in the equaltemperament tuning system the most common system

in the equal-temperament tuning system (the most common system today for tuning musical instruments), each half-step on the musical scale has a frequency that is 2^(1/12) higher than the previous note.

(a) Make a list of the frequencies of the 12 half-steps (C#, D, D#, E, F, F#, G, G#, A, A#, B, C) above middle C, given that A is defined to be 440 hz.

(b) Argue that any note 12 half steps above another note will have exactly twice the frquency of the lower note

(c) Certain combinations of notes sound \"harmonic\" because their frequencies are very nearly simple integer ratios of each other. As an example, consider a C major chord (C,E,G). What are the simplest ratios that are close to the actual ratios of the frequencies of E to C and G to C?(In the equal-temperament tuning system, these ratios are not quite exact. Other tuning systems make these ratios more pure in certain keys, but the equal-temperament system, because of its symmetry, has the advantage that no key is favored.)

(d) Sets of notes with simple frequency ratios also correspond to the harmonic frequencies of a single note. If C, E, and G are adjacent harmonics of some fundamental tone, what is the frequency of that tone.

Thanks for any help, I\'m pretty lost

Solution

a) The formula for the frequencies of the notes of equal tempered scale is,

f_n = f₀ * aⁿ

f₀ = frequency of one fixed note (A = 440Hz)

n - Number of half steps away from the fixed note

a - (2^1/12 ) = 1.0595 {12^th root of 2}

Middle C is three steps above A.

f₃ = 440 * (1.0595)³ = 523.305 Hz = C

Similarly C# is 8 steps below A.

f_-8 = 440 * (1.0595)^-8 = 277.105 Hz

D = 440 * (1.0595)^-7 = 293.593 Hz

D# = 440 * (1.0595)^-6 = 311.062 Hz

E = 440 * (1.0595)^-5 = 329.570 Hz

F = 440 * (1.0595)^-4 = 349.180 Hz

F# = 440 * (1.0595)^-3 = 369.956 Hz

G = 440 * (1.0595)^-2 = 391.968 Hz

G# = 440 * (1.0595)^-1 = 415.290 Hz

A# = 440 * (1.0595)¹ = 466.180 Hz

B = 440 * (1.0595)² = 493.918 Hz

C = 523.306 Hz

b) The note above C is, f₄ = f₀ * a⁴ = 440 * 1.0595⁴ = 554.442 Hz

     The lowest note C# or f_-8 = 277.105 Hz

So from these we can see that f₄ = 2f_-8

c) E to C = 329.570 : 523.306 = 7 : 11 ( originally it is 4 : 5)

    G to C = 391.968 : 523.306 = 3 : 4