List of pitch intervals

Below is a list of intervals expressible in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals.

For commonly encountered harmonic or melodic intervals between pairs of notes in contemporary Western music theory, without consideration of the way in which they are tuned, see Interval (music) § Main intervals.

Terminology

The prime limit henceforth referred to simply as the limit, is the largest prime number occurring in the factorizations of the numerator and denominator of the frequency ratio describing a rational interval. For instance, the limit of the just perfect fourth (4:3) is 3, but the just minor tone (10:9) has a limit of 5, because 10 can be factored into 2 × 5 (and 9 into 3 × 3). There exists another type of limit, the odd limit, a concept used by Harry Partch (bigger of odd numbers obtained after dividing numerator and denominator by highest possible powers of 2), but it is not used here. The term "limit" was devised by Partch.
By definition, every interval in a given limit can also be part of a limit of higher order. For instance, a 3-limit unit can also be part of a 5-limit tuning and so on. By sorting the limit columns in the table below, all intervals of a given limit can be brought together (sort backwards by clicking the button twice).
Pythagorean tuning means 3-limit intonation—a ratio of numbers with prime factors no higher than three.
Just intonation means 5-limit intonation—a ratio of numbers with prime factors no higher than five.
Septimal, undecimal, tridecimal, and septendecimal mean, respectively, 7, 11, 13, and 17-limit intonation.
Meantone refers to meantone temperament, where the whole tone is the mean of the major third. In general, a meantone is constructed in the same way as Pythagorean tuning, as a stack of fifths: the tone is reached after two fifths, the major third after four, so that as all fifths are the same, the tone is the mean of the third. In a meantone temperament, each fifth is narrowed ("tempered") by the same small amount. The most common of meantone temperaments is the quarter-comma meantone, in which each fifth is tempered by 1⁄4 of the syntonic comma, so that after four steps the major third (as C-G-D-A-E) is a full syntonic comma lower than the Pythagorean one. The extremes of the meantone systems encountered in historical practice are the Pythagorean tuning, where the whole tone corresponds to 9:8, i.e. ⁠(3:2)²/2⁠, the mean of the major third ⁠(3:2)⁴/4⁠, and the fifth (3:2) is not tempered; and the 1⁄3-comma meantone, where the fifth is tempered to the extent that three ascending fifths produce a pure minor third. (See meantone temperaments).
Equal-tempered refers to X-tone equal temperament with intervals corresponding to X divisions per octave.
Tempered intervals however cannot be expressed in terms of prime limits and, unless exceptions, are not found in the table below.
The table can also be sorted by frequency ratio, by cents, or alphabetically.
Superparticular ratios are intervals that can be expressed as the ratio of two consecutive integers.

List

Column	Legend
TET	X-tone equal temperament (12-tet, etc.).
Limit	3-limit intonation, or Pythagorean.
	5-limit "just" intonation, or just.
	7-limit intonation, or septimal.
	11-limit intonation, or undecimal.
	13-limit intonation, or tridecimal.
	17-limit intonation, or septendecimal.
	19-limit intonation, or novendecimal.
	Higher limits.
M	Meantone temperament or tuning.
S	Superparticular ratio (no separate color code).

