25/16

25/16, the classic(al) augmented fifth is the interval obtained by stacking two 5/4 major thirds, however, it gains additional isoharmonic identity from its position between 11/8 and 7/4, so it can frequently be used in conjunction with those, even in chords.

Interval information

Ratio	25/16
Factorization	2-4 × 52
Monzo	[-4 0 2⟩
Size in cents	772.6274¢
Names	classic(al) augmented fifth, diptolemaic augmented fifth
Color name	yy5, yoyo 5th
FJS name	[math]\displaystyle{ \text{A5}^{5,5} }[/math]
Special properties	reduced, reduced harmonic
Tenney norm (log2 nd)	8.64386
Weil norm (log2 max(n, d))	9.28771
Wilson norm (sopfr(nd))	18
[sound info]
Open this interval in xen-calc

While this interval has been referred to as the classic augmented fifth or classical augmented fifth for some time, the term diptolemaic was coined on Discord by Flora Canou while discussing a proposal for a consistent naming scheme for different 5-limit intervals with Aura. Specifically, since "diptolemaic" intervals have two instances of prime 5 in their factorization, this interval is also referred to as the diptolemaic augmented fifth.

Approximation

Edo approximations for 25/16 (772.63 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
3	2\3	800.00	+27.37	+6.84
11	7\11	763.64	-8.99	-8.24
14	9\14	771.43	-1.20	-1.40
17	11\17	776.47	+3.84	+5.44
25	16\25	768.00	-4.63	-9.64
28	18\28	771.43	-1.20	-2.80
31	20\31	774.19	+1.57	+4.05
42	27\42	771.43	-1.20	-4.20
45	29\45	773.33	+0.71	+2.65
48	31\48	775.00	+2.37	+9.49
56	36\56	771.43	-1.20	-5.59
59	38\59	772.88	+0.25	+1.25
62	40\62	774.19	+1.57	+8.09
70	45\70	771.43	-1.20	-6.99
73	47\73	772.60	-0.02	-0.15
76	49\76	773.68	+1.06	+6.69

See also

• 32/25 – its octave complement