List of pitch intervals

Last updated September 09, 2024

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.

Terminology

The prime limit ^[1] 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.^[1]
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). The music program Logic Pro uses also 1⁄2-comma meantone temperament.
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^[2]	1 : 1	1 : 1	play ^ⓘ Unison,^[3] monophony,^[4] perfect prime,^[3] tonic,^[5] 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 ^[3]^[6]		7		S
0.72	E +	2401 : 2400	7⁴ : 2⁵×3×5²	play ^ⓘ Breedsma ^[3]^[6]		7		S
1.00		2^1/1200	2^1/1200	play ^ⓘ Cent ^[7]	1200
1.20		2^1/1000	2^1/1000	play ^ⓘ Millioctave	1000
1.95	B♯++	32805 : 32768	3⁸×5 : 2¹⁵	play ^ⓘ Schisma ^[3]^[5]		5
1.96		3:2÷(2^7/12)	3 : 2^19/12	Grad, Werckmeister^[8]
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,^[3]^[6] marvel comma		7		S
8.11	B −	15625 : 15552	5⁶ : 2⁶×3⁵	play ^ⓘ Kleisma or semicomma majeur^[3]^[6]		5
10.06	A ++	2109375 : 2097152	3³×5⁷ : 2²¹	play ^ⓘ Semicomma,^[3]^[6] Fokker's comma^[3]		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 ^[3]^[9]		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,^[6] small septimal comma,^[3] starling comma		7		S
14.37	C♭↑↑−	121 : 120	11² : 2³×3×5	play ^ⓘUndecimal seconds comma^[3]		11		S
16.67	C↑^{[lower-alpha 1]}	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 --^[2]	2048 : 2025	2¹¹ : 3⁴×5²	play ^ⓘ Diaschisma,^[3]^[6] minor comma		5
21.51	C+^[2]	81 : 80	3⁴ : 2⁴×5	play ^ⓘ Syntonic comma,^[3]^[5]^[6] major comma, komma, chromatic diesis, or comma of Didymus^[3]^[6]^[10]^[11]		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,^[3]^[5]^[6]^[10]^[11] ditonic comma^[3]^[6]		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,^[5] 13th-partial chroma^[3]		13		S
27.26	C −	64 : 63	2⁶ : 3²×7	play ^ⓘ Septimal comma,^[3]^[6]^[11] Archytas' comma,^[3] 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,^[3] Ptolemy's enharmonic:^[5] difference between (11 : 8) and (7 : 5) tritone		11		S
33.33	C /D♭ ^{[lower-alpha 1]}	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^[3]^[6]		7		S
35.70	D ♭	49 : 48	7² : 2⁴×3	play ^ⓘ Septimal diesis, slendro diesis or septimal 1/6-tone^[3]		7		S
38.05	C	46 : 45	2×23 : 3²×5	play ^ⓘInferior quarter tone,^[5] 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,^[6] diminished second,^[5]^[6] minor diesis or diesis,^[3] 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,^[5] 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,^[3]^[6] septimal chroma,^[2] superior quarter tone^[5]		7		S
49.98		246 : 239	3×41 : 239	play ^ⓘJust quarter tone^[11]		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,^[5] lesser 17-limit quarter tone		17		S
50.72	B ♯++	59049 : 57344	3¹⁰ : 2¹³×7	play ^ⓘHarrison's comma (10 P5s – 1 H7)^[3]		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,^[5] undecimal comma, undecimal quarter tone		11		S
54.96	D ♭-	32 : 31	2⁵ : 31	play ^ⓘInferior quarter-tone,^[5] 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,^[5] 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^[5]		7		S
63.81		(3 : 2)^1/11	3^1/11 : 2^1/11	play ^ⓘ Beta scale step	18.75
65.34	C ♯+	27 : 26	3³ : 2×13	play ^ⓘ Chromatic diesis,^[12] tridecimal comma^[3]		13		S
