I haven’t been able to find much mention of it on this forum (edit: see the below comment), but for me personally, one of the biggest reasons to use base twelve (“dozenal”) positional fractions is for writing binary logarithms.

The approximations on which Western keyboard instruments and the chromatic scale are designed can be used for general-purpose approximate calculation.

Namely, when written dozenally,

lg 1 = 0

lg 2 = 1

lg 3 ~ 1.7 (off by about 0.1%)

lg 4 = 2

lg 5 ~ 2.4 (off by about 1%, for greater precision use 2.3x)

lg 6 ~ 2.7

lg 7 ~ 2.98

lg 8 = 3

lg 9 ~ 3.2

lg 10 ~ 3.4 [3.3x]

lg 11 ~ 3.56

lg 12 ~ 3.7

lg 13 ~ 3.85

lg 14 ~ 3.98

lg 15 ~ 3.e [3.xe]

lg 16 = 4

lg 17 ~ 4.11

lg 18 ~ 4.2

lg 19 ~ 4.30

lg 20 ~ 4.4 [4.3x]

lg 21 ~ 4.48

lg 22 ~ 4.56

...

This means that any simple fractions using powers of 2, 3, and 5 can be approximately expressed as a binary logarithm with only one dozenal digit after the radix point. (If you take multiple powers of 5, the approximation gets a bit less accurate each time.)

As a contrived example,

lg 2√(10)/27

= lg 2 + (1/2)(lg 2 + lg 5) – (3) lg 3

~ 1 + 3.4 / 2 – 3(1.7)

= 1 + 1.8 – 4.9

= –2.1

~ lg 15/64

~ lg 4/17 (if you prefer a smaller fraction)

Or as decimals,

2√(10)/27 ~ 0.234243

15/64 = 0.234375

4/17 ~ 0.235294

A bit lucky that we get an error of only ~0.06% or ~0.45% here – it could easily go up to a percent or two – and we avoided trickier estimates or interpolation, but this isn’t too atypical. Also note, all we need to do for a square or cube root is divide by two or three, which are easy with dozenal fractions.

* * *

I know some old engineers who memorized 3 or 4 digits of the common logarithms (base ten expressed as decimal fractions) for all the integers from 1 through about 50, so that they could do fast approximate computation.

This is the same general idea, but gets a lot more leverage out of memorizing only two basic facts, and only to 2 digits.

* * *

Of course, the general idea of writing base 2 logarithms using duodecimal notation isn’t new. Indeed, the music theory world multiplies base two logarithms by 1200 to arrive at “cents” as a basic unit – a cent is 1/100 of an equally tempered semitone, and an equal-tempered semitone is 1/12 of an octave, where an octave is a doubling of frequency.

After I had thought about this for a while, I also found this paper by Parry Moon along similar lines:

https://www.pdf-archive.com/2016/11/19/ ... 907071.pdf

And I found this patent application almost precisely describing the same kind of circular slide rule I intended to build.

https://www.google.com/patents/US2326413

https://patentimages.storage.googleapis.com...US2326413-1.png

But anyway, I think this is applicable far beyond music theory. For example, I think it would be great for constructing scales of wire gauges, paper sizes (as a generalization of ISO paper dimensions which already all fall on sizes with only one digit after the radix point on a log scale; the advantage of using the full duodecimal fractions is that we can get something very close to 2:3 or 5:4 or whatever, which means that we can easily break our paper up into columns; just using √2:1 as a paper shape sucks for general-purpose graphic design), or any kind of logarithmic scale.

The richter scale, decibels, etc. all suffer from the problem that powers of ten are not very relevant to human estimation or senses. Repeated doubling and halving is much easier to comprehend. That we can also easily pull in factors of three and five is an amazing bonus.

* * *

Note: I edited the title to mention the name “dublogs”.

There is a scale of 83, which corresponds to semitones.

lg2 = 12, lg 3 = 19, lg 5 = 28, lg 10 = 40, lg 12 = 43, lg 120 = 83, lg 180 = 90.

This is the semitone series. The decimal scale is R40, used for sizes of electrical components. The resolution is of the order of 18, which means it can tell 18 (50) from 19 (51) from 20 (52).

It is supposed to be the range people can tell apart.