Logarithms Base Two Written Dozenally (“dublogs”)

Logarithms Base Two Written Dozenally (“dublogs”)

jrus
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Nov 20 2016, 07:24 AM #1

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”.
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wendy.krieger
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Nov 20 2016, 07:46 AM #2

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.
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jrus
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Nov 20 2016, 07:57 AM #3

wendy.krieger @ Nov 20 2016, 07:46 AM wrote: 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.
I don’t know what you mean by “scale of 83”, but yes, I realize people sometimes use semitones (12 * lg x) as a scale as an alternative to directly writing binary logarithms or using cents (1200 * lg x). This is primarily done because fractions are written decimally in our culture, and twelfths don’t make nice decimal fractions. If we wrote fractions dozenally, there would be no reason to bother shifting the radix point one over to the right, we could just use binary logarithms directly. That was my main point.

Here’s a historical comparison of other logarithmic scales used for music, if anyone is curious: http://scitation.aip.org/content/asa/jo ... /1.1909997

R40, etc. (cf. wikipedia: preferred number) are ways to break up powers of ten into roughly equal steps. Obviously this makes sense in a predominantly decimal society. I think it’s much less pleasant overall than using powers of two, which are much easier to estimate by eye and much more comprehensible. The decimal scales also aren’t all that useful as a general-purpose tools for estimation / approximate arithmetic.

The nice thing about using dozenally notated binary logarithms is that if we want more precision (in case 12 steps per octave isn’t enough) we can very easily drop to the next digit and make it 16 steps per octave, or 18, or 24, or 36, or 144...
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Oschkar
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Nov 20 2016, 07:59 AM #4

Dozenal doesn’t even fail when factors of 7 are involved... Sure, they’re not exact (actually, they’re off by about 1 part in 109), but you can play a dominant seventh chord as D-F#-A-C and it doesn’t really sound wrong.
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jrus
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Nov 20 2016, 08:05 AM #5

Written dozenally,
lg 7 ~ 2.983, which isn’t quite as nice to work with as
lg 3 ~ 1.703 or
lg 5 ~ 2.3x4

But you can approximate it as 2.x if you want to.

Or if you want you can remember
lg 73 ~ 8.5
73 ~ 210 / 3
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Double sharp
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Nov 20 2016, 08:09 AM #6

@Oschkar: That's because classically speaking, the dominant seventh is not supposed to be a consonant 4:5:6:7 chord in theory. It is a fundamental dissonance theoretically involving the Pythagorean 16:9 as the minor seventh against the tonic, and this dissonant character is not lost until the time of the French Impressionists. (That is if we want to anachronistically shoehorn classical thinking to overtone-series thinking which does not have a lot to do with it.)

I would also dispute the idea that music is based on the overtone series. The fact that a minor triad can serve as well as a major triad as a tonic should speak against that already (though it is true that a major triad is less dissonant). Haydn, Mozart, and Beethoven were already working with a system based completely on 12-equal temperament, with only the use of triads being perhaps a last vestige of impractical just intonation (to be finally abolished by Schoenberg).
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jrus
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Nov 20 2016, 08:12 AM #7

Our keyboard instruments and 12-note scale are absolutely based on the approximations:

219 ~ 312 and
27 ~ 53

If those approximations didn’t exist, we’d need a substantially different setup.

Our “harmonic” instruments make pleasing chords with frequency ratios of approximately 3:2, 4:3, 5:4, etc. It’s a happy coincidence that twelfth roots of two happen to be close to those.

But anyway, that’s all somewhat irrelevant to my basic point.
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Oschkar
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Nov 20 2016, 08:14 AM #8

jrus @ Nov 20 2016, 07:57 AM wrote:I don’t know what you mean by “scale of 83”
Wendy assumes that base 120 should be the default, and 83 semitones correspond to a frequency ratio of 120 to 1.
R40, etc. (cf. wikipedia: preferred number) are ways to break up powers of ten into roughly equal steps. Obviously this makes sense in a predominantly decimal society. I think it’s much less pleasant overall than using powers of two, which are much easier to estimate by eye and much more comprehensible.
R40 is practically identical to the chromatic scale, though. Breaking up a ratio of ten into 40 equal steps is almost the same as breaking up a ratio of two into a dozen equal steps.

101/40 ≈ 1.059253725
21/12 ≈ 1.059463094

The dozenal equivalent to the R40 series would be the R37z series, with three dozen seven steps to the dozen. It’s not a round number anymore, but it preserves the fact that a doubling is twelve steps.
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Oschkar
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Nov 20 2016, 08:23 AM #9

Double sharp @ Nov 20 2016, 08:09 AM wrote:@Oschkar: That's because classically speaking, the dominant seventh is not supposed to be a consonant 4:5:6:7 chord in theory. It is a fundamental dissonance theoretically involving the Pythagorean 16:9 as the minor seventh against the tonic, and this dissonant character is not lost until the time of the French Impressionists.
Dominant sevenths in 12tet still sound ambiguous to me. They sound like 4:5:6:7 in isolation, but, say, the I7 leads to the IV or ii chords, which would only make sense if the seventh is in fact 16:9, and, say, the ii chord fills in the gap left by the I7 chord.