66.34	D ♭	133 : 128	7×19 : 2⁷	play ^ⓘOne-hundred-thirty-third harmonic		19
66.67	C ↑/C♯ ^{[lower-alpha 1]}	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^[5]		13		S
70.67	C♯^[2]	25 : 24	5² : 2³×3	play ^ⓘ Just chromatic semitone or minor chroma,^[3] lesser chromatic semitone, small (just) semitone^[11] or minor second,^[4] minor chromatic semitone,^[13] or minor semitone,^[5]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^[5]		67
80.54	C↑ -	22 : 21	2×11 : 3×7	play ^ⓘHard semitone,^[5] two-fifth tone small semitone		11		S
84.47	D ♭	21 : 20	3×7 : 2²×5	play ^ⓘ Septimal chromatic semitone, minor semitone^[3]		7		S
88.80	C ♯	20 : 19	2²×5 : 19	play ^ⓘNovendecimal augmented unison		19		S
90.22	D♭−−^[2]	256 : 243	2⁸ : 3⁵	play ^ⓘPythagorean minor second or limma,^[3]^[6]^[11] Pythagorean diatonic semitone, Low Semitone^[14]		3
92.18	C♯+^[2]	135 : 128	3³×5 : 2⁷	play ^ⓘGreater chromatic semitone, chromatic semitone, semitone medius, major chroma or major limma,^[3] small limma,^[11] major chromatic semitone,^[13] limma ascendant^[5]		5
93.60	D ♭-	19 : 18	19 : 2×9	Novendecimal minor second play ^ⓘ		19		S
97.36	D↓↓	128 : 121	2⁷ : 11²	play ^ⓘ121st subharmonic,^[5]^[6] undecimal minor second		11
98.95	D ♭	18 : 17	2×3² : 17	play ^ⓘJust minor semitone, Arabic lute index finger^[3]		17		S
100.00	C♯/D♭	2^1/12	2^1/12	play ^ⓘEqual-tempered minor second or semitone	12		M
104.96	C ♯^[2]	17 : 16	17 : 2⁴	play ^ⓘ Minor diatonic semitone, just major semitone, overtone semitone,^[5] 17th harmonic,^[3] 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)	25
111.73	D♭-^[2]	16 : 15	2⁴ : 3×5	play ^ⓘ Just minor second,^[15] just diatonic semitone, large just semitone or major second,^[4] major semitone,^[5] limma, minor diatonic semitone,^[3] diatonic second^[16] semitone,^[14] diatonic semitone,^[11]1⁄6-comma meantone minor second		5		S
113.69	C♯++	2187 : 2048	3⁷ : 2¹¹	play ^ⓘApotome^[3]^[11] or Pythagorean major semitone,^[6] 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,^[3] Cowell semitone^[5]		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^[17]		13		S
130.23	C ♯+	69 : 64	3×23 : 2⁶	play ^ⓘSixty-ninth harmonic^[5]		23
133.24	D♭	27 : 25	3³ : 5²	play ^ⓘ Semitone maximus, minor second, large limma or Bohlen-Pierce small semitone,^[3] high semitone,^[14] alternate Renaissance half-step,^[5] large limma, acute minor second^{[ citation needed ]}		5
133.33	C♯ /D♭ ^{[lower-alpha 1]}	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,^[17] Three-quarter tone^[5]		13		S
150.00	C /D	2^3/24	2^1/8	play ^ⓘEqual-tempered neutral second	8, 24
150.64	D↓^[2]	12 : 11	2²×3 : 11	play ^ⓘ3⁄4 tone or Undecimal neutral second,^[3]^[5] trumpet three-quarter tone,^[11] middle finger [between frets]^[14]		11		S
155.14	D	35 : 32	5×7 : 2⁵	play ^ⓘThirty-fifth harmonic^[5]		7
160.90	D−−	800 : 729	2⁵×5² : 3⁶	play ^ⓘGrave whole tone,^[3] neutral second, grave major second^{[ citation needed ]}		5
165.00	D↑♭−^[2]	11 : 10	11 : 2×5	play ^ⓘGreater undecimal minor/major/neutral second, 4/5-tone^[6] or Ptolemy's second^[3]		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^[5]		71
180.45	E −−−	65536 : 59049	2¹⁶ : 3¹⁰	play ^ⓘPythagorean diminished third,^[3]^[6] Pythagorean minor tone		3