List of musical intervals
Cents	Note (from C)	Freq. ratio	Prime factors	Interval name	TET	Limit	M	S
0.00	C	1 : 1	1 : 1	play^ⓘ Unison, monophony, perfect prime/first, tonic, or fundamental	1, 12	3	M
0.03		65537 : 65536	65537 : 2¹⁶	play^ⓘ Sixty-five-thousand-five-hundred-thirty-seventh harmonic		65537		S
0.40	C♯−	4375 : 4374	5⁴×7 : 2×3⁷	play^ⓘ Ragisma		7		S
0.72	E+	2401 : 2400	7⁴ : 2⁵×3×5²	play^ⓘ Breedsma		7		S
1.00		2^1/1200	2^1/1200	play^ⓘ Cent	1200
1.20		2^1/1000	2^1/1000	play^ⓘ Millioctave	1000
1.95	B♯++	32805 : 32768	3⁸×5 : 2¹⁵	play^ⓘ Schisma		5
1.96		3:2÷(2^7/12)	3 : 2^19/12	Grad, Werckmeister
3.99		10^1/1000	2^1/1000×5^1/1000	play^ⓘ Savart or eptaméride	301.03
7.71	B♯	225 : 224	3²×5² : 2⁵×7	play^ⓘ Septimal kleisma, marvel comma		7		S
8.11	B−	15625 : 15552	5⁶ : 2⁶×3⁵	play^ⓘ Kleisma or semicomma majeur		5
10.06	A++	2109375 : 2097152	3³×5⁷ : 2²¹	play^ⓘ Semicomma, Fokker's comma		5
10.85	C	160 : 159	2⁵×5 : 3×53	play^ⓘ Difference between 5:3 & 53:32		53		S
11.98	C	145 : 144	5×29 : 2⁴×3²	play^ⓘ Difference between 29:16 & 9:5		29		S
12.50		2^1/96	2^1/96	play^ⓘ Sixteenth tone	96
13.07	B−	1728 : 1715	2⁶×3³ : 5×7³	play^ⓘ Orwell comma		7
13.47	C	129 : 128	3×43 : 2⁷	play^ⓘ Hundred-twenty-ninth harmonic		43		S
13.79	D	126 : 125	2×3²×7 : 5³	play^ⓘ Small septimal semicomma, small septimal comma, starling comma		7		S
14.37	C♭↑↑−	121 : 120	11² : 2³×3×5	play^ⓘ Undecimal seconds comma		11		S
16.67	C↑	2^1/72	2^1/72	play^ⓘ 1 step in 72 equal temperament	72
18.13	C	96 : 95	2⁵×3 : 5×19	play^ⓘ Difference between 19:16 & 6:5		19		S
19.55	D--	2048 : 2025	2¹¹ : 3⁴×5²	play^ⓘ Diaschisma, minor comma		5
21.51	C+	81 : 80	3⁴ : 2⁴×5	play^ⓘ Syntonic comma, major comma, komma, chromatic diesis, or comma of Didymus		5		S
22.64		2^1/53	2^1/53	play^ⓘ Holdrian comma, Holder's comma, 1 step in 53 equal temperament	53
23.46	B♯+++	531441 : 524288	3¹² : 2¹⁹	play^ⓘ Pythagorean comma, ditonic comma, Pythagorean augmented seventh		3
25.00		2^1/48	2^1/48	play^ⓘ Eighth tone	48
26.84	C	65 : 64	5×13 : 2⁶	play^ⓘ Sixty-fifth harmonic, 13th-partial chroma		13		S
27.26	C−	64 : 63	2⁶ : 3²×7	play^ⓘ Septimal comma, Archytas' comma, 63rd subharmonic		7		S
29.27		2^1/41	2^1/41	play^ⓘ 1 step in 41 equal temperament	41
31.19	D♭↓	56 : 55	2³×7 : 5×11	play^ⓘ Undecimal diesis, Ptolemy's enharmonic: difference between (11 : 8) and (7 : 5) tritone		11		S
33.33	C/D♭	2^1/36	2^1/36	play^ⓘ Sixth tone	36, 72
34.28	C	51 : 50	3×17 : 2×5²	play^ⓘ Difference between 17:16 & 25:24		17		S
34.98	B♯-	50 : 49	2×5² : 7²	play^ⓘ Septimal sixth tone or jubilisma, Erlich's decatonic comma or tritonic diesis		7		S
35.70	D♭	49 : 48	7² : 2⁴×3	play^ⓘ Septimal diesis, slendro diesis or septimal 1/6-tone		7		S
38.05	C	46 : 45	2×23 : 3²×5	play^ⓘ Inferior quarter tone, difference between 23:16 & 45:32		23		S
38.71		2^1/31	2^1/31	play^ⓘ 1 step in 31 equal temperament or Normal Diesis	31
38.91	C↓♯+	45 : 44	3²×5 : 4×11	play^ⓘ Undecimal diesis or undecimal fifth tone		11		S
40.00		2^1/30	2^1/30	play^ⓘ Fifth tone	30
41.06	D−	128 : 125	2⁷ : 5³	play^ⓘ Enharmonic diesis or 5-limit limma, minor diesis, diminished second, minor diesis or diesis, 125th subharmonic		5
41.72	D♭	42 : 41	2×3×7 : 41	play^ⓘ Lesser 41-limit fifth tone		41		S
42.75	C	41 : 40	41 : 2³×5	play^ⓘ Greater 41-limit fifth tone		41		S
43.83	C♯	40 : 39	2³×5 : 3×13	play^ⓘ Tridecimal fifth tone		13		S
44.97	C	39 : 38	3×13 : 2×19	play^ⓘ Superior quarter-tone, novendecimal fifth tone		19		S
46.17	D-	38 : 37	2×19 : 37	play^ⓘ Lesser 37-limit quarter tone		37		S
47.43	C♯	37 : 36	37 : 2²×3²	play^ⓘ Greater 37-limit quarter tone		37		S
48.77	C	36 : 35	2²×3² : 5×7	play^ⓘ Septimal quarter tone, septimal diesis, septimal chroma, superior quarter tone		7		S
49.98		246 : 239	3×41 : 239	play^ⓘ Just quarter tone		239
50.00	C/D	2^1/24	2^1/24	play^ⓘ Equal-tempered quarter tone	24
50.18	D♭	35 : 34	5×7 : 2×17	play^ⓘ ET quarter-tone approximation, lesser 17-limit quarter tone		17		S
50.72	B♯++	59049 : 57344	3¹⁰ : 2¹³×7	play^ⓘ Harrison's comma (10 P5s – 1 H7)		7
51.68	C↓♯	34 : 33	2×17 : 3×11	play^ⓘ Greater 17-limit quarter tone		17		S
53.27	C↑	33 : 32	3×11 : 2⁵	play^ⓘ Thirty-third harmonic, undecimal comma, undecimal quarter tone		11		S
