31/16: Difference between revisions

Interval information
Ratio	31/16
Subgroup monzo	2.31 [-4 1⟩
Size in cents	1145.036¢
Names	tricesimoprimal ultramajor seventh,; tricesimoprimal semidiminished octave,; octave-reduced 31st harmonic
Color name	31o7, thiwo 7th
FJS name	[math]\displaystyle{ \text{M7}^{31} }[/math]
Special properties	reduced,; reduced harmonic
Tenney norm (log2 nd)	8.9542
Weil norm (log2 max(n, d))	9.90839
Wilson norm (sopfr(nd))	39
	https://en.xen.wiki/w/File:Jid_31_16_pluck_adu_dr220.mp3; [sound info]
	Open this interval in xen-calc

Latest revision as of 07:26, 25 February 2026

In 31-limit just intonation, 31/16 is either the tricesimoprimal ultramajor seventh or tricesimoprimal semidiminished octave, which is also the octave-reduced 31st harmonic. It is sharp of the Pythagorean major seventh (243/128) by 248/243, and flat of the octave by 32/31.

31/16 is very well approximated by 22edo, with 21\22 being 0.4 ¢ sharp of 31/16.

Edo approximations for 31/16 (1145.04 ¢)
*≤ 80edo, relative error ≤ 10%*
Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
2	2\2	1200.00	+54.96	+9.16
20	19\20	1140.00	-5.04	-8.39
21	20\21	1142.86	-2.18	-3.81
22	21\22	1145.45	+0.42	+0.77
23	22\23	1147.83	+2.79	+5.35
24	23\24	1150.00	+4.96	+9.93
42	40\42	1142.86	-2.18	-7.62
43	41\43	1144.19	-0.85	-3.04
44	42\44	1145.45	+0.42	+1.54
45	43\45	1146.67	+1.63	+6.12
64	61\64	1143.75	-1.29	-6.86
65	62\65	1144.62	-0.42	-2.28
66	63\66	1145.45	+0.42	+2.30
67	64\67	1146.27	+1.23	+6.88

@@ Line 1: / Line 1: @@
-'''31/16'''
+{{Infobox Interval
-|-4 0 0 0 0 0 0 0 0 0 1&gt;
+| Name = tricesimoprimal ultramajor seventh, tricesimoprimal semidiminished octave, octave-reduced 31st harmonic
+| Color name = 31o7, thiwo 7th
+| Sound = jid_31_16_pluck_adu_dr220.mp3
+}}
-.0356 cents
+In [[31-limit]] [[just intonation]], '''31/16''' is either the '''tricesimoprimal ultramajor seventh''' or '''tricesimoprimal semidiminished octave''', which is also the [[octave reduction|octave-reduced]] [[31/1|31st]] [[harmonic]]. It is sharp of the [[243/128|Pythagorean major seventh (243/128)]] by [[248/243]], and flat of the octave by [[32/31]].
-[[File:jid_31_16_pluck_adu_dr220.mp3]] [[:File:jid_31_16_pluck_adu_dr220.mp3|sound sample]]
+== Approximation ==
+/16 is very well approximated by [[22edo]], with 21\22 being 0.4 [[Cent|¢]] sharp of 31/16.
+{{Interval edo approximation}}
-The 31st harmonic, octave-reduced.
+== See also ==
-[[Category:todo:expand]]
+* [[32/31]] – its [[octave complement]]
+* [[48/31]] – its [[twelfth complement]]
+[[Category:Seventh]]
+[[Category:Supermajor seventh]]
+[[Category:Octave]]
+[[Category:Suboctave]]