182.40	D−^[2]	10 : 9	2×5 : 3²	play ^ⓘ Small just whole tone or major second,^[4] minor whole tone,^[3]^[5] lesser whole tone,^[16] minor tone,^[14] minor second,^[11] 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^[2]	9 : 8	3² : 2³	play ^ⓘPythagorean major second, Large just whole tone or major second^[11] (sesquioctavan),^[4]tonus, major whole tone,^[3]^[5] greater whole tone,^[16] major tone^[14]		3		S
215.89	D	145 : 128	5×29 : 2⁷	play ^ⓘHundred-forty-fifth harmonic		29
223.46	E −^[2]	256 : 225	2⁸ : 3²×5²	play ^ⓘJust diminished third,^[16] 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^[5]		73
231.17	D −^[2]	8 : 7	2³ : 7	play ^ⓘ Septimal major second,^[4] septimal whole tone^[3]^[5]		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^[3]		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^[5]		37
253.08	D♯−	125 : 108	5³ : 2²×3³	play ^ⓘSemi-augmented whole tone,^[3] semi-augmented second^{[ citation needed ]}		5
262.37	E↓♭	64 : 55	2⁶ : 5×11	play ^ⓘ55th subharmonic^[5]^[6]		11
266.87	E ♭^[2]	7 : 6	7 : 2×3	play ^ⓘ Septimal minor third ^[3]^[4]^[11] or Sub minor third^[14]		7		S
268.80	D	299 : 256	13×23 : 2⁸	play ^ⓘTwo-hundred-ninety-ninth harmonic		23
274.58	D♯^[2]	75 : 64	3×5² : 2⁶	play ^ⓘJust augmented second,^[16] Augmented tone,^[14] augmented second^[5]^[13]		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^[3]		13
294.13	E♭−^[2]	32 : 27	2⁵ : 3³	play ^ⓘ Pythagorean minor third ^[3]^[5]^[6]^[14]^[16] semiditone, or 27th subharmonic		3
297.51	E ♭^[2]	19 : 16	19 : 2⁴	play ^ⓘ19th harmonic,^[3] 19-limit minor third, overtone minor third^[5]		19
300.00	D♯/E♭	2^3/12	2^1/4	play ^ⓘEqual-tempered minor third	4, 12		M
301.85	D ♯-	25 : 21^[5]	5² : 3×7	play ^ⓘQuasi-equal-tempered minor third, 2nd 7-limit minor third, Bohlen-Pierce second^[3]^[6]		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	3.85
315.64	E♭^[2]	6 : 5	2×3 : 5	play ^ⓘ Just minor third,^[3]^[4]^[5]^[11]^[16] minor third,^[14]1⁄3-comma meantone minor third		5	M	S
317.60	D♯++	19683 : 16384	3⁹ : 2¹⁴	play ^ⓘPythagorean augmented second ^[3]^[6]		3
320.14	E ♭↑	77 : 64	7×11 : 2⁶	play ^ⓘSeventy-seventh harmonic^[5]		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^[18]		17
337.15	E♭+	243 : 200	3⁵ : 2³×5²	play ^ⓘAcute minor third^[3]		5
342.48	E ♭	39 : 32	3×13 : 2⁵	play ^ⓘThirty-ninth harmonic^[5]		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,^[5] septimal neutral third^[6]		7
347.41	E↑♭−^[2]	11 : 9	11 : 3²	play ^ⓘUndecimal neutral third ^[3]^[5]		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^[6] 12:11 X 9:8^[14]		11
359.47	E ^[2]	16 : 13	2⁴ : 13	play ^ⓘTridecimal neutral third ^[3]		13
364.54		79 : 64	79 : 2⁶	play ^ⓘSeventy-ninth harmonic^[5]		79
364.81	E−	100 : 81	2²×5² : 3⁴	play ^ⓘGrave major third^[3]		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,^[3]^[6] Pythagorean 'schismatic' third^[5]		3
386.31	E^[2]	5 : 4	5 : 2²	play ^ⓘ Just major third,^[3]^[4]^[5]^[11]^[16] major third,^[14] 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+^[2]	81 : 64	3⁴ : 2⁶	play ^ⓘPythagorean major third,^[3]^[5]^[6]^[14]^[16] ditone		3
417.51	F ↓+^[2]	14 : 11	2×7 : 11	play ^ⓘUndecimal diminished fourth or major third^[3]		11
425.00		2^17/48	2^17/48	play ^ⓘ17 steps in 48 equal temperament	48
427.37	F♭^[2]	32 : 25	2⁵ : 5²	play ^ⓘJust diminished fourth,^[16] diminished fourth,^[5]^[13] 25th subharmonic		5
429.06	E	41 : 32	41 : 2⁵	play ^ⓘForty-first harmonic^[5]		41
