List of superparticular intervals

This list of superparticular intervals ordered by prime limit. It reaches to the 101-limit and is complete up to the 23-limit.

Superparticular numbers are ratios of the form (n + 1)/n, or 1 + 1/n, where n is a whole number other than 1. They appear frequently in just intonation and harmonic series music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio 21/20. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common commas are superparticular ratios.

The list below is ordered by harmonic limit, or the largest prime involved in the prime factorization. 36/35, for instance, is an interval of the 7-limit, as it factors to (2²×3²)/(5×7), while 37/36 would belong to the 37-limit.

Størmer's theorem states that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8. OEIS: A002071 gives the number of superparticular ratios in each prime limit, OEIS: A145604 shows the increment from limit to limit, and OEIS: A117581 gives the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).

See also gallery of just intervals. Many of the names below come from the Scala website.

Ratio	Cents	Factorization	Monzo	Name(s)	Meta
2-limit (complete)
2/1	1200.000	2/1	[1⟩	octave, duple; after octave reduction: (perfect) unison, unity, perfect prime, tonic
3-limit (complete)
3/2	701.955	3/2	[-1 1⟩	perfect fifth, octave-reduced 3rd harmonic, diapente
4/3	498.045	22/3	[2 -1⟩	perfect fourth, octave-reduced 3rd subharmonic, diatessaron	S2
9/8	203.910	32/23	[-3 2⟩	(Pythagorean) (whole) tone, Pythagorean major second, major whole tone, octave-reduced 9th harmonic or harmonic ninth	S3
5-limit (complete)
5/4	386.314	5/22	[-2 0 1⟩	classic(al)/just major third, octave-reduced 5th harmonic
6/5	315.641	(2*3)/5	[1 1 -1⟩	classic(al)/just minor third
10/9	182.404	(2*5)/32	[1 -2 1⟩	classic(al) (whole) tone, classic major second, minor whole tone
16/15	111.731	24/(3*5)	[4 -1 -1⟩	classic(al)/just diatonic semitone, 15th subharmonic	S4
25/24	70.672	52/(23*3)	[-3 -1 2⟩	classic(al)/just chromatic semitone, chroma, Zarlinian semitone	S5
81/80	21.506	(3/2)4/5	[-4 4 -1⟩	syntonic comma, Didymus comma	S9
7-limit (complete)
7/6	266.871	7/(2*3)	[-1 -1 0 1⟩	(septimal) subminor third, septimal minor third
8/7	231.174	23/7	[3 0 0 -1⟩	(septimal) supermajor second, septimal whole tone, octave-reduced 7th subharmonic
15/14	119.443	(3*5)/(2*7)	[-1 1 1 -1⟩	septimal major semitone, septimal diatonic semitone
21/20	84.467	(3*7)/(22*5)	[-2 1 -1 1⟩	septimal minor semitone, large septimal chroma
