Piano key frequencies

This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 108-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A₄), tuned to 440 Hz (referred to as A440).^[1][2] Every octave is made of twelve steps called semitones. A jump from the lowest semitone to the highest semitone in one octave doubles the frequency (for example, the fifth A is 440 Hz and the sixth A is 880 Hz). The frequency of a pitch is derived by multiplying (ascending) or dividing (descending) the frequency of the previous pitch by the twelfth root of two (approximately 1.059463).^[1][2] For example, to get the frequency one semitone up from A₄ (A♯₄), multiply 440 Hz by the twelfth root of two. To go from A₄ up two semitones (one whole tone) to B₄, multiply 440 twice by the twelfth root of two (or once by the sixth root of two, approximately 1.122462). To go from A₄ up three semitones to C₅ (a minor third), multiply 440 Hz three times by the twelfth root of two (or once by the fourth root of two, approximately 1.189207). For other tuning schemes, refer to musical tuning.

This list of frequencies is for a theoretically ideal piano. On an actual piano, the ratio between semitones is slightly larger, especially at the high and low ends, where string stiffness causes inharmonicity, i.e., the tendency for the harmonic makeup of each note to run sharp. To compensate for this, octaves are tuned slightly wide, stretched according to the inharmonic characteristics of each instrument.^[3] This deviation from equal temperament is called the Railsback curve.

The following equation gives the frequency f (Hz) of the n^th key on the idealized standard piano with the 49th key tuned to A₄ at 440 Hz:

${\displaystyle f(n)=\left({\sqrt[{12}]{2}}\,\right)^{n-49}\times 440\,{\text{Hz}}\,=2^{\frac {n-49}{12}}\times 440\,{\text{Hz}}\,}$

where n is shown in the table below.^[1]

Conversely, the key number of a pitch with a frequency f (Hz) on the idealized standard piano is:

${\displaystyle n=12\,\log _{2}\left({\frac {f}{440\,{\text{Hz}}}}\right)+49}$

List

An 88-key piano, with the octaves numbered and Middle C (cyan) and A440 (yellow) highlighted

A printable version of the standard key frequencies (only including the 88 keys on a standard piano)

Values in bold are exact on an idealized standard piano. Keys shaded gray are rare and only appear on extended pianos. The normal 88 keys were numbered 1–88, with the extra low keys numbered 89–97 and the extra high keys numbered 98–108. A 108-key piano that extends from C₀ to B₈ was first built in 2018 by Stuart & Sons.^[4] (Note: these piano key numbers 1-108 are not the n keys in the equations or the table.)