54.96	D♭-	32 : 31	2⁵ : 31	play^ⓘ Inferior quarter-tone, thirty-first subharmonic		31		S
56.55	B♯+	529 : 512	23² : 2⁹	play^ⓘ Five-hundred-twenty-ninth harmonic		23
56.77	C	31 : 30	31 : 2×3×5	play^ⓘ Greater quarter-tone, difference between 31:16 & 15:8		31		S
58.69	C♯	30 : 29	2×3×5 : 29	play^ⓘ Lesser 29-limit quarter tone		29		S
60.75	C	29 : 28	29 : 2²×7	play^ⓘ Greater 29-limit quarter tone		29		S
62.96	D♭-	28 : 27	2²×7 : 3³	play^ⓘ Septimal minor second, small minor second, inferior quarter tone		7		S
63.81		(3 : 2)^1/11	3^1/11 : 2^1/11	play^ⓘ Beta scale step	18.80
65.34	C♯+	27 : 26	3³ : 2×13	play^ⓘ Chromatic diesis, tridecimal comma		13		S
66.34	D♭	133 : 128	7×19 : 2⁷	play^ⓘ One-hundred-thirty-third harmonic		19
66.67	C↑/C♯	2^1/18	2^1/18	play^ⓘ Third tone	18, 36, 72
67.90	D-	26 : 25	2×13 : 5²	play^ⓘ Tridecimal third tone, third tone		13		S
70.67	C♯	25 : 24	5² : 2³×3	play^ⓘ Just chromatic semitone or minor chroma, lesser chromatic semitone, small (just) semitone or minor second, minor chromatic semitone, or minor semitone, 2⁄7-comma meantone chromatic semitone, augmented unison		5		S
73.68	D♭-	24 : 23	2³×3 : 23	play^ⓘ Lesser 23-limit semitone		23		S
75.00		2^1/16	2^3/48	play^ⓘ 1 step in 16 equal temperament, 3 steps in 48	16, 48
76.96	C↓♯+	23 : 22	23 : 2×11	play^ⓘ Greater 23-limit semitone		23		S
78.00		(3 : 2)^1/9	3^1/9 : 2^1/9	play^ⓘ Alpha scale step	15.39
79.31		67 : 64	67 : 2⁶	play^ⓘ Sixty-seventh harmonic		67
80.54	C↑-	22 : 21	2×11 : 3×7	play^ⓘ Hard semitone, two-fifth tone small semitone		11		S
84.47	D♭	21 : 20	3×7 : 2²×5	play^ⓘ Septimal chromatic semitone, minor semitone		7		S
88.80	C♯	20 : 19	2²×5 : 19	play^ⓘ Novendecimal augmented unison		19		S
90.22	D♭−−	256 : 243	2⁸ : 3⁵	play^ⓘ Pythagorean minor second or limma, Pythagorean diatonic semitone, Low Semitone		3
92.18	C♯+	135 : 128	3³×5 : 2⁷	play^ⓘ Greater chromatic semitone, chromatic semitone, semitone medius, major chroma or major limma, small limma, major chromatic semitone, limma ascendant		5
93.60	D♭-	19 : 18	19 : 2×3²	Novendecimal minor secondplay^ⓘ		19		S
97.36	D↓↓	128 : 121	2⁷ : 11²	play^ⓘ 121st subharmonic, undecimal minor second		11
98.95	D♭	18 : 17	2×3² : 17	play^ⓘ Just minor semitone, Arabic lute index finger		17		S
100.00	C♯/D♭	2^1/12	2^1/12	play^ⓘ Equal-tempered minor second or semitone	12		M
104.96	C♯	17 : 16	17 : 2⁴	play^ⓘ Minor diatonic semitone, just major semitone, overtone semitone, 17th harmonic, limma^{[citation needed]}		17		S
111.45		²⁵√5	(5 : 1)^1/25	play^ⓘ Studie II interval (compound just major third, 5:1, divided into 25 equal parts)	10.77
111.73	D♭-	16 : 15	2⁴ : 3×5	play^ⓘ Just minor second, just diatonic semitone, large just semitone or major second, major semitone, limma, minor diatonic semitone, diatonic second semitone, diatonic semitone, 1⁄6-comma meantone minor second		5		S
113.69	C♯++	2187 : 2048	3⁷ : 2¹¹	play^ⓘ Apotome or Pythagorean major semitone, Pythagorean augmented unison, Pythagorean chromatic semitone, or Pythagorean apotome		3
116.72		(18 : 5)^1/19	2^1/19×3^2/19 : 5^1/19	play^ⓘ Secor	10.28
119.44	C♯	15 : 14	3×5 : 2×7	play^ⓘ Septimal diatonic semitone, major diatonic semitone, Cowell semitone		7		S
125.00		2^5/48	2^5/48	play^ⓘ 5 steps in 48 equal temperament	48
128.30	D	14 : 13	2×7 : 13	play^ⓘ Lesser tridecimal 2/3-tone		13		S
130.23	C♯+	69 : 64	3×23 : 2⁶	play^ⓘ Sixty-ninth harmonic		23
133.24	D♭	27 : 25	3³ : 5²	play^ⓘ Semitone maximus, minor second, large limma or Bohlen-Pierce small semitone, high semitone, alternate Renaissance half-step, large limma, acute minor second^{[citation needed]}		5
133.33	C♯/D♭	2^1/9	2^2/18	play^ⓘ Two-third tone	9, 18, 36, 72
138.57	D♭-	13 : 12	13 : 2²×3	play^ⓘ Greater tridecimal 2/3-tone, Three-quarter tone		13		S
150.00	C/D	2^3/24	2^1/8	play^ⓘ Equal-tempered neutral second	8, 24
150.64	D↓	12 : 11	2²×3 : 11	play^ⓘ 3⁄4 tone or Undecimal neutral second, trumpet three-quarter tone, middle finger [between frets]		11		S
155.14	D	35 : 32	5×7 : 2⁵	play^ⓘ Thirty-fifth harmonic		7
160.90	D−−	800 : 729	2⁵×5² : 3⁶	play^ⓘ Grave whole tone, neutral second, grave major second^{[citation needed]}		5
165.00	D↑♭−	11 : 10	11 : 2×5	play^ⓘ Greater undecimal minor/major/neutral second, 4/5-tone or Ptolemy's second		11		S
