21/20

21/20 is a small semitone of about 85 cents. It may be found in 7-limit just intonation as, for example, the difference between 4/3 and 7/5, 8/7 and 6/5, or 5/3 and 7/4.

Interval information

Ratio	21/20
Factorization	2-2 × 3 × 5-1 × 7
Monzo	[-2 1 -1 1⟩
Size in cents	84.46719¢
Names	septimal minor semitone, septimal chromatic semitone, large septimal chroma
Color name	zg2, zogu 2nd
FJS name	[math]\displaystyle{ \text{m2}^{7}_{5} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	8.71425
Weil norm (log2 max(n, d))	8.78463
Wilson norm (sopfr(nd))	19
Comma size	medium
S-expressions	S6⋅S7, S7⋅S8⋅S9
[sound info]
Open this interval in xen-calc

English Wikipedia has an article on:

Septimal chromatic semitone

In what is known as an authentic cadence, there is a resolution from the V chord to the I chord. If the V is a harmonic seventh chord, its harmonic seventh (21/16 above the tonic) resolves down to the major third of the I chord (5/4) by a step of 21/20.

Terminology

21/20 is traditionally called a chroma, perhaps for its proximity (and conflation in systems like septimal meantone) with the major chroma 135/128. However, it is a diatonic semitone in both Helmholtz–Ellis notation and Functional Just System, viewed as the Pythagorean minor second 256/243 altered by 5120/5103. Marc Sabat has taken to call it the minor diatonic semitone in the same material where 15/14 is also named as the major chromatic semitone.

Approximation

Edo approximations for 21/20 (84.47 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
13	1\13	92.31	+7.84	+8.49
14	1\14	85.71	+1.25	+1.45
15	1\15	80.00	-4.47	-5.58
27	2\27	88.89	+4.42	+9.95
28	2\28	85.71	+1.25	+2.91
29	2\29	82.76	-1.71	-4.13
42	3\42	85.71	+1.25	+4.36
43	3\43	83.72	-0.75	-2.67
44	3\44	81.82	-2.65	-9.71
56	4\56	85.71	+1.25	+5.82
57	4\57	84.21	-0.26	-1.22
58	4\58	82.76	-1.71	-8.26
70	5\70	85.71	+1.25	+7.27
71	5\71	84.51	+0.04	+0.24
72	5\72	83.33	-1.13	-6.80

Interval chain

An interval chain of 21/20's stacked on top of one another comes close to approximating some important JI intervals. The error between the approximation and the target JI interval may be tempered out in some regular temperaments.

Some examples include:

• A stack of two 21/20's upwards is ~4 ¢ from 11/10;
• A stack of seven 21/20's upwards is ~9 ¢ from 7/5;
• A stack of ten 21/20's upwards is ~4 ¢ from 13/8;
• A stack of twelve 21/20's upwards is ~4 ¢ from 9/5;

and

• A stack of six 21/20's downwards is ~10 ¢ from 3/2;
• A stack of nine 21/20's downwards is ~5 ¢ from 9/7;
• A stack of eleven 21/20's downwards is ~4 ¢ from 7/6.

When treated as a scale, this interval chain can be called the ambitonal sequence of 21/20 (AS21/20 or 1ed21/20).

1ed21/20 is equal to approximately 14.2067edo, and as a result of tethering between compressed 14 and heavily stretched 15. It is quite xenharmonic in its sound. It is related to the nautilus, sextilifourths and floral temperaments.

1ed21/20 offers a possible approximation of the no-3s 11-limit, or alternatively of the 2.9.5.7.11.17 subgroup.

Approximation of harmonics in 1ed21/20

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-17.5	+40.8	-34.9	+1.1	+23.3	+9.9	+32.1	-2.9	-16.4	-12.4	+5.9
	Relative (%)	-20.7	+48.3	-41.3	+1.3	+27.6	+11.7	+38.0	-3.4	-19.4	-14.7	+7.0
Step		14	23	28	33	37	40	43	45	47	49	51

Approximation of harmonics in 1ed21/20

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+36.2	-7.6	+41.9	+14.6	-5.9	-20.3	-29.5	-33.8	-33.8	-29.9	-22.4	-11.6
	Relative (%)	+42.9	-9.0	+49.6	+17.3	-6.9	-24.1	-34.9	-40.0	-40.0	-35.4	-26.5	-13.7
Step		53	54	56	57	58	59	60	61	62	63	64	65

See also

• 40/21 – its octave complement
• 10/7 – its fifth complement
• List of superparticular intervals
• Gallery of just intervals
• Septisemi temperaments, where it is tempered out