435.08	E ^[2]	9 : 7	3² : 7	play ^ⓘ Septimal major third,^[3]^[5] Bohlen-Pierce third,^[3] Super major Third^[14]		7
444.77	F↓	128 : 99	2⁷ : 9×11	play ^ⓘ99th subharmonic^[5]^[6]		11
450.00	E /F	2^9/24	2^9/24	play ^ⓘ9 steps in 24 equal temperament	24
450.05		83 : 64	83 : 2⁶	play ^ⓘEighty-third harmonic^[5]		83
454.21	F♭	13 : 10	13 : 2×5	play ^ⓘTridecimal major third or diminished fourth		13
456.99	E♯^[2]	125 : 96	5³ : 2⁵×3	play ^ⓘJust augmented third, augmented third^[5]		5
462.35	E -	64 : 49	2⁶ : 7²	play ^ⓘ49th subharmonic^[5]^[6]		7
470.78	F +^[2]	21 : 16	3×7 : 2⁴	play ^ⓘTwenty-first harmonic, narrow fourth,^[3] septimal fourth,^[5] 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^[3]		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^[5]		17
498.04	F^[2]	4 : 3	2² : 3	play ^ⓘPerfect fourth,^[3]^[5]^[16] Pythagorean perfect fourth, Just perfect fourth or diatessaron^[4]		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	18.75
511.52	F	43 : 32	43 : 2⁵	play ^ⓘForty-third harmonic^[5]		43
514.29		2^3/7	2^3/7	play ^ⓘ3 steps in 7 equal temperament	7
519.55	F+^[2]	27 : 20	3³ : 2²×5	play ^ⓘ5-limit wolf fourth, acute fourth,^[3] imperfect fourth^[16]		5
521.51	E♯+++	177147 : 131072	3¹¹ : 2¹⁷	play ^ⓘPythagorean augmented third ^[3]^[6] (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^[5]		29
536.95	F↓♯+	15 : 11	3×5 : 11	play ^ⓘUndecimal augmented fourth^[3]		11
550.00	F /G	2^11/24	2^11/24	play ^ⓘ11 steps in 24 equal temperament	24
551.32	F↑^[2]	11 : 8	11 : 2³	play ^ⓘ eleventh harmonic,^[5] undecimal tritone,^[5] lesser undecimal tritone, undecimal semi-augmented fourth^[3]		11
563.38	F ♯+	18 : 13	2×9 : 13	play ^ⓘTridecimal augmented fourth^[3]		13
568.72	F♯^[2]	25 : 18	5² : 2×3²	play ^ⓘJust augmented fourth^[3]^[5]		5
570.88		89 : 64	89 : 2⁶	play ^ⓘEighty-ninth harmonic^[5]		89
575.00		2^23/48	2^23/48	play ^ⓘ23 steps in 48 equal temperament	48
582.51	G ♭^[2]	7 : 5	7 : 5	play ^ⓘLesser septimal tritone, septimal tritone^[3]^[4]^[5] Huygens' tritone or Bohlen-Pierce fourth,^[3] septimal fifth,^[11] septimal diminished fifth^[19]		7
588.27	G♭−−	1024 : 729	2¹⁰ : 3⁶	play ^ⓘPythagorean diminished fifth,^[3]^[6] low Pythagorean tritone^[5]		3
590.22	F♯+^[2]	45 : 32	3²×5 : 2⁵	play ^ⓘJust augmented fourth, just tritone,^[4]^[11] tritone,^[6] diatonic tritone,^[3] 'augmented' or 'false' fourth,^[16] high 5-limit tritone,^[5]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^[5]		13
609.78	G♭−^[2]	64 : 45	2⁶ : 3²×5	play ^ⓘJust tritone,^[4] 2nd tritone,^[6] 'false' fifth,^[16] diminished fifth,^[13] low 5-limit tritone,^[5] 45th subharmonic		5
611.73	F♯++	729 : 512	3⁶ : 2⁹	play ^ⓘPythagorean tritone,^[3]^[6] Pythagorean augmented fourth, high Pythagorean tritone^[5]		3
617.49	F♯ ^[2]	10 : 7	2×5 : 7	play ^ⓘGreater septimal tritone, septimal tritone,^[4]^[5] Euler's tritone^[3]		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,^[5] classic diminished fifth^{[ citation needed ]}		23
631.28	G♭^[2]	36 : 25	2²×3² : 5²	play ^ⓘJust diminished fifth ^[5]		5
646.99	F ♯+	93 : 64	3×31 : 2⁶	play ^ⓘNinety-third harmonic^[5]		31
648.68	G↓^[2]	16 : 11	2⁴ : 11	play ^ⓘ` undecimal semi-diminished fifth^[3]		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^[5]		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 ^[3]^[6]		3
