Fundamental Frequency question! Please help!

Question

4) Calculate 5 different frequencies for G6 rounding each, if necessary, to the nearest 0.001 Hz. Show your work and name the tuning system used in each case. For each of these, A4 must have a frequency of 440 Hz, and an octave must have 12 half-steps (as is normal).

Answer 1

Hello, Kelci! This is a really hard challenge. I'm closely related to this due to some of my music production activities. This kind of calculations are used in sound design for virtual synths. I hope you find this useful!

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Have a great week. Take care!

Carlos

I'd use these:

1.) Equal Temperament Tuning:

   440 ∙ 2(22/12) = 1,567.982 Hz 

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2.) Pythagorean Tuning: 

   440 ∙ (3/2)–2 ∙ 23 = 1,564.444 Hz 

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3.) Harmonic Series (key of A):

   440 ∙ (7/4) ∙ 21 = 1,540 Hz 

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4.) Harmonic series: you could use the Key of Ab:

Equation #1: Ab4 x (12/11) = 440 Hz. (which is A)

(you have Ab4 and need to go up 1 minor second, which is 12/11 in Harmonic Series).

Step 1.) Getting Ab4 

First, you’ll need to divide both sides of Equation #1 by 12/11 (which means multiplying by 11/12) = 440 x 11/12 = 403.33333 Hz. 

This is, Ab4 = 403.33333 Hz

Step 2.) Finding G6: 

For this, you need to take your Ab4 and multiply by the fraction corresponding to a major seventh , then multiply again by 21^.  

Thus, it would be 403.33333 ∙ (11/6) ∙ 21 = 1,478.889 Hz 

G6 (according to the Harmonic Series in the key of Ab) = 1,478.889 Hz

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5.) Just Intonation: You could use the Key of Ab again.

Ab4 ∙ (16/15) = 440 Hz. (which is A)

Step 1.) You should divide both sides by 16/15 (this is, multiply by 15/16) to get A4b

When you do this, you’ll get Ab4 = 412.5 Hz 

Step 2.) Finding G6: 

You will take Ab4 and multiply by 15/8, which is a major seventh (in Just Intonation), then multiplying by 21

Thus, 412.5 ∙ (15/8) ∙ 21 = 1,546.875 Hz (this is G6 according to Just Intonation)