more accurate 5-limit intervals are wanted. Both are supported by 21edo, which, along with 14edo, supports the omnidiatonic ternary diatonic scale.

Equitetrachordal scale

The equitetrachordal scale is the scale sssLsss where L represents 9/8, and s represents the cube root of 4/3. If s is taken to be 11/10 and s+s to be 6/5, the result is Porcupine temperament, the most common temperament interpretation of the equitetrachordal scale. It is tuned close to just in 29edo.

Just equiheptatonic scale

This is Ptolemy's equable diatonic, the otonal or utonal series 18:20:22:24:27:30:36, which has steps of 10/9, 11/10, and 12/11 in each tetrachord and 9/8 separating them. It is equated to the former in Porcupine temperament.

Other equiheptatonic scales

Any heptatonic MOS has a range of equiheptatonic tunings. The following is a table of equiheptatonic MOSes with related temperaments, expressed here as temperaments of the 2.3.5 subgroup (though often, other more accurate interpretations are available). Unlike equipentatonic systems, the scale is usually not tuned quite as equally, so extensions to 7 are more common.

MOS	Pattern	Temperament	EDO
6L 1s	LLLLLLs	Tetracot	27
5L 2s	LLsLLLs	Meantone	26
4L 3s	LLsLsLs	Sixix	25
3L 4s	LssLsLs	Dicot	24
2L 5s	LssLsss	Mavila	23
1L 6s	Lssssss	Porcupine	22