171.43		2^1/7	2^1/7	play^ⓘ 1 step in 7 equal temperament	7
175.00		2^7/48	2^7/48	play^ⓘ 7 steps in 48 equal temperament	48
179.70		71 : 64	71 : 2⁶	play^ⓘ Seventy-first harmonic		71
180.45	E−−−	65536 : 59049	2¹⁶ : 3¹⁰	play^ⓘ Pythagorean diminished third, Pythagorean minor tone		3
182.40	D−	10 : 9	2×5 : 3²	play^ⓘ Small just whole tone or major second, minor whole tone, lesser whole tone, minor tone, minor second, half-comma meantone major second		5		S
200.00	D	2^2/12	2^1/6	play^ⓘ Equal-tempered major second	6, 12		M
203.91	D	9 : 8	3² : 2³	play^ⓘ Pythagorean major second, Large just whole tone or major second (sesquioctavan), tonus, major whole tone, greater whole tone, major tone		3		S
215.89	D	145 : 128	5×29 : 2⁷	play^ⓘ Hundred-forty-fifth harmonic		29
223.46	E−	256 : 225	2⁸ : 3²×5²	play^ⓘ Just diminished third, 225th subharmonic		5
225.00		2^3/16	2^9/48	play^ⓘ 9 steps in 48 equal temperament	16, 48
227.79		73 : 64	73 : 2⁶	play^ⓘ Seventy-third harmonic		73
231.17	D−	8 : 7	2³ : 7	play^ⓘ Septimal major second, septimal whole tone		7		S
240.00		2^1/5	2^1/5	play^ⓘ 1 step in 5 equal temperament	5
247.74	D♯	15 : 13	3×5 : 13	play^ⓘ Tridecimal 5⁄4 tone		13
250.00	D/E	2^5/24	2^5/24	play^ⓘ 5 steps in 24 equal temperament	24
251.34	D♯	37 : 32	37 : 2⁵	play^ⓘ Thirty-seventh harmonic		37
253.08	D♯−	125 : 108	5³ : 2²×3³	play^ⓘ Semi-augmented whole tone, semi-augmented second^{[citation needed]}		5
262.37	E↓♭	64 : 55	2⁶ : 5×11	play^ⓘ 55th subharmonic		11
266.87	E♭	7 : 6	7 : 2×3	play^ⓘ Septimal minor third or Sub minor third		7		S
268.80	D	299 : 256	13×23 : 2⁸	play^ⓘ Two-hundred-ninety-ninth harmonic		23
274.58	D♯	75 : 64	3×5² : 2⁶	play^ⓘ Just augmented second, Augmented tone, augmented second		5
275.00		2^11/48	2^11/48	play^ⓘ 11 steps in 48 equal temperament	48
289.21	E↓♭	13 : 11	13 : 11	play^ⓘ Tridecimal minor third		13
294.13	E♭−	32 : 27	2⁵ : 3³	play^ⓘ Pythagorean minor third semiditone, or 27th subharmonic		3
297.51	E♭	19 : 16	19 : 2⁴	play^ⓘ 19th harmonic, 19-limit minor third, overtone minor third		19
300.00	D♯/E♭	2^3/12	2^1/4	play^ⓘ Equal-tempered minor third	4, 12		M
301.85	D♯-	25 : 21	5² : 3×7	play^ⓘ Quasi-equal-tempered minor third, 2nd 7-limit minor third, Bohlen-Pierce second		7
310.26		6:5÷(81:80)^1/4	2² : 5^3/4	play^ⓘ Quarter-comma meantone minor third			M
311.98		(3 : 2)^4/9	3^4/9 : 2^4/9	play^ⓘ Alpha scale minor third	15.39
315.64	E♭	6 : 5	2×3 : 5	play^ⓘ Just minor third, minor third, 1⁄3-comma meantone minor third		5	M	S
317.60	D♯++	19683 : 16384	3⁹ : 2¹⁴	play^ⓘ Pythagorean augmented second		3
320.14	E♭↑	77 : 64	7×11 : 2⁶	play^ⓘ Seventy-seventh harmonic		11
325.00		2^13/48	2^13/48	play^ⓘ 13 steps in 48 equal temperament	48
336.13	D♯-	17 : 14	17 : 2×7	play^ⓘ Superminor third		17
337.15	E♭+	243 : 200	3⁵ : 2³×5²	play^ⓘ Acute minor third		5
342.48	E♭	39 : 32	3×13 : 2⁵	play^ⓘ Thirty-ninth harmonic		13
342.86		2^2/7	2^2/7	play^ⓘ 2 steps in 7 equal temperament	7
342.91	E♭-	128 : 105	2⁷ : 3×5×7	play^ⓘ 105th subharmonic, septimal neutral third		7
347.41	E↑♭−	11 : 9	11 : 3²	play^ⓘ Undecimal neutral third		11
350.00	D/E	2^7/24	2^7/24	play^ⓘ Equal-tempered neutral third	24
354.55	E↓+	27 : 22	3³ : 2×11	play^ⓘ Zalzal's wosta 12:11 X 9:8		11
359.47	E	16 : 13	2⁴ : 13	play^ⓘ Tridecimal neutral third		13
364.54		79 : 64	79 : 2⁶	play^ⓘ Seventy-ninth harmonic		79
364.81	E−	100 : 81	2²×5² : 3⁴	play^ⓘ Grave major third		5
375.00		2^5/16	2^15/48	play^ⓘ 15 steps in 48 equal temperament	16, 48
384.36	F♭−−	8192 : 6561	2¹³ : 3⁸	play^ⓘ Pythagorean diminished fourth, Pythagorean 'schismatic' third		3
386.31	E	5 : 4	5 : 2²	play^ⓘ Just major third, major third, quarter-comma meantone major third		5	M	S
397.10	E+	161 : 128	7×23 : 2⁷	play^ⓘ One-hundred-sixty-first harmonic		23
400.00	E	2^4/12	2^1/3	play^ⓘ Equal-tempered major third	3, 12		M
402.47	E	323 : 256	17×19 : 2⁸	play^ⓘ Three-hundred-twenty-third harmonic		19
407.82	E+	81 : 64	3⁴ : 2⁶	play^ⓘ Pythagorean major third, ditone		3
417.51	F↓+	14 : 11	2×7 : 11	play^ⓘ Undecimal diminished fourth or major third		11
425.00		2^17/48	2^17/48	play^ⓘ 17 steps in 48 equal temperament	48
427.37	F♭	32 : 25	2⁵ : 5²	play^ⓘ Just diminished fourth, diminished fourth, 25th subharmonic		5