28/27	62.961	(22*7)/33	[2 -3 0 1⟩	septimal 1/3-tone, small septimal chroma, (septimal) subminor second, septimal minor second, trienstonic comma
36/35	48.770	(22*32)/(5*7)	[2 2 -1 -1⟩	septimal 1/4-tone, septimal diesis	S6
49/48	35.697	72/(24*3)	[-4 -1 0 2⟩	slendro diesis, large septimal diesis, large septimal 1/6-tone	S7
50/49	34.976	2*(5/7)2	[1 0 2 -2⟩	jubilisma, small septimal diesis, small septimal 1/6-tone, tritonic diesis, Erlich's decatonic comma
64/63	27.264	26/(32*7)	[6 -2 0 -1⟩	septimal comma, Archytas' comma	S8
126/125	13.795	(2*32*7)/53	[1 2 -3 1⟩	starling comma, septimal semicomma
225/224	7.7115	(3*5)2/(25*7)	[-5 2 2 -1⟩	marvel comma, septimal kleisma	S15
2401/2400	0.72120	74/(25*3*52)	[-5 -1 -2 4⟩	breedsma	S49
4375/4374	0.39576	(54*7)/(2*37)	[-1 -7 4 1⟩	ragisma
11-limit (complete)
11/10	165.004	11/(2*5)	[-1 0 -1 0 1⟩	(large) undecimal neutral second, undecimal submajor second, Ptolemy's second
12/11	150.637	(22*3)/11	[2 1 0 0 -1⟩	(small) undecimal neutral second
22/21	80.537	(2*11)/(3*7)	[1 -1 0 -1 1⟩	undecimal minor semitone
33/32	53.273	(3*11)/25	[-5 1 0 0 1⟩	undecimal 1/4-tone, undecimal diesis, al-Farabi's 1/4-tone, octave-reduced 33rd harmonic
45/44	38.906	(3/2)2*(5/11)	[-2 2 1 0 -1⟩	undecimal 1/5-tone
55/54	31.767	(5*11)/(2*33)	[-1 -3 1 0 1⟩	telepathma, eleventyfive comma, undecimal diasecundal comma
56/55	31.194	(23*7)/(5*11)	[3 0 -1 1 -1⟩	undecimal tritonic comma, konbini comma
99/98	17.576	(3/7)2*(11/2)	[-1 2 0 -2 1⟩	mothwellsma, small undecimal comma
100/99	17.399	(2*5/3)2/11)	[2 -2 2 0 -1⟩	ptolemisma, Ptolemy's comma	S10
121/120	14.376	112/(23*3*5)	[-3 -1 -1 0 2⟩	biyatisma, undecimal seconds comma	S11
176/175	9.8646	(24*11)/(52*7)	[4 0 -2 -1 1⟩	valinorsma
243/242	7.1391	35/(2*112)	[-1 5 0 0 -2⟩	rastma, neutral thirds comma
385/384	4.5026	(5*7*11)/(27*3)	[-7 -1 1 1 1⟩	keenanisma
441/440	3.9302	(3*7)2/(23*5*11)	[-3 2 -1 2 -1⟩	werckisma, Werckmeister's undecimal septenarian schisma	S21
540/539	3.2090	(2/7)2*33*5/11	[2 3 1 -2 -1⟩	swetisma, Swets' comma
3025/3024	0.57240	(5*11)2/(24*32*7)	[-4 -3 2 -1 2⟩	lehmerisma	S55
9801/9800	0.17665	(11/(5*7))2*34/23	[-3 4 -2 -2 2⟩	kalisma, Gauss comma	S99
13-limit (complete)
13/12	138.573	13/(22*3)	[-2 -1 0 0 0 1⟩	(large) tridecimal 2/3-tone, tridecimal neutral second
14/13	128.298	(2*7)/13	[1 0 0 1 0 -1⟩	(small) tridecimal 2/3-tone, trienthird
26/25	67.900	(2*13)/52	[1 0 -2 0 0 1⟩	(large) tridecimal 1/3-tone
27/26	65.337	33/(2*13)	[-1 3 0 0 0 -1⟩	(small) tridecimal 1/3-tone
40/39	43.831	(23*5)/(3*13)	[3 -1 1 0 0 -1⟩	tridecimal minor diesis
