Microtonal Tunning Comparisons

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Starting with a pitch

let's start with selecting a pitch

• in orchestra, it's 440Hz

• in Astralica Traditional Music, it's 84.6627802711Hz

• of course you can pick any number you like

this is your fundamental frequency

let's notate it as "f"

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Octave

let's multiply f by some specific number, say "Ω"

for different species or individuals, Ω maybe different

Ω is selected, such that fΩ sounds ALMOST the same as f but higher

by using the same logic, f, fΩ, fΩΩ, ..., fΩ^n, all of these sounds the same for us, but higher than the previous ones

for normal humans, Ω = 2

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In An Octave

by intuition, we know that there must be other frequencies in between f and fΩ

assume that we need to multiply a number "p" with f by some integer amount "E", such that you'll end up with fΩ

in other words

at this point you might realise that the equation above can be interpreted as

This means the following are available frequencies in between f and fΩ :

for example : take our normal black and white piano

for us normal humans, Ω = 2, and E = 12

you can know this because we have 12 keys in an octave

so :

and all the notes in an octave are

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other ways to cut the Octave

E does not has to get stuck with 12, it can be other numbers as well,

For example,

Astralica Traditional Music let E = 16

so all available pitches are

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How good it is ?

For humans, a "good" pitch has a "simple" ratio to the fundamental frequency f.

by "simple", we mean "the smaller the number, the better it is"

For example : 1/2 is way way way "better" than 37/41

For humans, a ratio of 1/2 is an octave

The next ratio we can look for is 1f:3f, however 3 runs outside of the range between 1 and 2, but for humans, 3f and 1.5f sounds the same but lower

We can try some of the simple ratios :

factors	octave clamped factors	music theory names
3	3/2 = 1.5	Perfect 5th
5	5/4 = 1.25	Major 3rd
7	7/4 = 1.75
11	11/8 = 1.375
13	13/8 = 1.625

Now we can adjust the equation as

where

P is the required exact pitch

F is the desired Octave Clamped Factor

Assume we want to see how "good" our Perfect 5th in 12EDO

We can do the calculation:

Here we can treat 1.5 as our desired target, but 1.4983 is what we have to stick with, the distance will be

The smaller the distance, the "better" it is

How can we find P ? by given Ω, E, and F ?

here's how :

 let's tidy things a little bit :

where

P is the ideal key

F is the ideal Octave Clamped Frequency

where

round(P) is the closest key that an EDO can give us

And Fhat is the factor we have to deal with

where D is distance

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There's a tool for that ?!

Here is a tool to do all of that hard work for you :

https://memo2007ultra.wixsite.com/m27u-m2016/keybowoard-extra