I have several questions on the subject of note frequencies and their ranges:
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:
the geometric mean:
square root (164.81 * 155.56)
the arithmetic mean:
average (164.81 , 155.5)
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:
square root (155.56 * 164.81) to square root (164.81 * 174.61)
= 160.11 to 169.63
average (155.56 , 164.81) to average (164.81 , 174.61)
= 160.15 to 169.71