65/64	26.841	(5*13)/26	[-6 0 1 0 0 1⟩	wilsorma, 13th-partial chroma
66/65	26.432	(2*3*11)/(5*13)	[1 1 -1 0 1 -1⟩	winmeanma
78/77	22.339	(2*3*13)/(7*11)	[1 1 0 -1 -1 1⟩	negustma
91/90	19.130	(7*13)/(2*32*5)	[-1 -2 -1 1 0 1⟩	Biome comma, superleap comma
105/104	16.567	(3*5*7)/(23*13)	[-3 1 1 1 0 -1⟩	animist comma, small tridecimal comma
144/143	12.064	(22*3)2/(11*13)	[4 2 0 0 -1 -1⟩	grossma	S12
169/168	10.274	132/(23*3*7)	[-3 -1 0 -1 0 2⟩	buzurgisma, dhanvantarisma	S13
196/195	8.8554	(2*7)2/(3*5*13)	[2 -1 -1 2 0 -1⟩	mynucuma	S14
325/324	5.3351	(52*13)/(22*34)	[-2 -4 2 0 0 1⟩	marveltwin comma
351/350	4.9393	(3/5)2*13/(2*7)	[-1 3 -2 -1 0 1⟩	ratwolfsma
352/351	4.9253	(25*11)/(32*13)	[5 -3 0 0 1 -1⟩	minthma
364/363	4.7627	(2/11)2*7*13/3	[2 -1 0 1 -2 1⟩	gentle comma
625/624	2.7722	(5/2)4/(3*13)	[-4 -1 4 0 0 -1⟩	tunbarsma	S25
676/675	2.5629	(2*13/5)2/33	[2 -3 -2 0 0 2⟩	island comma	S26
729/728	2.3764	(32/2)3/(7*13)	[-3 6 0 -1 0 -1⟩	squbema	S27
1001/1000	1.7304	7*11*13/(2*5)3	[-3 0 -3 1 1 1⟩	sinbadma
1716/1715	1.0092	22*3*11*13/(5*73)	[2 1 -1 -3 1 1⟩	lummic comma
2080/2079	0.83252	25*5*13/(33*7*11)	[5 -3 1 -1 -1 1⟩	ibnsinma
4096/4095	0.42272	(26/3)2/(5*7*13)	[12 -2 -1 -1 0 -1⟩	schismina, tridecimal schisma	S65
4225/4224	0.40981	(5*13)2/(27*3*11)	[-7 -1 2 0 -1 2⟩	leprechaun comma	S66
6656/6655	0.26012	(23/11)3*13/5	[9 0 -1 0 -3 1⟩	jacobin comma
10648/10647	0.16260	(2*11)3/((3*13)2*7)	[3 -2 0 -1 3 -2⟩	harmonisma
123201/123200	0.014052	(3/2)6*(13/5)2/(7*11)	[-6 6 -2 -1 -1 2⟩	chalmersia	S351
17-limit (complete)
17/16	104.955	17/24	[-4 0 0 0 0 0 1⟩	large septendecimal semitone, octave-reduced 17th harmonic
18/17	98.955	(2*32)/17	[1 2 0 0 0 0 -1⟩	small septendecimal semitone, Arabic lute index finger
34/33	51.682	(2*17)/(3*11)	[1 -1 0 0 -1 0 1⟩	large septendecimal 1/4-tone
35/34	50.184	(5*7)/(2*17)	[-1 0 1 1 0 0 -1⟩	small septendecimal 1/4-tone
51/50	34.283	(3*17)/(2*52)	[-1 1 -2 0 0 0 1⟩	large septendecimal 1/6-tone
52/51	33.617	(22*13)/(3*17)	[2 -1 0 0 0 1 -1⟩	small septendecimal 1/6-tone
85/84	20.488	(5*17)/(22*3*7)	[-2 -1 1 -1 0 0 1⟩	septendecimal comma (?)
120/119	14.487	(23*3*5)/(7*17)	[3 1 1 -1 0 0 -1⟩	lynchisma
136/135	12.777	(2/3)3*17/5	[3 -3 -1 0 0 0 1⟩	septendecimal major second comma
154/153	11.278	(2*7*11)/(32*17)	[1 -2 0 1 1 0 -1⟩
170/169	10.214	(2*5*17)/132	[1 0 1 0 0 -2 1⟩
221/220	7.8514	(13*17)/(22*5*11)	[-2 0 -1 0 -1 1 1⟩
256/255	6.7759	28/(3*5*17)	[8 -1 -1 0 0 0 -1⟩	septendecimal kleisma, octave-reduced 255th subharmonic	S16
273/272	6.3532	(3*7*13)/(24*17)	[-4 1 0 1 0 1 -1⟩	tannisma
289/288	6.0008	(17/3)2/25	[-5 -2 0 0 0 0 2⟩	septendecimal 6-cent comma	S17