Code: Select all

  . . B  (G7)
F . G D

  A E .  (Am)
. C . .
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jrus
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Nov 20 2016, 08:27 AM #10

Fair enough. I don’t know anyone who does general-purpose computation by interpolating between values of the R40 scale, noting the value of the logarithm, doing simple arithmetic among the logs, and then interpolating back. But I suppose someone could do so. Remembering all 40 facts out to 3 digits seems like a pain.

* * *

I think trying to explicitly fit n steps into each power of twelve would be a mistake. As I said, I think powers of two are much easier for humans to reason about.
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wendy.krieger
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Nov 20 2016, 10:14 AM #11

When you place the third-ripple regular mantissa in base 120 onto the table of 83, all but six of the spaces are unoccupied. These fall in the fourth ripple. Here is a sample filled out from memory. The yellow spots are one of a dual. The second examples are 105 210 420, in the second row 1,1340 2,2680 4,5340, in the third set 2680, 5340, V680, and in the fourth set, 2896, 5772, E524. In base 180, this is R90. It's like 10, and 45 in that the one-place logs include the regulars.
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Shaun
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Nov 20 2016, 10:15 AM #12

Dublogs are described in Tom Pendlebury's TGM, chapter 8.

TGM booklet

and tables from

Dublog Tables
I use the following conventions for dozenal numbers in my posts.

* prefixes a dozenal number, e.g. *50 = 60.
The apostrophe (') is used as a dozenal point, e.g. 0'6 = 0.5.
T and E stand for ten and eleven respectively.
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wendy.krieger
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Nov 20 2016, 10:39 AM #13

Even the decimal logs can be extralopated to reasonable precision from the decibel scale.

One supposes, for example that log 2 is 0.3 + x, and log 5 is 0.7 - x. The table gives a form in C, by average, where the x's cancel. So the first row would be 2.1+7x, and 2.1-3x. We multiply the first by 7, and the second by 3, and divide by 10 to get 125.9, which is the mantissa of 2.1, The calculated value is in C, the exact value by logrithm is in D.

Code: Select all

          A        B          C             D
   0    1        1          1000         1000
   1    128      125        1259         1258.925
   2    16       15625      1585         1584.893
   3    2        1953125    1995.3125    1995.2623
   4    256      25         2512         2511.8864
   5    32       3125       3162.5       3162.2776
   6    4        390625     3981.25      3981.071
   7    512      5          5012         5011.872
   8    64       625        6310         5309.573
   9    8        78125      7943.75      7943.282
This is the start of the binary logs, written to base 12. The interval here is from 6 to 12. Each one represents a power of 3 on the left, normalised to the range 6-12. So the 9 represents '3', since it is 3/4. 69 represents 9/16, and the powers of 3 count in multiples of 7 in the first column. If you add the two on the 6-row, you get

.86623+.851768 = 1.4E7998 where sqrt(2) is 1.4E7917

For the column, add (5n A+7n B ) where n is the semitone number, reduced modulo 12, and A and B are the left adjusted number. You will get the 2^0.1 to 2^0.E to four places. For row 2, you would get (5*2)*.69+(7*2)*.67VE... = V*.69+2*.67VE...

Code: Select all

   0    6
   1    64V783     63V28
   2    69         67VE00EV28
   3    725E7946   714
   4    7716       75V94714
   5    813873069  8
   6    86623      851768
   7    9          8V668139768
   8    973E646    9594
   9    V16        9EV461594
   V    V98E5809   V8
   E    E483       E2V20V8
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jrus
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Nov 20 2016, 12:55 PM #14

Thanks, Shaun.

I didn’t know the “dublog” keyword. That turns up a few more threads:
http://z13.invisionfree.com/DozensOnlin ... =414&st=30
http://z13.invisionfree.com/DozensOnlin ... wtopic=827
http://z13.invisionfree.com/DozensOnlin ... wtopic=948
http://z13.invisionfree.com/DozensOnlin ... topic=1334

Still surprisingly few, and not the fullest discussion.