429.06	E	41 : 32	41 : 2⁵	play^ⓘ Forty-first harmonic		41
435.08	E	9 : 7	3² : 7	play^ⓘ Septimal major third, Bohlen-Pierce third, Super major Third		7
444.77	F↓	128 : 99	2⁷ : 3²×11	play^ⓘ 99th subharmonic		11
450.00	E/F	2^9/24	2^9/24	play^ⓘ 9 steps in 24 equal temperament	8, 24
450.05		83 : 64	83 : 2⁶	play^ⓘ Eighty-third harmonic		83
454.21	F♭	13 : 10	13 : 2×5	play^ⓘ Tridecimal major third or diminished fourth		13
456.99	E♯	125 : 96	5³ : 2⁵×3	play^ⓘ Just augmented third, augmented third		5
462.35	E-	64 : 49	2⁶ : 7²	play^ⓘ 49th subharmonic		7
470.78	F+	21 : 16	3×7 : 2⁴	play^ⓘ Twenty-first harmonic, narrow fourth, septimal fourth, wide augmented third,^{[citation needed]} H7 on G		7
475.00		2^19/48	2^19/48	play^ⓘ 19 steps in 48 equal temperament	48
478.49	E♯+	675 : 512	3³×5² : 2⁹	play^ⓘ Six-hundred-seventy-fifth harmonic, wide augmented third		5
480.00		2^2/5	2^2/5	play^ⓘ 2 steps in 5 equal temperament	5
491.27	E♯	85 : 64	5×17 : 2⁶	play^ⓘ Eighty-fifth harmonic		17
498.04	F	4 : 3	2² : 3	play^ⓘ Perfect fourth, Pythagorean perfect fourth, Just perfect fourth or diatessaron		3		S
500.00	F	2^5/12	2^5/12	play^ⓘ Equal-tempered perfect fourth	12		M
501.42	F+	171 : 128	3²×19 : 2⁷	play^ⓘ One-hundred-seventy-first harmonic		19
510.51		(3 : 2)^8/11	3^8/11 : 2^8/11	play^ⓘ Beta scale perfect fourth	18.80
511.52	F	43 : 32	43 : 2⁵	play^ⓘ Forty-third harmonic		43
514.29		2^3/7	2^3/7	play^ⓘ 3 steps in 7 equal temperament	7
519.55	F+	27 : 20	3³ : 2²×5	play^ⓘ 5-limit wolf fourth, acute fourth, imperfect fourth		5
521.51	E♯+++	177147 : 131072	3¹¹ : 2¹⁷	play^ⓘ Pythagorean augmented third (F+ (pitch))		3
525.00		2^7/16	2^21/48	play^ⓘ 21 steps in 48 equal temperament	16, 48
531.53	F+	87 : 64	3×29 : 2⁶	play^ⓘ Eighty-seventh harmonic		29
536.95	F↓♯+	15 : 11	3×5 : 11	play^ⓘ Undecimal augmented fourth		11
550.00	F/G	2^11/24	2^11/24	play^ⓘ 11 steps in 24 equal temperament	24
551.32	F↑	11 : 8	11 : 2³	play^ⓘ eleventh harmonic, undecimal tritone, lesser undecimal tritone, undecimal semi-augmented fourth		11
563.38	F♯+	18 : 13	2×9 : 13	play^ⓘ Tridecimal augmented fourth		13
568.72	F♯	25 : 18	5² : 2×3²	play^ⓘ Just augmented fourth		5
570.88		89 : 64	89 : 2⁶	play^ⓘ Eighty-ninth harmonic		89
575.00		2^23/48	2^23/48	play^ⓘ 23 steps in 48 equal temperament	48
582.51	G♭	7 : 5	7 : 5	play^ⓘ Lesser septimal tritone, septimal tritone Huygens' tritone or Bohlen-Pierce fourth, septimal fifth, septimal diminished fifth		7
588.27	G♭−−	1024 : 729	2¹⁰ : 3⁶	play^ⓘ Pythagorean diminished fifth, low Pythagorean tritone		3
590.22	F♯+	45 : 32	3²×5 : 2⁵	play^ⓘ Just augmented fourth, just tritone, tritone, diatonic tritone, 'augmented' or 'false' fourth, high 5-limit tritone, 1⁄6-comma meantone augmented fourth		5
595.03	G♭	361 : 256	19² : 2⁸	play^ⓘ Three-hundred-sixty-first harmonic		19
600.00	F♯/G♭	2^6/12	2^1/2=√2	play^ⓘ Equal-tempered tritone	2, 12		M
609.35	G♭	91 : 64	7×13 : 2⁶	play^ⓘ Ninety-first harmonic		13
609.78	G♭−	64 : 45	2⁶ : 3²×5	play^ⓘ Just tritone, 2nd tritone, 'false' fifth, diminished fifth, low 5-limit tritone, 45th subharmonic		5
611.73	F♯++	729 : 512	3⁶ : 2⁹	play^ⓘ Pythagorean tritone, Pythagorean augmented fourth, high Pythagorean tritone		3
617.49	F♯	10 : 7	2×5 : 7	play^ⓘ Greater septimal tritone, septimal tritone, Euler's tritone		7
625.00		2^25/48	2^25/48	play^ⓘ 25 steps in 48 equal temperament	48
628.27	F♯+	23 : 16	23 : 2⁴	play^ⓘ Twenty-third harmonic, classic diminished fifth^{[citation needed]}		23
631.28	G♭	36 : 25	2²×3² : 5²	play^ⓘ Just diminished fifth		5
646.99	F♯+	93 : 64	3×31 : 2⁶	play^ⓘ Ninety-third harmonic		31
648.68	G↓	16 : 11	2⁴ : 11	play^ⓘ Undecimal semi-diminished fifth		11
650.00	F/G	2^13/24	2^13/24	play^ⓘ 13 steps in 24 equal temperament	24
665.51	G	47 : 32	47 : 2⁵	play^ⓘ Forty-seventh harmonic		47
675.00		2^9/16	2^27/48	play^ⓘ 27 steps in 48 equal temperament	16, 48
678.49	A−−−	262144 : 177147	2¹⁸ : 3¹¹	play^ⓘ Pythagorean diminished sixth, Pythagorean wolf fifth		3
680.45	G−	40 : 27	2³×5 : 3³	play^ⓘ 5-limit wolf fifth, or diminished sixth, grave fifth, imperfect fifth,		5
683.83	G	95 : 64	5×19 : 2⁶	play^ⓘ Ninety-fifth harmonic		19
684.82	E++	12167 : 8192	23³ : 2¹³	play^ⓘ 12167th harmonic		23
685.71		2^4/7 : 1		play^ⓘ 4 steps in 7 equal temperament	7