375/374	4.6228	(3*53)/(2*11*17)	[-1 1 3 0 -1 0 -1⟩	ursulisma
442/441	3.9213	(2*13*17)/(3*7)2	[1 -2 0 -2 0 1 1⟩
561/560	3.0887	(3*11*17)/(24*5*7)	[-4 1 -1 -1 1 0 1⟩
595/594	2.9121	(5*7*17)/(2*33*11)	[-1 -3 1 1 -1 0 1⟩	dakotisma
715/714	2.4230	(5*11*13)/(2*3*7*17)	[-1 -1 1 -1 1 1 -1⟩	september comma, septembrisma
833/832	2.0796	(72*17)/(26*13)	[-6 0 0 2 0 -1 1⟩	horizma, horizon comma
936/935	1.8506	(23*32*13)/(5*11*17)	[3 2 -1 0 -1 1 -1⟩	ainos comma, ainma
1089/1088	1.5905	(3*11)2/(26*17)	[-6 2 0 0 2 0 -1⟩	twosquare comma	S33
1156/1155	1.4983	(2*17)2/(3*5*7*11)	[2 -1 -1 -1 -1 0 2⟩	septendecimal 1/4-tones comma	S34
1225/1224	1.4138	(5*7)2/(23*32*17)	[-3 -2 2 2 0 0 -1⟩	noellisma	S35
1275/1274	1.3584	(3*52*17)/(2*72*13)	[-1 1 2 -2 0 -1 1⟩
1701/1700	1.0181	(35*7)/[(2*5)2*17]	[-2 5 -2 1 0 0 -1⟩	palingenetic comma, palingenesis
2058/2057	0.84143	(2*3*73)/(112*17)	[1 1 0 3 -2 0 -1⟩	xenisma
2431/2430	0.71230	(11*13*17)/(2*35*5)	[-1 -5 -1 0 1 1 1⟩
2500/2499	0.69263	(2*52)2/(3*72*17)	[2 -1 4 -2 0 0 -1⟩	sperasma	S50
2601/2600	0.66573	(3*17)2/(23*52*13)	[-3 2 -2 0 0 -1 2⟩	septendecimal 1/6-tones comma	S51
4914/4913	0.35234	(2*33*7*13)/173	[1 3 0 1 0 1 -3⟩
5832/5831	0.29688	(2*32)3/(73*17)	[3 6 0 -3 0 0 -1⟩	chlorisma
12376/12375	0.13989	(23*7*13*17)/(32*53*11)	[3 -2 -3 1 -1 1 1⟩	flashma
14400/14399	0.12023	(23*3*5)2/(7*112*17)	[6 2 2 -1 -2 0 -1⟩	sparkisma	S120
28561/28560	0.060616	134/(24*3*5*7*17)	[-4 -1 -1 -1 0 4 -1⟩		S169
31213/31212	0.055466	(74*13)/(22*33*172)	[-2 -3 0 4 0 1 -2⟩
37180/37179	0.046564	(22*5*11*132)/(37*17)	[2 -7 1 0 1 2 -1⟩
194481/194480	0.008902	(3*7)4/(24*5*11*13*17)	[-4 4 -1 4 -1 -1 -1⟩	scintillisma	S441
336141/336140	0.005150	(32*133*17)/(22*5*75)	[-2 2 -1 -5 0 3 1⟩
19-limit (complete)
19/18	93.603	19/(2*32)	[-1 -2 0 0 0 0 0 1⟩	large undevicesimal semitone
20/19	88.801	(22*5)/19	[2 0 1 0 0 0 0 -1⟩	small undevicesimal semitone
39/38	44.970	(3*13)/(2*19)	[-1 1 0 0 0 1 0 -1⟩	undevicesimal 2/9-tone
57/56	30.642	(3*19)/(23*7)	[-3 1 0 -1 0 0 0 1⟩	hendrix comma
76/75	22.931	(22*19)/(3*52)	[2 -1 -2 0 0 0 0 1⟩	large undevicesimal 1/9-tone
77/76	22.631	(7*11)/(22*19)	[-2 0 0 1 1 0 0 -1⟩	small undevicesimal 1/9-tone
96/95	18.128	(25*3)/(5*19)	[5 1 -1 0 0 0 0 -1⟩	19th-partial chroma
133/132	13.066	(19*7)/(22*3*11)	[-2 -1 0 1 -1 0 0 1⟩
153/152	11.352	(32*17)/(23*19)	[-3 2 0 0 0 0 1 -1⟩	ganassisma, Ganassi's comma
171/170	10.154	(32*19)/(2*5*17)	[-1 2 -1 0 0 0 -1 1⟩
190/189	9.1358	(2*5*19)/(33*7)	[1 -3 1 -1 0 0 0 1⟩
209/208	8.3033	(11*19)/(24*13)	[-4 0 0 0 1 -1 0 1⟩	yama comma
