8afdo

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Prime factorization	23
Fifth	12/8 (701.955c)

8afdo (arithmetic frequency division of the octave), or 8odo (otonal division of the octave), divides the octave into eight parts of 1/8 each. It is a superset of 7afdo and a subset of 9afdo. As a scale it may be known as mode 8 of the harmonic series or the Over-8 scale. Dante Rosati calls this the "Diatonic harmonic series scale". Because 8 is a power of 2, 8afdo corresponds to the 8th to 16th harmonics octave-reduced. 8afdo is a highly effective scale. Above its root it contains many commonly-used intervals such as 9/8, 5/4, 3/2, 7/4, 15/8, and 2/1.

The smallest edo that maintains 25% or lower relative error on all intervals of 8afdo is 87edo.

Intervals

#	Cents	Ratio	Decimal	Interval name	Audio
0	0	1/1	1.0000	perfect unison
1	203.91	9/8	1.1250	whole tone
2	386.31	5/4	1.2500	just major third
3	551.32	11/8	1.3750	undecimal superfourth
4	701.96	3/2	1.5000	just perfect fifth
5	840.53	13/8	1.6250	lesser tridecimal neutral sixth
6	968.83	7/4	1.7500	harmonic seventh
7	1088.27	15/8	1.8750	just major seventh
8	1200.00	2/1	2.0000	perfect octave

Scales

• Otonalpentad: 9/8-10/8-12/8-14/8-16/8 (a.k.a. springwater^{[idiosyncratic term]})
• Otonalpentad with sharper third degree: 9/8-11/8-12/8-14/8-16/8
• Otonalpentad with sharper fifth degree: 9/8-10/8-12/8-15/8-16/8
• Dark, vaguely Hirajoshi-like pentad: 9/8-11/8-12/8-13/8-16/8
• Bright, vaguely blues scale-like pentad: 10/8-11/8-12/8-14/8-16/8

Music

Dante Rosati

• Paracelsus for Diatonic Harmonic Guitar
• No Snow for Diatonic Harmonic Guitar