Piano key number	MIDI note number	Helmholtz name[5]	Scientific pitch name[5]	n	Frequency f(n) (Hz) (Equal temperament) [6]	Corresponding open strings on other instruments						Vocal Ranges
						Violin	Viola	Cello	Bass	Guitar	Ukulele	Soprano	Mezzo-soprano	Contralto	Tenor	Baritone	Bass
108	119	b′′′′′	B8	99	7902.133
107	118	a♯′′′′′/b♭′′′′′	A♯8/B♭8	98	7458.620
106	117	a′′′′′	A8	97	7040.000
105	116	g♯′′′′′/a♭′′′′′	G♯8/A♭8	96	6644.875
104	115	g′′′′′	G8	95	6271.927
103	114	f♯′′′′′/g♭′′′′′	F♯8/G♭8	94	5919.911
102	113	f′′′′′	F8	93	5587.652
101	112	e′′′′′	E8	92	5274.041
100	111	d♯′′′′′/e♭′′′′′	D♯8/E♭8	91	4978.032
99	110	d′′′′′	D8	90	4698.636
98	109	c♯′′′′′/d♭′′′′′	C♯8/D♭8	89	4434.922
88	108	c′′′′′ 5-line octave	C8 Eighth octave	88	4186.009
87	107	b′′′′	B7	87	3951.066
86	106	a♯′′′′/b♭′′′′	A♯7/B♭7	86	3729.310
85	105	a′′′′	A7	85	3520.000
84	104	g♯′′′′/a♭′′′′	G♯7/A♭7	84	3322.438
83	103	g′′′′	G7	83	3135.963
82	102	f♯′′′′/g♭′′′′	F♯7/G♭7	82	2959.955
81	101	f′′′′	F7	81	2793.826
80	100	e′′′′	E7	80	2637.020
79	99	d♯′′′′/e♭′′′′	D♯7/E♭7	79	2489.016
78	98	d′′′′	D7	78	2349.318
77	97	c♯′′′′/d♭′′′′	C♯7/D♭7	77	2217.461
76	96	c′′′′ 4-line octave	C7 Double high C	76	2093.005
75	95	b′′′	B6	75	1975.533
74	94	a♯′′′/b♭′′′	A♯6/B♭6	74	1864.655
73	93	a′′′	A6	73	1760.000
72	92	g♯′′′/a♭′′′	G♯6/A♭6	72	1661.219
71	91	g′′′	G6	71	1567.982
70	90	f♯′′′/g♭′′′	F♯6/G♭6	70	1479.978
69	89	f′′′	F6	69	1396.913
68	88	e′′′	E6	68	1318.510
67	87	d♯′′′/e♭′′′	D♯6/E♭6	67	1244.508
66	86	d′′′	D6	66	1174.659
65	85	c♯′′′/d♭′′′	C♯6/D♭6	65	1108.731
64	84	c′′′ 3-line octave	C6 Soprano C (High C)	64	1046.502
63	83	b′′	B5	63	987.7666
62	82	a♯′′/b♭′′	A♯5/B♭5	62	932.3275
61	81	a′′	A5	61	880.0000
60	80	g♯′′/a♭′′	G♯5/A♭5	60	830.6094
59	79	g′′	G5	59	783.9909
58	78	f♯′′/g♭′′	F♯5/G♭5	58	739.9888
57	77	f′′	F5	57	698.4565
56	76	e′′	E5	56	659.2551	E	E (5 String Viola)
55	75	d♯′′/e♭′′	D♯5/E♭5	55	622.2540
54	74	d′′	D5	54	587.3295
53	73	c♯′′/d♭′′	C♯5/D♭5	53	554.3653
52	72	c′′ 2-line octave	C5 Tenor C	52	523.2511
51	71	b′	B4	51	493.8833					High B (Optional for 12 String Guitar)
50	70	a♯′/b♭′	A♯4/B♭4	50	466.1638
49	69	a′	A4 A440	49	440.0000	A	A			High A (Optional)	A
48	68	g♯′/a♭′	G♯4/A♭4	48	415.3047				High Ab (12 Single String Bass)
47	67	g′	G4	47	391.9954						High G
46	66	f♯′/g♭′	F♯4/G♭4	46	369.9944
45	65	f′	F4	45	349.2282
44	64	e′	E4	44	329.6276			High E (5 String Cello)		High E	E
43	63	d♯′/e♭′	D♯4/E♭4	43	311.1270				High Eb (12 String Single String Bass)
42	62	d′	D4	42	293.6648	D	D
41	61	c♯′/d♭′	C♯4/D♭4	41	277.1826
40	60	c′ 1-line octave	C4 Middle C	40	261.6256						C
39	59	b	B3	39	246.9417					B
38	58	a♯/b♭	A♯3/B♭3	38	233.0819
37	57	a	A3	37	220.0000			A
36	56	g♯/a♭	G♯3/A♭3	36	207.6523
35	55	g	G3	35	195.9977	G	G			G	Low G