691.20		3:2÷(81:80)^1/2	2×5^1/2 : 3	play^ⓘ Half-comma meantone perfect fifth			M
694.79		3:2÷(81:80)^1/3	2^1/3×5^1/3 : 3^1/3	play^ⓘ 1⁄3-comma meantone perfect fifth			M
695.81		3:2÷(81:80)^2/7	2^1/7×5^2/7 : 3^1/7	play^ⓘ 2⁄7-comma meantone perfect fifth			M
696.58		3:2÷(81:80)^1/4	5^1/4	play^ⓘ Quarter-comma meantone perfect fifth			M
697.65		3:2÷(81:80)^1/5	3^1/5×5^1/5 : 2^1/5	play^ⓘ 1⁄5-comma meantone perfect fifth			M
698.37		3:2÷(81:80)^1/6	3^1/3×5^1/6 : 2^1/3	play^ⓘ 1⁄6-comma meantone perfect fifth			M
700.00	G	2^7/12	2^7/12	play^ⓘ Equal-tempered perfect fifth	12		M
701.89		2^31/53	2^31/53	play^ⓘ 53-TET perfect fifth	53
701.96	G	3 : 2	3 : 2	play^ⓘ Perfect fifth, Pythagorean perfect fifth, Just perfect fifth or diapente, fifth, Just fifth		3		S
702.44		2^24/41	2^24/41	play^ⓘ 41-TET perfect fifth	41
703.45		2^17/29	2^17/29	play^ⓘ 29-TET perfect fifth	29
719.90		97 : 64	97 : 2⁶	play^ⓘ Ninety-seventh harmonic		97
720.00		2^3/5 : 1		play^ⓘ 3 steps in 5 equal temperament	5
721.51	A−	1024 : 675	2¹⁰ : 3³×5²	play^ⓘ Narrow diminished sixth		5
725.00		2^29/48	2^29/48	play^ⓘ 29 steps in 48 equal temperament	48
729.22	G-	32 : 21	2⁴ : 3×7	play^ⓘ 21st subharmonic, septimal diminished sixth		7
733.23	F+	391 : 256	17×23 : 2⁸	play^ⓘ Three-hundred-ninety-first harmonic		23
737.65	A♭+	49 : 32	7×7 : 2⁵	play^ⓘ Forty-ninth harmonic		7
743.01	A	192 : 125	2⁶×3 : 5³	play^ⓘ Classic diminished sixth		5
750.00	G/A	2^15/24	2^15/24	play^ⓘ 15 steps in 24 equal temperament	8, 24
755.23	G↑	99 : 64	3²×11 : 2⁶	play^ⓘ Ninety-ninth harmonic		11
764.92	A♭	14 : 9	2×7 : 3²	play^ⓘ Septimal minor sixth		7
772.63	G♯	25 : 16	5² : 2⁴	play^ⓘ Just augmented fifth		5
775.00		2^31/48	2^31/48	play^ⓘ 31 steps in 48 equal temperament	48
781.79		$π$ : 2		play^ⓘ Wallis product
782.49	G↑-	11 : 7	11 : 7	play^ⓘ Undecimal minor sixth, undecimal augmented fifth, Lucas numbers		11
789.85		101 : 64	101 : 2⁶	play^ⓘ Hundred-first harmonic		101
792.18	A♭−	128 : 81	2⁷ : 3⁴	play^ⓘ Pythagorean minor sixth, 81st subharmonic		3
798.40	A♭+	203 : 128	7×29 : 2⁷	play^ⓘ Two-hundred-third harmonic		29
800.00	G♯/A♭	2^8/12	2^2/3	play^ⓘ Equal-tempered minor sixth	3, 12		M
806.91	G♯	51 : 32	3×17 : 2⁵	play^ⓘ Fifty-first harmonic		17
813.69	A♭	8 : 5	2³ : 5	play^ⓘ Just minor sixth		5
815.64	G♯++	6561 : 4096	3⁸ : 2¹²	play^ⓘ Pythagorean augmented fifth, Pythagorean 'schismatic' sixth		3
823.80		103 : 64	103 : 2⁶	play^ⓘ Hundred-third harmonic		103
825.00		2^11/16	2^33/48	play^ⓘ 33 steps in 48 equal temperament	16, 48
832.18	G♯+	207 : 128	3²×23 : 2⁷	play^ⓘ Two-hundred-seventh harmonic		23
833.09		(5^1/2+1)/2	$φ$ : 1	play^ⓘ Golden ratio (833 cents scale)
835.19	A♭+	81 : 50	3⁴ : 2×5²	play^ⓘ Acute minor sixth		5
840.53	A♭	13 : 8	13 : 2³	play^ⓘ Tridecimal neutral sixth,Fibonacci numbers, overtone sixth, thirteenth harmonic		13
848.83	A♭↑	209 : 128	11×19 : 2⁷	play^ⓘ Two-hundred-ninth harmonic		19
850.00	G/A	2^17/24	2^17/24	play^ⓘ Equal-tempered neutral sixth	24
852.59	A↓+	18 : 11	2×3² : 11	play^ⓘ Undecimal neutral sixth, Zalzal's neutral sixth		11
857.09	A+	105 : 64	3×5×7 : 2⁶	play^ⓘ Hundred-fifth harmonic		7
857.14		2^5/7	2^5/7	play^ⓘ 5 steps in 7 equal temperament	7
862.85	A−	400 : 243	2⁴×5² : 3⁵	play^ⓘ Grave major sixth		5
873.50	A	53 : 32	53 : 2⁵	play^ⓘ Fifty-third harmonic		53
875.00		2^35/48	2^35/48	play^ⓘ 35 steps in 48 equal temperament	48
879.86	A↓	128 : 77	2⁷ : 7×11	play^ⓘ 77th subharmonic		11
882.40	B−−−	32768 : 19683	2¹⁵ : 3⁹	play^ⓘ Pythagorean diminished seventh		3
884.36	A	5 : 3	5 : 3	play^ⓘ Just major sixth, Bohlen-Pierce sixth, 1⁄3-comma meantone major sixth		5	M
889.76		107 : 64	107 : 2⁶	play^ⓘ Hundred-seventh harmonic		107
892.54	B	6859 : 4096	19³ : 2¹²	play^ⓘ 6859th harmonic		19
900.00	A	2^9/12	2^3/4	play^ⓘ Equal-tempered major sixth	4, 12		M
902.49	A	32 : 19	2⁵ : 19	play^ⓘ 19th subharmonic		19
905.87	A+	27 : 16	3³ : 2⁴	play^ⓘ Pythagorean major sixth		3
921.82		109 : 64	109 : 2⁶	play^ⓘ Hundred-ninth harmonic		109
925.00		2^37/48	2^37/48	play^ⓘ 37 steps in 48 equal temperament	48
925.42	B−	128 : 75	2⁷ : 3×5²	play^ⓘ Just diminished seventh, diminished seventh, 75th subharmonic		5
925.79	A+	437 : 256	19×23 : 2⁸	play^ⓘ Four-hundred-thirty-seventh harmonic		23
