10/9

Interval information

Ratio	10/9
Factorization	2 × 3-2 × 5
Monzo	[1 -2 1⟩
Size in cents	182.4037¢
Names	small whole tone, classic(al) whole tone, ptolemaic whole tone
Color name	y2, yo 2nd
FJS name	[math]\displaystyle{ \text{M2}^{5} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	6.49185
Weil norm (log2 max(n, d))	6.64386
Wilson norm (sopfr(nd))	13
[sound info]
Open this interval in xen-calc

English Wikipedia has an article on:

Major second

In 5-limit just intonation, 10/9 is the small, classic(al), or ptolemaic whole tone of about 182.4¢. It is a superparticular interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is 9/8 (about 203.9¢), which is 81/80 (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to 12edo's single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, 19edo, 31edo and other meantone systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

Approximations

Edo approximations for 10/9 (182.40 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
6	1\6	200.00	+17.60	+8.80
7	1\7	171.43	-10.98	-6.40
13	2\13	184.62	+2.21	+2.40
20	3\20	180.00	-2.40	-4.01
26	4\26	184.62	+2.21	+4.79
33	5\33	181.82	-0.59	-1.61
39	6\39	184.62	+2.21	+7.19
40	6\40	180.00	-2.40	-8.01
46	7\46	182.61	+0.20	+0.79
52	8\52	184.62	+2.21	+9.58
53	8\53	181.13	-1.27	-5.62
59	9\59	183.05	+0.65	+3.18
66	10\66	181.82	-0.59	-3.22
72	11\72	183.33	+0.93	+5.58
73	11\73	180.82	-1.58	-9.62
79	12\79	182.28	-0.13	-0.82

Temperaments

The following linear temperaments are generated by a ~10/9:

• Porcupine
• Minortone

See also

• Decaononic - temperament which sets the tone to this interval, instead of to 9/8
• 9/5 – its octave complement
• 27/20 – its fifth complement
• 6/5 – its fourth complement
• Gallery of just intervals