210/209	8.2637	(2*3*5*7)/(11*19)	[1 1 1 1 -1 0 0 -1⟩	spleen comma
286/285	6.0639	(2*11*13)/(3*5*19)	[1 -1 -1 0 1 1 0 -1⟩
324/323	5.3516	(2*32)2/(17*19)	[2 4 0 0 0 0 -1 -1⟩	nusu comma	S18
343/342	5.0547	73/(2*32*19)	[-1 -2 0 3 0 0 0 -1⟩
361/360	4.8023	192/(23*32*5)	[-3 -2 -1 0 0 0 0 2⟩	go comma	S19
400/399	4.3335	(22*5)2/(3*7*19)	[4 -1 2 -1 0 0 0 -1⟩		S20
456/455	3.8007	(23*3*19)/(5*7*13)	[3 1 -1 -1 0 -1 0 1⟩
476/475	3.6409	(22*7*17)/(52*19)	[2 0 -2 1 0 0 1 -1⟩
495/494	3.5010	(32*5*11)/(2*13*19)	[-1 2 1 0 1 -1 0 -1⟩
513/512	3.3780	(33*19)/29	[-9 3 0 0 0 0 0 1⟩	undevicesimal comma, undevicesimal schisma, Boethius' comma, 513th harmonic
969/968	1.7875	(3*17*19)/(23*112)	[-3 1 0 0 -2 0 1 1⟩
1216/1215	1.4243	(26*19)/(35*5)	[6 -5 -1 0 0 0 0 1⟩	password comma, Eratosthenes' comma
1331/1330	1.3012	113/(2*5*7*19)	[-1 0 -1 -1 3 0 0 -1⟩
1445/1444	1.1985	5*(17/(2*19))2	[-2 0 1 0 0 0 2 -2⟩	aureusma
1521/1520	1.1386	(3*13)2/(24*5*19)	[-4 2 -1 0 0 2 0 -1⟩	pinkanberry	S39
1540/1539	1.1245	(22*5*7*11)/(34*19)	[2 -4 1 1 1 0 0 -1⟩
1729/1728	1.0016	(7*13*19)/(22*3)3	[-6 -3 0 1 0 1 0 1⟩	ramanujanisma
2376/2375	0.7288	(23*33*11)/(53*19)	[3 3 -3 0 1 0 0 -1⟩
2432/2431	0.7120	(27*19)/(11*13*17)	[7 0 0 0 -1 -1 -1 1⟩	Blumeyer comma
2926/2925	0.5918	(2*7*11*19)/(32*52*13)	[1 -2 -2 1 1 -1 0 1⟩
3136/3135	0.5521	(23*7)2/(3*5*11*19)	[6 -1 -1 2 -1 0 0 -1⟩		S56
3250/3249	0.5328	(2*53*13)/(3*19)2	[1 -2 3 0 0 1 0 -2⟩
4200/4199	0.4123	(23*3*52*7)/(13*17*19)	[3 1 2 1 0 -1 -1 -1⟩
5776/5775	0.2998	(22*19)2/(3*52*7*11)	[4 -1 -2 -1 -1 0 0 2⟩		S76
5929/5928	0.2920	(7*11)2/(23*3*13*19)	[-3 -1 0 2 2 -1 0 -1⟩		S77
5985/5984	0.2893	(32*5*7*19)/(25*11*17)	[-5 2 1 1 -1 0 -1 1⟩
6175/6174	0.2804	(52*13*19)/(2*32*73)	[-1 -2 2 -3 0 1 0 1⟩
6860/6859	0.2524	(22*5*73)/193	[2 0 1 3 0 0 0 -3⟩
10241/10240	0.1691	(72*11*19)/(211*5)	[-11 0 -1 2 1 0 0 1⟩
10830/10829	0.1599	(2*3*5*192)/(72*13*17)	[1 1 1 -2 0 -1 -1 2⟩
12636/12635	0.1370	(22*35*13)/(5*7*192)	[2 5 -1 -1 0 1 0 -2⟩
13377/13376	0.1294	(3*73*13)/(26*11*19)	[-6 1 0 3 -1 1 0 -1⟩
14080/14079	0.1230	(28*5*11)/(3*13*192)	[8 -1 1 0 1 -1 0 -2⟩
14365/14364	0.1205	(5*132*17)/(22*33*7*19)	[-2 -3 1 -1 0 1 1 -1⟩
23409/23408	0.07396	(32*17)2/(24*7*11*19)	[-4 4 0 -1 -1 0 1 -1⟩		S153
27456/27455	0.06306	(26*3*11)/(5*172*19)	[6 1 -1 0 1 0 -2 -1⟩
28900/28899	0.05991	(2*5*17)2/(32*132*19)	[2 -2 2 0 0 -2 2 -1⟩		S170
43681/43680	0.03963	(11*19)2/(25*3*5*7*13)	[-5 -1 -1 -1 2 -1 0 2⟩		S209
89376/89375	0.01937	(25*3*72*19)/(54*11*13)	[5 1 -4 2 -1 -1 0 1⟩
104976/104975	0.01649	(2*32)4/(52*13*17*19)	[4 8 -2 0 0 0 -1 -1 -1⟩		S324
