Is there a mathematical formula or a list of frequencies (Hz) of notes?

Question

I want to create an app that generates the notes played on a standard bass guitar (4 strings, standard tuning). For this, I need the Hz value of the notes from low E (41 Hz, I think) on up.

Is this a simple math problem, where each half-tone is a certain value greater than the previous note? If not, then is there a chart somewhere that shows the Hz values of what I'm thinking of as "E1" (open E string) on up to "F4" (G string, fret 22)?

If that's confusing, here's a chart that shows how I've got them numbered:

Answer 1

it's very simple, here it is in php. assuming you're using A4 = 440Hz and a bass guitar (12 tone equal temperament) which is in the question.

<?php
class Pitch extends IntegerValue
 {
    private const A4_FREQUENCY = 440.0;
    private const A4_DEGREE = 4*12+9;
    public function getFrequency(): float
     {
        $n = Pitch::A4_DEGREE - $this->getValue();
        $f = ((2.0**(1.0/12.0))**$n) * Pitch::A4_FREQUENCY;
        return $f;
     }
 }

or here in csv

C0,16.35
C#0,17.32
D0,18.35
D#0,19.45
E0,20.60
F0,21.83
F#0,23.12
G0,24.50
G#0,25.96
A0,27.50
A#0,29.14
B0,30.87
C1,32.70
C#1,34.65
D1,36.71
D#1,38.89
E1,41.20
F1,43.65
F#1,46.25
G1,49.00
G#1,51.91
A1,55.00
A#1,58.27
B1,61.74
C2,65.41
C#2,69.30
D2,73.42
D#2,77.78
E2,82.41
F2,87.31
F#2,92.50
G2,98.00
G#2,103.83
A2,110.00
A#2,116.54
B2,123.47
C3,130.81
C#3,138.59
D3,146.83
D#3,155.56
E3,164.81
F3,174.61
F#3,185.00
G3,196.00
G#3,207.65
A3,220.00
A#3,233.08
B3,246.94
C4,261.63
C#4,277.18
D4,293.66
D#4,311.13
E4,329.63
F4,349.23
F#4,369.99
G4,392.00
G#4,415.30
A4,440.00
A#4,466.16
B4,493.88
C5,523.25
C#5,554.37
D5,587.33
D#5,622.25
E5,659.26
F5,698.46
F#5,739.99
G5,783.99
G#5,830.61
A5,880.00
A#5,932.33
B5,987.77
C6,1046.50
C#6,1108.73
D6,1174.66
D#6,1244.51
E6,1318.51
F6,1396.91
F#6,1479.98
G6,1567.98
G#6,1661.22
A6,1760.00
A#6,1864.66
B6,1975.53
C7,2093.00
C#7,2217.46
D7,2349.32
D#7,2489.02
E7,2637.02
F7,2793.83
F#7,2959.96
G7,3135.96
G#7,3322.44
A7,3520.00
A#7,3729.31
B7,3951.07
C8,4186.01
C#8,4434.92
D8,4698.64
D#8,4978.03
E8,5274.04
F8,5587.65
F#8,5919.91
G8,6271.93
G#8,6644.88
A8,7040.00
A#8,7458.62
B8,7902.13

Answer 2

In the 12-TET (12 Tone Equal Temperament) tuning system, we want the ratio of pitches in an octave to be 2:1, and we also want 12 (exponentially) equally spaced semitones per octave. So, the frequency (f_n) of a certain pitch n semitones above the frequency of an initial pitch (f₀) is defined by the following formula:

f_n = f₀ x 2^n/12

(This means that the note 1 semitone above the initial pitch will have a frequency that is approximately 1.06 times greater, as 2^1/12 = 1.05946….)

Our initial pitch usually sets the frequency of A₄ (the A above middle C) to be f₀ = 440 Hz.

---

For the lowest string on our bass guitar (E₁), the note E is 5 semitones below A, but to get from A₄ to E₁ we need to go down a further 3 octaves, n = –5 – 3 x 12 = –41. Putting this n into our formula, we have the frequency of E₁:

E₁ = f_–41 = 440 Hz x 2^–41/12 = 41.2 Hz

Similarly, we can find the frequency of the other notes by first calculating n, and plugging this into our formula. For example, for our highest note (F#₄): n = –3, and so:

F#₄ = f_–3 = 440 Hz x 2^–3/12 = 370 Hz

Answer 3

Musical tuning is a very complex subject. As a starting point, you should surely take equal temperament as your basic approach. Each fret raises the frequency by a factor of the twelfth root of two.

You can experiment with having the open strings tuned acoustically (the ratio of frequencies being 4:3) or in equal temperament (the ratio being 2^(5/12)).

Specific bass guitars will further deviate from theoretical equal temperament because of design factors such as fret placement, bridge placement, and the like, as well as individual factors such as the weight and therefore tension of the chosen strings, and other adjustable parameters. These variations are likely to be irrelevant to most applications of the sort you describe, and in any event many musicians will see them as problematic rather than desirable in many contexts.

All of the other things mentioned in comments and other answers amount to problems of emulating human performance. Bass players can modify the pitch of their instrument by temporarily changing the tuning of a string or by using the fretting hand to push the string sideways, raising its tension, or with an effects pedal.

Depending on the application's requirements, this may or may not be significant. If you find that your initial efforts are unsatisfactory, it may be that one reason for this is the difference between theoretical calculated frequencies and the actual frequencies produced by human musicians, but there are many more factors and details that are difficult to incorporate into computer synthesis that could be at play, for example volume, variable articulation, and tone color.

Answer 4

I know nothing about guitar tuning and this answer may not apply to guitars, but it's worth noting that pianos are not tuned in frequency ratios of 2^1/12 even though they are nominally an equal-tempered instrument. The reason is that what makes two or more notes sound consonant (in tune) to the ear is alignment of the harmonics, not the ratio of the base frequencies. If the strings were ideal mathematical oscillators then the harmonics would be exact integer multiples of the base frequency and the two definitions would be equivalent, but the harmonics of real piano strings are slightly higher, so in the best key-agnostic compromise tuning (which is what equal temperament is), the ratio between consecutive base frequencies is slightly larger than 2^1/12. It varies with the pitch and also from piano to piano. There is software that aids in piano tuning, but you can't just tune each string to a base frequency determined by a mathematical formula and expect to get a good sound. You have to listen to the piano (or at least have the software do it).

See the Wikipedia article on stretched tuning for more.

Answer 5

There isn't a simple single formula as two different principles are being used. First is the frequency of the base (not the bass) note. For example, radio stations use A=440hz for the note A above middle C (different numbers are used by physicists and MIDI). Some orchestras tune to 442 or even 443 as their conductor may feel it sounds better.

Second is the ratio of notes to each other. These depend on the style of tuning. In equal temperament (called 12-tet in the tuning business for the case here) each half-step raised the frequency by the 12th root of 2, approximately 19/18 (as suggested by Vincenzo Galilei.) Other possibilities are tuning to "just" intervals like 5/4 for a major third, 3/2 for a perfect fifth, etc., or tuning all fifths to a ratio of 3/2. "You pays your money and you takes your choice."