680.45	G−	40 : 27	2³×5 : 3³	play ^ⓘ5-limit wolf fifth,^[5] or diminished sixth, grave fifth,^[3]^[6]^[11] imperfect fifth,^[16]		5
683.83	G	95 : 64	5×19 : 2⁶	play ^ⓘNinety-fifth harmonic^[5]		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
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^[2]	3 : 2	3 : 2	play ^ⓘ Perfect fifth,^[3]^[5]^[16] Pythagorean perfect fifth, Just perfect fifth or diapente,^[4] fifth,^[14] Just fifth^[11]		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^[5]		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^[3]		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,^[5]^[6] 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^[5]		7
743.01	A	192 : 125	2⁶×3 : 5³	play ^ⓘClassic diminished sixth ^[3]		5
750.00	G /A	2^15/24	2^15/24	play ^ⓘ15 steps in 24 equal temperament	24
755.23	G↑	99 : 64	3²×11 : 2⁶	play ^ⓘNinety-ninth harmonic^[5]		11
764.92	A ♭^[2]	14 : 9	2×7 : 3²	play ^ⓘ Septimal minor sixth ^[3]^[5]		7
772.63	G♯	25 : 16	5² : 2⁴	play ^ⓘJust augmented fifth ^[5]^[16]		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 ↑-^[2]	11 : 7	11 : 7	play ^ⓘ Undecimal minor sixth,^[5] undecimal augmented fifth,^[3] Fibonacci numbers		11
789.85		101 : 64	101 : 2⁶	play ^ⓘHundred-first harmonic^[5]		101
792.18	A♭−^[2]	128 : 81	2⁷ : 3⁴	play ^ⓘPythagorean minor sixth,^[3]^[5]^[6] 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^[5]		17
813.69	A♭^[2]	8 : 5	2³ : 5	play ^ⓘJust minor sixth ^[3]^[4]^[11]^[16]		5
815.64	G♯++	6561 : 4096	3⁸ : 2¹²	play ^ⓘPythagorean augmented fifth,^[3]^[6] Pythagorean 'schismatic' sixth^[5]		3
823.80		103 : 64	103 : 2⁶	play ^ⓘHundred-third harmonic^[5]		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^[3]		5
840.53	A ♭^[2]	13 : 8	13 : 2³	play ^ⓘTridecimal neutral sixth,^[3] overtone sixth,^[5] 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↓+^[2]	18 : 11	2×3² : 11	play ^ⓘUndecimal neutral sixth,^[3]^[5] Zalzal's neutral sixth		11
857.09	A +	105 : 64	3×5×7 : 2⁶	play ^ⓘHundred-fifth harmonic^[5]		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^[3]		5
873.50	A	53 : 32	53 : 2⁵	play ^ⓘFifty-third harmonic^[5]		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^[5]^[6]		11
882.40	B −−−	32768 : 19683	2¹⁵ : 3⁹	play ^ⓘPythagorean diminished seventh ^[3]^[6]		3
884.36	A^[2]	5 : 3	5 : 3	play ^ⓘJust major sixth,^[3]^[4]^[5]^[11]^[16] Bohlen-Pierce sixth,^[3]1⁄3-comma meantone major sixth		5	M
889.76		107 : 64	107 : 2⁶	play ^ⓘHundred-seventh harmonic^[5]		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 ^[5]^[6]		19
905.87	A+^[2]	27 : 16	3³ : 2⁴	play ^ⓘPythagorean major sixth ^[3]^[5]^[11]^[16]		3
921.82		109 : 64	109 : 2⁶	play ^ⓘHundred-ninth harmonic^[5]		109
925.00		2^37/48	2^37/48	play ^ⓘ37 steps in 48 equal temperament	48
925.42	B −^[2]	128 : 75	2⁷ : 3×5²	play ^ⓘJust diminished seventh,^[16] diminished seventh,^[5]^[13] 75th subharmonic		5
925.79	A +	437 : 256	19×23 : 2⁸	play ^ⓘFour-hundred-thirty-seventh harmonic		23
933.13	A ^[2]	12 : 7	2²×3 : 7	play ^ⓘ Septimal major sixth ^[3]^[4]^[5]		7
937.63	A↑	55 : 32	5×11 : 2⁵	play ^ⓘFifty-fifth harmonic^[5]^[20]		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^[5]		37
955.03	A♯^[2]	125 : 72	5³ : 2³×3²	play ^ⓘJust augmented sixth ^[5]		5
957.21		(3 : 2)^15/11	3^15/11 : 2^15/11	play ^ⓘ15 steps in Beta scale	18.75