933.13	A	12 : 7	2²×3 : 7	play^ⓘ Septimal major sixth		7
937.63	A↑	55 : 32	5×11 : 2⁵	play^ⓘ Fifty-fifth harmonic		11
950.00	A/B	2^19/24	2^19/24	play^ⓘ 19 steps in 24 equal temperament	24
953.30	A♯+	111 : 64	3×37 : 2⁶	play^ⓘ Hundred-eleventh harmonic		37
955.03	A♯	125 : 72	5³ : 2³×3²	play^ⓘ Just augmented sixth		5
957.21		(3 : 2)^15/11	3^15/11 : 2^15/11	play^ⓘ 15 steps in Beta scale	18.80
960.00		2^4/5	2^4/5	play^ⓘ 4 steps in 5 equal temperament	5
968.83	B♭	7 : 4	7 : 2²	play^ⓘ Septimal minor seventh, harmonic seventh, augmented sixth^{[citation needed]}		7
975.00		2^13/16	2^39/48	play^ⓘ 39 steps in 48 equal temperament	16, 48
976.54	A♯+	225 : 128	3²×5² : 2⁷	play^ⓘ Just augmented sixth		5
984.21		113 : 64	113 : 2⁶	play^ⓘ Hundred-thirteenth harmonic		113
996.09	B♭−	16 : 9	2⁴ : 3²	play^ⓘ Pythagorean minor seventh, Small just minor seventh, lesser minor seventh, just minor seventh, Pythagorean small minor seventh		3
999.47	B♭	57 : 32	3×19 : 2⁵	play^ⓘ Fifty-seventh harmonic		19
1000.00	A♯/B♭	2^10/12	2^5/6	play^ⓘ Equal-tempered minor seventh	6, 12		M
1014.59	A♯+	115 : 64	5×23 : 2⁶	play^ⓘ Hundred-fifteenth harmonic		23
1017.60	B♭	9 : 5	3² : 5	play^ⓘ Greater just minor seventh, large just minor seventh, Bohlen-Pierce seventh		5
1019.55	A♯+++	59049 : 32768	3¹⁰ : 2¹⁵	play^ⓘ Pythagorean augmented sixth		3
1025.00		2^41/48	2^41/48	play^ⓘ 41 steps in 48 equal temperament	48
1028.57		2^6/7	2^6/7	play^ⓘ 6 steps in 7 equal temperament	7
1029.58	B♭	29 : 16	29 : 2⁴	play^ⓘ Twenty-ninth harmonic, minor seventh^{[citation needed]}		29
1035.00	B↓	20 : 11	2²×5 : 11	play^ⓘ Lesser undecimal neutral seventh, large minor seventh		11
1039.10	B♭+	729 : 400	3⁶ : 2⁴×5²	play^ⓘ Acute minor seventh		5
1044.44	B♭	117 : 64	3²×13 : 2⁶	play^ⓘ Hundred-seventeenth harmonic		13
1044.86	B♭-	64 : 35	2⁶ : 5×7	play^ⓘ 35th subharmonic, septimal neutral seventh		7
1049.36	B↑♭−	11 : 6	11 : 2×3	play^ⓘ 21⁄4-tone or Undecimal neutral seventh, undecimal 'median' seventh		11
1050.00	A/B	2^21/24	2^7/8	play^ⓘ Equal-tempered neutral seventh	8, 24
1059.17		59 : 32	59 : 2⁵	play^ⓘ Fifty-ninth harmonic		59
1066.76	B−	50 : 27	2×5² : 3³	play^ⓘ Grave major seventh		5
1071.70	B♭-	13 : 7	13 : 7	play^ⓘ Tridecimal neutral seventh		13
1073.78	B	119 : 64	7×17 : 2⁶	play^ⓘ Hundred-nineteenth harmonic		17
1075.00		2^43/48	2^43/48	play^ⓘ 43 steps in 48 equal temperament	48
1086.31	C′♭−−	4096 : 2187	2¹² : 3⁷	play^ⓘ Pythagorean diminished octave		3
1088.27	B	15 : 8	3×5 : 2³	play^ⓘ Just major seventh, small just major seventh, 1⁄6-comma meantone major seventh		5
1095.04	C♭	32 : 17	2⁵ : 17	play^ⓘ 17th subharmonic		17
1100.00	B	2^11/12	2^11/12	play^ⓘ Equal-tempered major seventh	12		M
1102.64	B↑↑♭-	121 : 64	11² : 2⁶	play^ⓘ Hundred-twenty-first harmonic		11
1107.82	C′♭−	256 : 135	2⁸ : 3³×5	play^ⓘ Octave − major chroma, 135th subharmonic, narrow diminished octave^{[citation needed]}		5
1109.78	B+	243 : 128	3⁵ : 2⁷	play^ⓘ Pythagorean major seventh		3
1116.88		61 : 32	61 : 2⁵	play^ⓘ Sixty-first harmonic		61
1125.00		2^15/16	2^45/48	play^ⓘ 45 steps in 48 equal temperament	16, 48
1129.33	C′♭	48 : 25	2⁴×3 : 5²	play^ⓘ Classic diminished octave, large just major seventh		5
1131.02	B	123 : 64	3×41 : 2⁶	play^ⓘ Hundred-twenty-third harmonic		41
1137.04	B	27 : 14	3³ : 2×7	play^ⓘ Septimal major seventh		7
1138.04	C♭	247 : 128	13×19 : 2⁷	play^ⓘ Two-hundred-forty-seventh harmonic		19
1145.04	B	31 : 16	31 : 2⁴	play^ⓘ Thirty-first harmonic, augmented seventh^{[citation needed]}		31
1146.73	C↓	64 : 33	2⁶ : 3×11	play^ⓘ 33rd subharmonic		11
1150.00	B/C	2^23/24	2^23/24	play^ⓘ 23 steps in 24 equal temperament	24
1151.23	C	35 : 18	5×7 : 2×3²	play^ⓘ Septimal supermajor seventh, septimal quarter tone inverted		7
1158.94	B♯	125 : 64	5³ : 2⁶	play^ⓘ Just augmented seventh, 125th harmonic		5
1172.74	C+	63 : 32	3²×7 : 2⁵	play^ⓘ Sixty-third harmonic		7
1175.00		2^47/48	2^47/48	play^ⓘ 47 steps in 48 equal temperament	48
1178.49	C′−	160 : 81	2⁵×5 : 3⁴	play^ⓘ Octave − syntonic comma, semi-diminished octave^{[citation needed]}		5
1179.59	B↑	253 : 128	11×23 : 2⁷	play^ⓘ Two-hundred-fifty-third harmonic		23
1186.42		127 : 64	127 : 2⁶	play^ⓘ Hundred-twenty-seventh harmonic		127
1200.00	C′	2 : 1	2 : 1	play^ⓘ Octave, perfect eighth or diapason	1, 12	3	M	S

