Logarithmic scale approximations

Logarithmic scale approximations are approximations of a logarithmic scale. Certain primes are represented by a certain amount of steps on the scale, allowing the rationals made of those primes to be approximated on the scale.

This is also known as Equal-step Tuning (ET) in Xenharmonic. In particular, the case with exact octaves is known as an EDO (Equal Divisions of the Octave).

To get started with logarithmic approximations, approximate logarithm of primes like this:

$\log_{2}(2)=12\div12$

$\log_{2}(3)\approx19\div12$

$\log_{2}(5)\approx28\div12$

$\log_{2}(7)\approx34\div12$

Then the primes can be combined to create composite numbers:

$\log_{2}(4)=\log_{2}(2\times2)=(12+12)\div12=24\div12$

$\log_{2}(6)=\log_{2}(2\times3)\approx(12+19)\div12=31\div12$

$\log_{2}(8)=\log_{2}(2\times2\times2)=(12+12+12)\div12=36\div12$

$\log_{2}(9)=\log_{2}(3\times3)\approx(19+19)\div12=38\div12$

$\log_{2}(10)=\log_{2}(2\times5)\approx(12+28)\div12=40\div12$

$\log_{2}(12)=\log_{2}(2\times2\times3)\approx(12+12+19)\div12=43\div12$

$\log_{2}(14)=\log_{2}(2\times7)\approx(12+34)\div12=46\div12$

$\log_{2}(15)=\log_{2}(3\times5)\approx(19+28)\div12=47\div12$

$\log_{2}(16)=\log_{2}(2\times2\times2\times2)=(12+12+12+12)\div12=48\div12$

$\log_{2}(18)=\log_{2}(2\times3\times3)\approx(12+19+19)\div12=50\div12$

$\log_{2}(20)=\log_{2}(2\times2\times5)\approx(12+12+28)\div12=52\div12$

$\log_{2}(21)=\log_{2}(3\times7)\approx(19+34)\div12=53\div12$

And similarly, handling fractions (3÷2, etc.) is by subtracting steps on logarithmic scale.

To get more precision and handle higher primes (11, 13, 17) check out the following sections.

41edo logarithmic scale

The logarithmic scale may be approximated in terms of 41edo. In the following table, primes (2, 3, 5, 7, 11, 13) are shown, allowing 13-smooth rationals to be approximated.

x	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
41log₂x	0	41	65	82	95	106	115	123	130	136	142	147	152	156	160	164

41edo has the property that any fraction with the odd factors of the numerator and denominator being no more than 15, rounded on the logarithmic scale has the same steps as rounding the prime factors individually and adding the steps, making it consistent in the 15-odd-limit.

Now to map steps on a logarithmic scale to fractions use Big 41 logarithmic scale.

46edo logarithmic scale

Alternatively, 46edo could be used. In the following table the primes (2, 3, 5, 7, 11, 13, 17) are shown.

x	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
46log₂x	0	46	73	92	107	119	129	138	146	153	159	165	170	175	180	184	188	192

Other scales

Various other EDO could be used to approximate numbers on a logarithmic scale, for example 12edo, 19edo, 22edo, 26edo, 31edo, 53edo, 58edo, 68edo, 72edo, 80edo, 87edo, 94edo, 111edo, 130edo, 140edo. Furthermore, Edt (equal divisions of the tritave) scales have exact 3 instead of exact 2. (73edt resembles 46edo, etc.)

Example of approximation: $\sqrt{4.9}$

In https://www.youtube.com/watch?v=Wx7lNCNN1y8 , the math problem of approximating the square root of 4.9 is presented.

4.9 can also be expressed as 2⁻¹×5⁻¹×7², or as 7×7÷10. As such, on a logarithmic scale add the steps for 7 twice and subtract the steps of 10.

41log₂4.9=41log₂(7×7÷10)≈115+115-136=94

Then the square root is halving the steps on the scale, so we arrive at 47 steps, which corresponds to 20÷9 or 11÷5.