960.00		2^4/5	2^4/5	play ^ⓘ4 steps in 5 equal temperament	5
968.83	B ♭^[2]	7 : 4	7 : 2²	play ^ⓘ Septimal minor seventh,^[4]^[5]^[11] harmonic seventh,^[3]^[11] 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♯+^[2]	225 : 128	3²×5² : 2⁷	play ^ⓘJust augmented sixth ^[16]		5
984.21		113 : 64	113 : 2⁶	play ^ⓘHundred-thirteenth harmonic^[5]		113
996.09	B♭−^[2]	16 : 9	2⁴ : 3²	play ^ⓘPythagorean minor seventh,^[3] Small just minor seventh,^[4] lesser minor seventh,^[16] just minor seventh,^[11] Pythagorean small minor seventh^[5]		3
999.47	B ♭	57 : 32	3×19 : 2⁵	play ^ⓘFifty-seventh harmonic^[5]		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^[5]		23
1017.60	B♭^[2]	9 : 5	3² : 5	play ^ⓘGreater just minor seventh,^[16] large just minor seventh,^[4]^[5] Bohlen-Pierce seventh^[3]		5
1019.55	A♯+++	59049 : 32768	3¹⁰ : 2¹⁵	play ^ⓘPythagorean augmented sixth ^[3]^[6]		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,^[5] minor seventh^{[ citation needed ]}		29
1035.00	B↓^[2]	20 : 11	2²×5 : 11	play ^ⓘLesser undecimal neutral seventh, large minor seventh^[3]		11
1039.10	B♭+	729 : 400	3⁶ : 2⁴×5²	play ^ⓘAcute minor seventh^[3]		5
1044.44	B ♭	117 : 64	3²×13 : 2⁶	play ^ⓘHundred-seventeenth harmonic^[5]		13
1044.86	B ♭-	64 : 35	2⁶ : 5×7	play ^ⓘ35th subharmonic,^[5] septimal neutral seventh^[6]		7
1049.36	B↑♭−^[2]	11 : 6	11 : 2×3	play ^ⓘ21⁄4-tone or Undecimal neutral seventh,^[3] undecimal 'median' seventh^[5]		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^[5]		59
1066.76	B−	50 : 27	2×5² : 3³	play ^ⓘGrave major seventh^[3]		5
1071.70	B ♭-	13 : 7	13 : 7	play ^ⓘTridecimal neutral seventh^[21]		13
1073.78	B	119 : 64	7×17 : 2⁶	play ^ⓘHundred-nineteenth harmonic^[5]		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]^[6]		3
1088.27	B^[2]	15 : 8	3×5 : 2³	play ^ⓘJust major seventh,^[3]^[5]^[11]^[16] small just major seventh,^[4]1⁄6-comma meantone major seventh		5
1095.04	C ♭	32 : 17	2⁵ : 17	play ^ⓘ17th subharmonic^[5]^[6]		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^[5]		11
1107.82	C′♭−	256 : 135	2⁸ : 3³×5	play ^ⓘOctave − major chroma,^[3] 135th subharmonic, narrow diminished octave^{[ citation needed ]}		5
1109.78	B+^[2]	243 : 128	3⁵ : 2⁷	play ^ⓘPythagorean major seventh ^[3]^[5]^[6]^[11]		3
1116.88		61 : 32	61 : 2⁵	play ^ⓘSixty-first harmonic^[5]		61
1125.00		2^15/16	2^45/48	play ^ⓘ45 steps in 48 equal temperament	16, 48
1129.33	C′♭^[2]	48 : 25	2⁴×3 : 5²	play ^ⓘClassic diminished octave,^[3]^[6] large just major seventh ^[4]		5
1131.02	B	123 : 64	3×41 : 2⁶	play ^ⓘHundred-twenty-third harmonic^[5]		41
1137.04	B	27 : 14	3³ : 2×7	play ^ⓘSeptimal major seventh^[5]		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,^[5] augmented seventh^{[ citation needed ]}		31
1146.73	C↓	64 : 33	2⁶ : 3×11	play ^ⓘ33rd subharmonic^[6]		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♯^[2]	125 : 64	5³ : 2⁶	play ^ⓘJust augmented seventh,^[5] 125th harmonic		5
1172.74	C +	63 : 32	3²×7 : 2⁵	play ^ⓘSixty-third harmonic^[5]		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,^[3] semi-diminished octave^{[ citation needed ]}		5
1179.59	B ↑	253 : 128	11×23 : 2⁷	play ^ⓘTwo-hundred-fifty-third harmonic^[5]		23
1186.42		127 : 64	127 : 2⁶	play ^ⓘHundred-twenty-seventh harmonic^[5]		127
1200.00	C′	2 : 1	2 : 1	play ^ⓘ Octave ^[3]^[11] or diapason^[4]	1, 12	3	M	S