Notes

1 2 3 4 Maneri-Sims notation

References

1 2 Fox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1–2. (Abingdon, Oxfordshire, UK: Routledge): p. 13.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–137.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 "List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Partch, Harry (1979). Genesis of a Music. Hachette Books. pp. 68–69. ISBN 978-0-306-80106-8.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 "Anatomy of an Octave", Kyle Gann (1998). Gann leaves off "just" but includes "5-limit". He uses "median" for "neutral".
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Haluška, Ján (2003). The Mathematical Theory of Tone Systems, pp. xxv–xxix. ISBN 978-0-8247-4714-5.
↑ Ellis, Alexander J.; Hipkins, Alfred J. (1884). "Tonometrical Observations on Some Existing Non-Harmonic Musical Scales". Proceedings of the Royal Society of London. 37 (232–234): 368–385. doi:10.1098/rspl.1884.0041. JSTOR 114325. S2CID 122407786.
↑ "Logarithmic Interval Measures", Huygens-Fokker Foundation. Accessed 2015-06-06.
↑ "Orwell Temperaments", Xenharmony.org.
1 2 Partch 1979, p. 70
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Alexander John Ellis (March 1885). On the musical scales of various nations, p. 488. Journal of the Society of Arts, vol. XXXII, no. 1688
↑ William Smythe Babcock Mathews (1895). Pronouncing Dictionary and Condensed Encyclopedia of Musical Terms, p. 13. ISBN 1-112-44188-3.
1 2 3 4 5 6 Anger, Joseph Humfrey (1912). A Treatise on Harmony, with Exercises, Volume 3, pp. xiv–xv. W. Tyrrell.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Hermann Ludwig F. von Helmholtz (Alexander John Ellis, trans.) (1875). "Additions by the translator", On the sensations of tone as a physiological basis for the theory of music, p. 644. [ISBN unspecified]
↑ A. R. Meuss (2004). Intervals, Scales, Tones and the Concert Pitch C. Temple Lodge Publishing. p. 15. ISBN 1902636465.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Paul, Oscar (1885). A Manual of Harmony for Use in Music-schools and Seminaries and for Self-instruction, p. 165. Theodore Baker, trans. G. Schirmer. Paul uses "natural" for "just".
1 2 "13th-harmonic", 31et.com.
↑ Brabner, John H. F. (1884). The National Encyclopaedia, vol. 13, p. 182. London. [ISBN unspecified]
↑ Sabat, Marc and von Schweinitz, Wolfgang (2004). "The Extended Helmholtz-Ellis JI Pitch Notation" [PDF], NewMusicBox. Accessed: 15 March 2014.
↑ "The Mathematical Structure of Pythagorean-like Tuning Systems". Math Intelligencer. 48: 58. December 2025. doi:10.1007/s00283-025-10478-y.
↑ Hermann L. F. von Helmholtz (2007). On the Sensations of Tone, p. 456. ISBN 978-1-60206-639-7.
↑ "Gallery of Just Intervals", Xenharmonic Wiki.

External links

"Names of seven-limit commas", XenHarmony.org. (Archived copy)
"List of Overtones", Xenharmonic Wiki.
"All Known Musical Intervals" (by Dale Pond), Svpvril.com."

[MSN-10] 1 2 3 4 Maneri-Sims notation

[Fox13-1] 1 2 Fox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1–2. (Abingdon, Oxfordshire, UK: Routledge): p. 13.

[Fonville-2] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–137.

[HF-3] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 "List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".

[Partch-4] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Partch, Harry (1979). Genesis of a Music. Hachette Books. pp. 68–69. ISBN 978-0-306-80106-8.

[Gann-5] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 "Anatomy of an Octave", Kyle Gann (1998). Gann leaves off "just" but includes "5-limit". He uses "median" for "neutral".

[Haluska-6] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Haluška, Ján (2003). The Mathematical Theory of Tone Systems, pp. xxv–xxix. ISBN 978-0-8247-4714-5.

[7] Ellis, Alexander J.; Hipkins, Alfred J. (1884). "Tonometrical Observations on Some Existing Non-Harmonic Musical Scales". Proceedings of the Royal Society of London. 37 (232–234): 368–385. doi:10.1098/rspl.1884.0041. JSTOR 114325. S2CID 122407786.

[8] "Logarithmic Interval Measures", Huygens-Fokker Foundation. Accessed 2015-06-06.

[9] "Orwell Temperaments", Xenharmony.org.

[Partch_70-11] 1 2 Partch 1979, p. 70

[Ellis-12] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Alexander John Ellis (March 1885). On the musical scales of various nations, p. 488. Journal of the Society of Arts, vol. XXXII, no. 1688

[13] William Smythe Babcock Mathews (1895). Pronouncing Dictionary and Condensed Encyclopedia of Musical Terms, p. 13. ISBN 1-112-44188-3.

[Anger-14] 1 2 3 4 5 6 Anger, Joseph Humfrey (1912). A Treatise on Harmony, with Exercises, Volume 3, pp. xiv–xv. W. Tyrrell.

[Helmholtz-15] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Hermann Ludwig F. von Helmholtz (Alexander John Ellis, trans.) (1875). "Additions by the translator", On the sensations of tone as a physiological basis for the theory of music, p. 644. [ISBN unspecified]

[Meuss-16] A. R. Meuss (2004). Intervals, Scales, Tones and the Concert Pitch C. Temple Lodge Publishing. p. 15. ISBN 1902636465.

[Paul-17] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Paul, Oscar (1885). A Manual of Harmony for Use in Music-schools and Seminaries and for Self-instruction, p. 165. Theodore Baker, trans. G. Schirmer. Paul uses "natural" for "just".

[31et.com-18] 1 2 "13th-harmonic", 31et.com.

[19] Brabner, John H. F. (1884). The National Encyclopaedia, vol. 13, p. 182. London. [ISBN unspecified]

[NMB-20] Sabat, Marc and von Schweinitz, Wolfgang (2004). "The Extended Helmholtz-Ellis JI Pitch Notation" [PDF], NewMusicBox. Accessed: 15 March 2014.

[21] "The Mathematical Structure of Pythagorean-like Tuning Systems". Math Intelligencer. 48: 58. December 2025. doi:10.1007/s00283-025-10478-y.

[22] Hermann L. F. von Helmholtz (2007). On the Sensations of Tone, p. 456. ISBN 978-1-60206-639-7.

[23] "Gallery of Just Intervals", Xenharmonic Wiki.

v t e Consonance and dissonance
Argument Avoid note Beating Cadence Chord Interval Musical note C F Nonchord tone Cambiata Changing tones Pedal point Preparation Resolution Spectra
Perfect Consonances	Perfect unison Perfect fourth Perfect fifth Perfect octave
Imperfect Consonances	Minor third Major third Minor sixth Major sixth
Dissonances	Minor second Major second Augmented fourth Diminished fifth Minor seventh Major seventh
List of musical intervals