34	54	f♯/g♭	F♯3/G♭3	34	184.9972
33	53	f	F3	33	174.6141				High F (7 String)
32	52	e	E3	32	164.8138				High E (5th tuning, 5 String Bass)
31	51	d♯/e♭	D♯3/E♭3	31	155.5635
30	50	d	D3	30	146.8324			D		D
29	49	c♯/d♭	C♯3/D♭3	29	138.5913
28	48	c small octave	C3	28	130.8128	C (5 String)	C		C (6 string)
27	47	B	B2	27	123.4708
26	46	A♯/B♭	A♯2/B♭2	26	116.5409
25	45	A	A2	25	110.0000				A (5th tuning Upright)	A
24	44	G♯/A♭	G♯2/A♭2	24	103.8262
23	43	G	G2	23	97.99886			G	G
22	42	F♯/G♭	F♯2/G♭2	22	92.49861
21	41	F	F2	21	87.30706	Low F (6 String)	Low F (6 String)
20	40	E	E2	20	82.40689					Low E
19	39	D♯/E♭	D♯2/E♭2	19	77.78175
18	38	D	D2	18	73.41619				D
17	37	C♯/D♭	C♯2/D♭2	17	69.29566
16	36	C great octave	C2 Deep C	16	65.40639			C
15	35	B͵	B1	15	61.73541					Low B (7 string)
14	34	A♯͵/B♭͵	A♯1/B♭1	14	58.27047
13	33	A͵	A1	13	55.00000				A
12	32	G♯͵/A♭͵	G♯1/A♭1	12	51.91309
11	31	G͵	G1	11	48.99943				G (5th tuning Upright)
10	30	F♯͵/G♭͵	F♯1/G♭1	10	46.24930					Low F♯ (8 string)
9	29	F͵	F1	9	43.65353			Low F (6 String)
8	28	E͵	E1	8	41.20344				E
7	27	D♯͵/E♭͵	D♯1/E♭1	7	38.89087
6	26	D͵	D1	6	36.70810
5	25	C♯͵/D♭͵	C♯1/D♭1	5	34.64783					Low C#(9 String)
4	24	C͵ contra-octave	C1 Pedal C	4	32.70320				C (Upright Extension or 5th tuning)
3	23	B͵͵	B0	3	30.86771				B (5 string)
2	22	A♯͵͵/B♭͵͵	A♯0/B♭0	2	29.13524
1	21	A͵͵	A0	1	27.50000
97	20	G♯͵͵/A♭͵͵	G♯0/A♭0	0	25.95654					Low G# (10 String)
96	19	G͵͵	G0	-1	24.49971
95	18	F♯͵͵/G♭͵͵	F♯0/G♭0	-2	23.12465
94	17	F͵͵	F0	-3	21.82676
93	16	E͵͵	E0	-4	20.60172
92	15	D♯͵͵/E♭͵͵	D♯0/E♭0	-5	19.44544
91	14	D͵͵	D0	-6	18.35405
90	13	C♯͵͵/D♭͵͵	C♯0/D♭0	-7	17.32391
89	12	C͵͵ sub-contra-octave	C0 Double Pedal C	-8	16.35160

See also

• Piano tuning
• Scientific pitch notation
• Music and mathematics

References

1. Weisstein, Eric. "Equal Temperament -- from Eric Weisstein's Treasure Trove of Music". Eric Weisstein's Treasure Trove of Music. Archived from the original on 2019-06-14. Retrieved 2019-12-26.
2. Nov, Yuval. "Explaining the Equal Temperament". www.yuvalnov.org. Archived from the original on 2019-05-26. Retrieved 2019-12-26.
3. Citak, Ray. "Information on Piano Tuning". www.pianotechnician.com. Archived from the original on 2019-02-26. Retrieved 2019-12-26.
4. Wills, Oscar; King, Rosie (2018-09-15). "Australian behind world's grandest piano". ABC News. Australia. Archived from the original on 2019-06-11. Retrieved 2019-12-26.
5. Goss, Clint (2019-02-18). "Octave Notation". Flutopedia. Archived from the original on 2019-05-12. Retrieved 2019-12-26.
6. Suits, Bryan (1998). "Frequencies of Musical Notes, A4 = 440 Hz". Physics of Music — Notes. Michigan Tech University. Archived from the original on 2019-12-16. Retrieved 2019-12-26.