Calculating the frequency of quarter tones

Question

I have several questions on the subject of note frequencies and their ranges:

First Question

How should I calculate the frequency of a quarter tone, such as the half-flat (half-bemol) that is found in Arabic/Eastern music such as for the Oud?

I have two theories, and I want to know which one is correct.

Let's take E3 (Mi3) for an example here.
I want to calculate its half-flat (half-bemol) frequency value, which is a quarter note under the E3 value.
We know that E3's frequency is 164.81 Hz, and Eb3 is half step lower with frequency of 155.56 Hz.
So if I want to calculate the value of the E(half-flat)3, will it be:

1. the geometric mean:
   square root (164.81 * 155.56)
   = 160.11
   or

2. the arithmetic mean:
   average (164.81 , 155.5)
   = 160.15

Second Question

If I want to cover all the frequencies with notes, i.e. map all frequencies to some note, where will the separation point be? (I mean the start of the range and end of the range)

Should I use the geometric or arithmetic mean to find the start and end of the range?

Let's again take E3 (164.81 Hz) for an example, and we have Eb3 (155.56 Hz) below it and F3 (174.61 Hz) above it.

Is it more accurate to say that E3 ranges between:

1. square root (155.56 * 164.81) to square root (164.81 * 174.61)
   = 160.11 to 169.63
   or

2. average (155.56 , 164.81) to average (164.81 , 174.61)
   = 160.15 to 169.71

Answer 1

1. The frequency of a quarter tone:

Your first method, the square root of the product or geometric mean is correct. If you're using a spreadsheet like Excel or Google Sheets, you can use the =GEOMEAN (f1,f2) function as a shortcut. This will actually calculate the midpoint between any two pitches, for example with A4 = 440 Hz and A5 = 880 Hz, the midpoint is √(440 * 880) = 622.25 Hz, which is E♭5.

2. Map each frequency I get to a Note:

You have to select a base frequency first, and calculate the number of quarter-tones difference from that base, using:

Q = 24 × (log (f / f₀) / log (2))

Where Q is the number of quarter-tones, f is the frequency, and f₀ is the base frequency. Q will be negative for pitches below f₀.

For your example pitch of 160.11 Hz and A4=440 Hz as the base frequency, I get -35 quarter-tones, or an octave and 11 quarter-tones down, which is correct for E𝄳 (E half-flat).

If your calculator or software can do logarithms in any base, it's simpler if you use the base 2 logarithm and this form: Q = 24 × log₂ (f / f₀). In Excel, it would be =LOG(F1/F0,2).

If you start with a frequency in between two quarter-tones, you won't get an integer. You can round if you like.

It might be helpful to go the other way around and generate a list of frequencies for each note. You would use

f = f₀ × 2 ^{Q / 24}.

If you increment Q by 0.5 you can get the (eighth-tone) boundary between two quarter-tone pitches if that's useful.

Answer 2

How to Calculate the frequency of a quarter tone, half flat - half bemol. that is found in Arabic/Eastern music such as Oud?

If you're using equal temperament, that is to say, dividing the octave into 12 equal half steps and then dividing each of those in two to get 24 equal half steps, the quarter tone is given by the geometric mean of the two pitches that it falls between. The ratio between any two pitches separated by a quarter tone is the 24th root of two.

Whether it's appropriate to use equal temperament is questionable. I somehow doubt that it's been in use for more than several decades.

do i need to calculate the square root of multiplying two notes, and that will be the end range of the first and the start range of the other ? or do i need to do the average between them and that will be the range.

The relationship between pitch and frequency is logarithmic (or should that be "between frequency and pitch"?), so, as above, you want to find the pitch midpoint by calculating the geometric mean of the frequencies: multiply the two frequencies and take the square root.

in some tuners i see that they pick a little different numbers for the Notes, that is also based on 440hz, is there any difference between notes other than the 440hz.

If they give a different frequency for some note despite both being set to A=440 Hz, then they could be using more precision or they could be using some unequal temperament. There are many of these in western classical music of the 17th and 18th centuries, and the intervals of just intonation also appear in Indian music theory some millennia earlier. If you ask about some specific frequencies, we may be able to recognize where they come from.