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 amount "E", such that you'll end up with fΩ

in other words

f × p × p × p × ... × p = fΩ

f × p ^ E = fΩ

p ^ E = Ω

p = Ω ^ (1/E)

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

p = 2 ^ (1/EDO)

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

• f × p

• f × p^2

• f × p^3

• ...

• f × p^(E-1)

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 :

p = 2 ^ (1/12)

and all the notes in an octave are

• 440Hz × 2 ^ (1/12) : A

• 440Hz × 2 ^ (2/12) : A#

• 440Hz × 2 ^ (3/12) : B

• ...

• 440Hz × 2 ^ (11/12) : G#

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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

{ f × 2 ^ (n/16) | n ∈ N , 0 ≤ n < 16 }

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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

Ω ^ (P/E) = F

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:

2 ^ (7/12) = 1.4983... ≈ 1.5

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

 1.5 - 1.4983... = 0.00169...

The smaller the distance, the "better" it is

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

here's how :

 Ω ^ (P/E) = F

log(F, base Ω) = P/E

log(F) / log(Ω) × E = P

 let's tidy things a little bit :

Ω ^ (P/E) = F

where P is the ideal key

Ω ^ (round(P)/E) = Fhat

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

And Fhat is the factor we have to deal with

D = | F - Fhat |

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