165376/165375	0.01047	(29*17*19)/(33*53*72)	[9 -3 -3 -2 0 0 1 1⟩	decimillisma
228096/228095	0.007590	(28*34*11)/(5*74*19)	[8 4 -1 -4 1 0 0 -1⟩
601426/601425	0.002879	(2*72*17*192)/(37*52*11)	[2 -7 -2 2 -1 0 1 2⟩
633556/633555	0.002733	(22*7*113*17)/(33*5*13*192)	[2 -3 -1 1 3 -1 1 -2⟩
709632/709631	0.002440	(210*32*7*11)/(133*17*19)	[10 2 0 1 1 -3 -1 -1⟩
5909761/5909760	0.0002929	(11*13*17)2/(28*35*5*19)	[-8 -5 -1 0 2 2 2 -1⟩		S2431
11859211/11859210	0.0001460	(7*13*194)/(2*34*5*114)	[-1 -4 -1 1 -4 1 0 4⟩
23-limit (complete)
23/22	76.956	23/(2*11)	[-1 0 0 0 -1 0 0 0 1⟩	greater vicesimotertial semitone
24/23	73.681	(23*3)/23	[3 1 0 0 0 0 0 0 -1⟩	small vicesimotertial semitone
46/45	38.051	(2*23)/(32*5)	[1 -2 -1 0 0 0 0 0 1⟩	vicesimotertial 1/5-tone
69/68	25.274	(3*23)/(22*17)	[-2 1 0 0 0 0 -1 0 1⟩	large vicesimotertial 1/8-tone
70/69	24.910	(2*5*7)/(3*23)	[1 -1 1 1 0 0 0 0 -1⟩	small vicesimotertial 1/8-tone
92/91	18.921	(22*23)/(7*13)	[2 0 0 -1 0 -1 0 0 1⟩
115/114	15.120	(5*23)/(2*3*19)	[-1 -1 1 0 0 0 0 -1 1⟩
161/160	10.787	(7*23)/(25*5)	[-5 0 -1 1 0 0 0 0 1⟩
162/161	10.720	(2*34)/(7*23)	[1 4 0 -1 0 0 0 0 -1⟩
208/207	8.3433	(24*13)/(32*23)	[4 -2 0 0 0 1 0 0 -1⟩
231/230	7.5108	(3*7*11)/(2*5*23)	[-1 1 -1 1 1 0 0 0 -1⟩
253/252	6.8564	(11*23)/((2*3)2*7)	[-2 -2 0 -1 1 0 0 0 1⟩
276/275	6.2840	(22*3*23)/(52*11)	[2 1 -2 0 -1 0 0 0 1⟩
300/299	5.7804	((2*5)2*3)/(13*23)	[2 1 2 0 0 -1 0 0 -1⟩
323/322	5.3682	(17*19)/(2*7*23)	[-1 0 0 -1 0 0 1 1 -1⟩
391/390	4.4334	(17*23)/(2*3*5*13)	[-1 -1 -1 0 0 -1 1 0 1⟩
392/391	4.4221	(23*72)/(17*23)	[3 0 0 2 0 0 -1 0 -1⟩
460/459	3.7676	(22*5*23)/(33*17)	[2 -3 1 0 0 0 -1 0 1⟩
484/483	3.5806	(2*11)2/(3*7*23)	[2 -1 0 -1 2 0 0 0 -1⟩		S22
507/506	3.4180	(3*132)/(2*11*23)	[-1 1 0 0 -1 2 0 0 -1⟩
529/528	3.2758	232/(24*3*11)	[-4 -1 0 0 -1 0 0 0 2⟩		S23
576/575	3.0082	(23*3)2/(23*52)	[6 2 -2 0 0 0 0 0 -1⟩		S24
736/735	2.3538	(25*23)/(3*5*72)	[5 -1 -1 -2 0 0 0 0 1⟩
760/759	2.2794	(23*5*19)/(3*11*23)	[3 -1 1 0 -1 0 0 1 -1⟩
875/874	1.9797	(53*7)/(2*19*23)	[-1 0 3 1 0 0 0 -1 -1⟩
897/896	1.9311	(3*13*23)/(27*7)	[-7 1 0 -1 0 1 0 0 1⟩
1105/1104	1.5674	(5*13*17)/(24*3*23)	[-4 -1 1 0 0 1 1 0 -1⟩
1197/1196	1.4469	(32*17*19)/(22*13*23)	[-2 2 0 0 0 -1 1 1 -1⟩
1288/1287	1.3446	(23*7*23)/(32*11*13)	[3 -2 0 1 -1 -1 0 0 1⟩
1496/1495	1.1576	(23*11*17)/(5*13*23)	[3 0 -1 0 1 -1 1 0 -1⟩
1863/1862	0.92952	(34*23)/(2*72*19)	[-1 4 0 -2 0 0 0 -1 1⟩
2024/2023	0.85556	(23*11*23)/(7*172)	[3 0 0 -1 1 0 -2 0 1⟩
2025/2024	0.85514	(32*5)2/(23*11*23)	[-3 4 2 0 -1 0 0 0 -1⟩		S45