Notes

1 2 3 4 Maneri-Sims notation

Related Research Articles

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.

In music, there are two common meanings for tuning:

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths which are "pure" or perfect, with ratio $. This is chosen because it is the next harmonic of a vibrating string, after the octave, and hence is the next most consonant "pure" interval, and the easiest to tune by ear. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is \approx 702 cents wide.$

Meantone temperaments are musical temperaments; that is, a variety of tuning systems constructed, similarly to Pythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within the same octave. But rather than using perfect fifths, consisting of frequency ratios of value $, these are tempered by a suitable factor that narrows them to ratios that are slightly less than, in order to bring the major or minor thirds closer to the just intonation ratio of or, respectively. A regular temperament is one in which all the fifths are chosen to be of the same size.$

In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125). Two notes that differ by this interval would sound different from each other even to untrained ears, but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third to a just major third.

In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including Pythagorean and most meantone temperaments.

A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale, visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C♯; the interval between them is a semitone.

In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions. The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A. Coincidentally, there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.

A regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most commonly used in the Western world, the generator is a tempered fifth, which is the basis behind the circle of fifths.

A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.

12 equal temperament (12-ET) is the musical system that divides the octave into 12 parts, all of which are equally tempered on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step.

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 2^1⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.

In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 3¹/ 2¹) and the perfect fourth with ratio 4/3 (equivalent to 2²/ 3¹) are Pythagorean intervals.

In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the process of altering the size of an interval by making it narrower or wider than pure. "Any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds is called a temperament." Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation, Pythagorean tuning and meantone temperament meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.

The harmonic seventh interval, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio (about 969 cents). This is somewhat narrower than and is, "particularly sweet", "sweeter in quality" than an "ordinary" just minor seventh, which has an intonation ratio of 9:5 (about 1018 cents).

In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps. Each step represents a frequency ratio of 2^1/41, or 29.27 cents, an interval close in size to the septimal comma. 41-ET can be seen as a tuning of the schismatic, magic and miracle temperaments. It is the second smallest equal temperament, after 29-ET, whose perfect fifth is closer to just intonation than that of 12-ET. In other words, $is a better approximation to the ratio than either or .$

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2⁻³·3¹·5¹ = 15/8.

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 . 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.
↑ 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."

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

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

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v t e Consonance and dissonance
Argument Avoid note Beating Cadence Chord Interval Musical note Nonchord tone Cambiata Changing tones Pedal point Preparation Resolution Spectra
Consonances	Unison Minor third Major third Perfect fourth Perfect fifth Minor sixth Major sixth Octave
Dissonances	Minor second Major second Tritone Minor seventh Major seventh
List of musical intervals