* * *

For anyone who doesn’t use “dublogs” in daily life or routine estimation, let me recommend it. Also a great quick tool to sanity-check computations done some other way.
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Kodegadulo
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Nov 20 2016, 02:01 PM #15

Sorry to go off topic, but how did you manage to do a search of DozensOnline to find references to "dublogs", jrus? I've found that the Search function on this site is pretty useless.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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jrus
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Nov 20 2016, 02:04 PM #16

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jrus
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Nov 20 2016, 02:11 PM #17

By the way, working with logarithms is one of those cases where using a “balanced” base is really helpful. That is, using digits from –6 to +6 instead of from 0 to e. In particular, when the characteristic goes negative the mantissa doesn’t change, but you don’t need a weird convention like putting a bar over the characteristic. And yet you still get an obvious additive inverse relationship between the logarithms of reciprocals.
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DavidKennedy
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Dec 4 2016, 06:04 PM #18

jrus, Thanks for this post about how dozenal can be used with logarithms to simplify the calculations of estimates. This is one of the best arguments for uncial that I have come across.

With a knowledge of the number of semitones corresponding to where the harmonics occur in the musical scales, it is possible to use this technique to estimate such calculations as the extraction of roots by simple division, and multiplication or division by simply adding or subtracting the number of semitones of the harmonics, using the laws of logarithms, and a knowledge of the frequency ratios corresponding to the intervals to convert back from logarithms to the estimate as a fraction.
wendy.krieger @ Nov 20 2016, 07:46 AM wrote: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,
What Wendy means by lg is the logarithm whose base is the semitone or the twelfth root of two, or twelve times the logarithm to the base two.
Oschkar @ Nov 20 2016, 08:14 AM wrote:R40 is practically identical to the chromatic scale, though. Breaking up a ratio of ten into 40 equal steps is almost the same as breaking up a ratio of two into a dozen equal steps.

101/40 ≈ 1.059253725
21/12 ≈ 1.059463094
An approximation to the semitone can be formed by any number having a root approximating the number two; for example, the seventh root of the gross. This has a connection to efficient bases for denominations of weights, as I explained at Base Denomination Efficiencies, for weights and currencies.
-- David Kennedy
Double negative is a plus
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jrus
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Dec 5 2016, 08:11 AM #19

DavidKennedy @ Dec 4 2016, 06:04 PM wrote:With a knowledge of the number of semitones corresponding to where the harmonics occur in the musical scales, it is possible to use this technique to estimate such calculations as the extraction of roots by simple division, and multiplication or division by simply adding or subtracting the number of semitones of the harmonics, using the laws of logarithms, and a knowledge of the frequency ratios corresponding to the intervals to convert back from logarithms to the estimate as a fraction.
Right. Exactly so.

I’m somewhat curious what other tricks would be developed for working with "dublogs", e.g. converting from dublogs <-> regular duodecimal fractions, after a few years of tinkering and practice.

In general the greatest practical difficulty with logarithms is figuring out how to add them, even approximately.

To find log(x + y) given log(x) and log(y) requires something like:

z = log y – log x
log(x + y) = log(x) + log(1 + exp(z))

Being able to get a good quick estimate accurate to 1% or so for log(1 + exp(z)) given any dublog z would be nice. Basic estimation skills can be learned through practice, but achieving accuracy will be a bit tricky.

If the numbers represent simple ratios (say, 1:1 or 3:2) it’s of course going to be easy in our dublog framework. But I dunno about arbitrary inputs without having a substantial table or slide rule at hand.
What Wendy means by lg is the logarithm whose base is the semitone or the twelfth root of two,
Right. Though to be precise I think Wendy means the 83rd root of 120 as the base, rather than the 12th root of 2 or the 40th root of 10. These three numbers are of course very close, 1.059377d, 1.059463d, & 1.059254d, respectively.
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wendy.krieger
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Dec 5 2016, 09:04 AM #20

The 83-scale is a twelfty version of R40, the dozenal version would be 3.7 in order (three-tulf seven).

These are meant, not as much for musical scales, but as sizes for things like resistors etc, the documentation on R40 gives intent.

The decibel scale is replicated by the the scale of 21, which consists of the 15 proper divisors of 120, along with 1¼, 1½, 2½, 48, 80 and 96. This divides the circle into 21 partitions, at ratios of 6/5, 5/4 and 4/3.

The scale of seven is the points 1, 2, 4, 8, 15, 30, and 60, as used of weights.

The scale of six are the coins (1, 2, 5, 10, 30, 60)

The scale of three is 1, 5, 24.

The scales go even finer too.

The scale of 1181 reckons 171 divisions to the octave, includes 7-smooth numbers.

The scale of 712577 is the finest scale i normally deal with. This includes the seven-smooth numbers too, but its range is some 40,000,000, that is, it is 126 decimals before the fine bands intersect.
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jrus
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Dec 6 2016, 04:53 AM #21

Wendy: For me, such a scale (ideally fitting a whole number of steps into a doubling/octave) is just perfect if it just doesn’t bother with the integer rounding in terms of 10 / 120 / whatever. The rounding is a helpful memory aid, but I want the actual measurements to remain precise to the log scale. That is, take the concept of equal temperament from music and apply it to whatever other field. [Funny enough, music itself is one area where I like the sound of other temperaments better.]
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