2185/2184	0.79251	(5*19*23)/(23*3*7*13)	[-3 -1 1 -1 0 -1 0 1 1⟩
2300/2299	0.75287	(22*52*23)/(112*19)	[2 0 2 0 -2 0 0 -1 1⟩
2646/2645	0.65441	(2*33*72)/(5*232)	[1 3 -1 2 0 0 0 0 -2⟩
2737/2736	0.63265	(7*17*23)/(24*32*19)	[-4 -2 0 1 0 0 1 -1 1⟩
3060/3059	0.56586	(22*32*5*17)/(7*19*23)	[2 2 1 -1 0 0 1 -1 -1⟩
3381/3380	0.51212	(3*72*23)/(22*5*132)	[-2 1 -1 2 0 -2 0 0 1⟩
3520/3519	0.49190	(26*5*11)/(32*17*23)	[6 -2 1 0 1 0 -1 0 -1⟩
3888/3887	0.44533	(24*35)/(132*23)	[4 5 0 0 0 -2 0 0 -1⟩
4693/4692	0.36893	(13*192)/(22*3*17*23)	[-2 -1 0 0 0 1 -1 2 -1⟩
4761/4760	0.36367	(3*23)2/(23*5*7*17)	[-3 2 -1 -1 0 0 -1 0 2⟩		S69
5083/5082	0.34063	(13*17*23)/(2*3*7*112)	[-1 -1 0 -1 -2 1 1 0 1⟩
7866/7865	0.22010	(2*32*19*23)/(5*112*13)	[1 2 -1 0 -2 -1 0 1 1⟩
8281/8280	0.20907	(7*13)2/(23*32*5*23)	[-3 -2 -1 2 0 2 0 0 -1⟩		S91
8625/8624	0.20073	(3*53*23)/(24*72*11)	[-4 1 3 -2 -1 0 0 0 1⟩
10626/10625	0.16293	(2*3*7*11*23)/(54*17)	[1 1 -4 1 1 0 -1 0 1⟩
11271/11270	0.15361	(3*13*172)/(2*5*72*23)	[-1 1 -1 -2 0 1 2 0 -1⟩
11662/11661	0.14846	(2*73*17)/(3*132*23)	[1 0 0 3 0 -2 1 0 -1⟩
12168/12167	0.14228	(23*32*132)/(233)	[3 2 0 0 0 2 0 0 -3⟩
16929/16928	0.10227	(34*11*19)/(25*232)	[-5 4 0 0 1 0 0 1 -2⟩
19551/19550	0.088552	(3*73*19)/(2*52*17*23)	[-1 1 -2 3 0 0 -1 1 -1⟩
21505/21504	0.080506	(5*11*17*23)/(210*3*7)	[-10 -1 1 -1 1 0 1 0 1⟩
21736/21735	0.079650	(23*11*13*19)/(33*5*7*23)	[3 -3 -1 -1 1 1 0 1 -1⟩
23276/23275	0.074380	(22*11*232)/(52*72*19)	[2 0 -2 -2 1 0 0 -1 2⟩
25025/25024	0.069182	(52*7*11*13)/(26*17*23)	[-6 0 2 1 1 1 -1 0 -1⟩
25921/25920	0.066790	(7*23)2/(26*34*5)	[-6 -4 -1 2 0 0 0 0 2⟩		S161
43264/43263	0.040016	(24*13)2/(32*11*19*23)	[8 -2 0 0 -1 2 0 -1 -1⟩		S208
52326/52325	0.033086	(2*34*17*19)/(52*7*13*23)	[1 4 -2 -1 0 -1 1 1 -1⟩
71875/71874	0.024087	(55*23)/(2*33*113)	[-1 -3 5 0 -3 0 0 0 1⟩
75141/75140	0.023040	(33*112*23)/(22*5*13*172)	[-2 3 -1 0 2 -1 -2 0 1⟩
76545/76544	0.022617	(37*5*7)/(28*13*23)	[-8 7 1 1 0 -1 0 0 -1⟩
104329/104328	0.016594	(17*19)2/(23*34*7*23)	[-3 -4 0 -1 -1 0 2 2 -1⟩		S323
122452/122451	0.014138	(22*113*23)/(3*74*17)	[2 -1 0 -4 3 0 -1 0 1⟩
126225/126224	0.013716	(33*52*11*17)/(24*73*23)	[-4 3 2 -3 1 0 1 0 -1⟩
152881/152880	0.011324	(17*23)2/(24*3*5*72*13)	[-4 -1 -1 -2 0 -1 2 0 2⟩		S391
202125/202124	0.0085652	(3*53*72*11)/(22*133*23)	[-2 1 3 2 1 -3 0 0 -1⟩
264385/264384	0.0065482	(5*112*19*23)/(26*35*17)	[-6 -5 1 0 2 0 -1 1 1⟩
282625/282624	0.0061256	(53*7*17*19)/(212*3*23)	[-12 -1 3 1 0 0 1 1 -1⟩
328510/328509	0.0052700	(2*5*7*13*192)/(3*23)3	[1 -3 1 1 0 1 0 0 -3⟩
2023425/2023424	0.00085560	(32*52*17*232)/(213*13*19)	[-13 2 2 0 0 -1 1 -1 2⟩
4096576/4096575	0.00042261	(23*11*23)2/(34*52*7*172)	[6 -4 -2 -1 2 0 -2 0 2⟩		S2024
5142501/5142500	0.00033665	(33*72*132*23)/(22*54*112*17)	[-2 3 -4 2 -2 2 -1 0 1⟩
29-limit (incomplete)
29/28	60.751	29/(22*7)		Large vicesimononal 1/4 tone
30/29	58.692	(2*3*5)/29		Small vicesimononal 1/4 tone
58/57	30.109	(2*29)/(3*19)
88/87	19.786	(23*11)/(3*29)
116/115	14.989	(22*29)/(5*23)
117/116	14.860	(33*13)/(22*29)
145/144	11.981	(5*29)/(24*32)
175/174	9.9211	(52*7)/(2*3*29)
204/203	8.5073
232/231	7.4783
261/260	6.6458
290/289	5.9801
320/319	5.4186
378/377	4.5861
406/405	4.2694
494/493	3.5081
551/550	3.1448
552/551	3.1391
609/608	2.8451
638/637	2.7157
726/725	2.3863
31-limit (incomplete)
31/30	56.767	31/(2*3*5)		large tricesimoprimal 1/4-tone
32/31	54.964	25/31		small tricesimoprimal 1/4-tone, 31st subharmonic
63/62	27.700	(32*7)/(2*31)
93/92	18.716	(3*31)/(22*23)
125/124	13.906	(53)/(22*31)		twizzler
621/620	2.7901	(3³*23)/(2²*5*31)		owowhatsthisma
3969/3968	0.43624	(34*72)/(27*31)		yunzee comma	S63
37-limit (incomplete)
37/36	47.434	37/(22*32)		Large 37-limit quarter tone, 37th-partial chroma
38/37	46.169	(2*19)/37		Small 37-limit quarter tone
75/74	23.238	(3*52)/(2*37)
41-limit (incomplete)
41/40	42.749	41/(23*5)		Large 41-limit fifth tone
42/41	41.719	(2*3*7)/41		Small 41-limit fifth tone
82/81	21.242	(2*41)/34		41st-partial chroma
43-limit (incomplete)
43/42	40.737	43/(2*3*7)		Large 43-limit fifth tone
44/43	39.800	(22*11)/43		Small 43-limit fifth tone
86/85	20.249	(2*43)/(5*17)
87/86	20.014	(3*29)/(2*43)
129/128	13.473	(3*43)/27		43rd-partial chroma
47-limit (incomplete)
47/46	37.232	47/(2*23)
48/47	36.448	(24*3)/47
94/93	18.516	(2*47)/(3*31)
95/94	18.320	(5*19)/(2*47)
53-limit (incomplete)
53/52	32.977	53/(22*13)
54/53	32.360	(2*33)/53
59-limit (incomplete)
59/58	29.594	59/(2*29)
60/59	29.097	(22*3*5)/59
61-limit (incomplete)
61/60	28.616	61/(22*3*5)
62/61	28.151	(2*31)/61
67-limit (incomplete)
67/66	26.034	67/(2*3*11)
68/67	25.648	(22*17)/67
71-limit (incomplete)
71/70	24.557	71/(2*5*7)
72/71	24.213	(23*32)/71
73-limit (incomplete)
73/72	23.879	73/(23*32)
74/73	23.555	(2*37)/73
79-limit (incomplete)
79/78	22.054	79/(2*3*13)
80/79	21.777	(24*5)/79
83-limit (incomplete)
83/82	20.985	83/(2*41)
84/83	20.734	(22*3*7)/83
89-limit (incomplete)
89/88	19.562	89/(23*11)
90/89	19.344	(2*32*5)/89
97-limit (incomplete)
97/96	17.940	97/(25*3)
98/97	17.756	(2*72)/97
101-limit (incomplete)
101/100	17.226	101/(22*52)
102/101	17.057	(2*3*17)/101
7777/7776	0.223	7*11*101/(25*35)