Ratios of Decimal and Dozenal Powers

Whereas when a single base of numeration may be used, its basic arithmetic could be done mentally quickly provided certain tables such as for multiplication be learnt, while two bases co-exist such as when one is being for a period of time supplanted by the other, there would be a benefit to more rapid mental calculation in learning additional facts, over those of a single base, for the purpose of conversion between the two bases until one of them becomes the exclusive base.

While conversion between powers of the two bases could be done using traditional methods or by computing devices, the use of those methods might hinder the development of effective numerical ability and comprehension of the relative meanings of various orders of magnitude that could be expected to be encountered in commerce and science. Some of this knowledge could be expected to be acquired vaguely with experience in trade and conversion. However, to speed up the facility with interconversion for applications requiring somewhat more reliable results, learning some values from the start could provide an accelerated advantage.

The power of the base twelve raised to a same exponent as the base ten can differ substantially from the decimal power. When dozenal is being introduced, people ought to understand what a dozenal power would be worth in terms of the decimal quantities with which they were beforehand acquainted. For example, a person ought to be able to fairly instantaneously gauge how much a million is in dozenal.

For fast comparison of decimal orders with dozenal orders, memorisation of some of their approximate ratios could be helpful for calculating their correspondences rounded to about one and a half positional places of accuracy or significant figures. Two numerical characters may be an appropriate number of significant figures for numbers to be considered acceptably rounded in a base of numeration.

Here follow some suggested values of ratios between dozenal and decimal powers. The values are chosen for their utility through simplicity in calculation. In many cases, there are alternative values that could have been used, and there may be circumstances in which one ratio or another might be more effective or convenient.

In these tables, cells the backgrounds of which are coloured blue are those which are most recommended to be memorised. Those in red can be memorised too, although they are less accurate. The ratios in cells with backgrounds of other colours can be derived from the values in the blue or red cells. Numbers in green cells can be derived vertically from numbers in blue cells by dividing by twelve. Chartreuse cells contain ratios derived horizontally from blue or green cells through changing by a factor of ten. Contents of orange cells are derived from red cells. Yellow cells are derived from chartreuse or orange cells.

The ratios of decimal to dozenal powers in the first table are the easiest to remember. There, the ratios of powers of the base ten with even exponents or indices to the dozenal power of index one unit less approximately follow the sequence of powers of the seventh root of twelve. These as denominations rounded, in most cases to the nearest whole number, are about {8, 6, 4, 3, 2, ~4/3, 1}.

First Table of Ratios of Decimal and Dozenal Powers

Exponent of Twelve	Exponent of Ten
0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	5:6	8:1
2		7:10	7:1
3			3:5	6:1
4				1:2	5:1
5					2:5	4:1
6						1:3	10:3
7							3:10	3:1
8								1:4	5:2
9									1:5	2:1
10										1:6	5:3
11											2:15	4:3
12												1:9	9:8
13													1:10	1:1
14														1:12

Second Table of Ratios of Decimal and Dozenal Powers

Exponent of Twelve	Exponent of Ten
	15	16	17	18	19	20	21	22	23	24	25	26	27	28
13	9:1
14	4:5	8:1
15		2:3	20:3
16			5:9	16:3
17				4:9	9:2
18					3:8	4:1
19						1:3	10:3
20							4:15	8:3
21								2:9	9:4
22									3:16	15:8
23										3:10	3:2
24											1:8	5:4
25												1:10	1:1
26													1:12	7:8

Third Table of Ratios of Decimal and Dozenal Powers

Exponent of Twelve	Exponent of Ten
	29	30	31	32	33	34	35	36	37	38	39	40	41	42
26	9:1
27	3:4	15:2
28		3:5	6:1
29			1:2	5:1
30				5:12	21:5
31					7:20	7:2
32						3:10	3:1
33							1:4	5:2
34								1:5	2:1
35									1:6	5:3
36										1:7	4:3
37											1:9	6:5
38												1:10	1:1
39													1:12	4:5
40														1:15

A more efficient method is to use order-of-magnitude at semitone level.

A dozen (43) is 3 semitones more than a decimal (40), and you could start with the semitone value of the number.

So 22 million, (decimal), is 2.2 E7. E7 needs 21 semitones to give D7, but there's about 13 available. So it's 8 deficient. 43-8 gives 35 st, which is 7½ (1 semitone less than 8), and then it's D6, since we borrowed a place.

Semitones are about the finest ratio people can discriminate, and they cover all sorts of different bases, including base 120, 10 and 12. The count up to the base is identical in all cases, so 6 is always 31 semitones, and 5 is 28 semitones.

wendy.krieger wrote:E7 needs 21 semitones to give D7,

"E7" meant ten to the power of seven, in the style of some calculator displays. The nearest semitone of the scale of twelve geometrically equal notes in the octave to the tenth harmonic is the fortieth over the fundamental frequency, that is, an interval of three octaves of twelve semitones each plus four more semitones. Wrapping this interval around the octaves gives a major third compounded on the three octaves. Letting the fundamental note be C musically, the tenth harmonic would be at the note E, approximately in equal temperament, but exactly a major third over C in just intonation. The justly intoned major third is diminished from the equally tempered major third by about a sixth to eighth root of a semitone, or a hundredth part, where a hundred is taken to mean the second power of some base around nine to twelve. In the raising of ten to the power of seven, this discrepancy accumulates each time and the interval drifts from the number of semitones, four times seven, of the compounded major third note such that by the time of a cycle of seven major thirds about a semitone has been lost. So, it is now about a minor third, at the note D sharp, compounded.

"D7" means the seventh power of twelve. The twelfth harmonic is at about three dozen plus seven semitones above the fundamental note. Wrapping this around the octaves gives a perfect fifth compounded with the three octaves. The number of semitones for twelve to the power of seven is given by a cycle of perfect fifths, and this gives the note C sharp, ten semitones plus an octave equalling 22, not 21, semitones above the note D sharp from ten to the power of seven.

It is of course much easier to see where in the scale of notes a number of semitones will be if that number be written dozenally. So, 22 semitones is a dozen plus ten semitones, and the interval is ten semitones compounded with one octave of twelve semitones, and that is a minor seventh interval compounded.

If one weren't multiplying by 2.2 and just wanted to know how many seventh powers of ten there are to seventh powers of twelve, the frequency ratio of the interval corresponding to 22 semitones being about a minor seventh compounded is two to the power of 22 twelfths, or two to the power of eleven sixths, a sixth root of two being a whole tone. The fourth harmonic is at two dozen semitones over the fundamental harmonic, and the third harmonic is at about 19 semitones above the fundamental frequency, so 22 semitones has a ratio of frequencies between three and four. One could guess three-and-a-half, but knowledge of the ratios of frequencies of intervals in just intonation provides a surer estimate from a minor seventh compounded with the octave as twice the square of the ratio of a perfect fourth interval. Alternatively, a simpler ratio to use for this interval is twice nine fifths, but this to match semitones ought to be diminished by about a hundredth part. These rational numbers become estimates for the ratio of the seventh powers of ten and twelve. Ten to the power of seven as a multiple of the seventh power of twelve is then the reciprocal of this rational number of times, so ten to the power of seven is approximately nine over the fifth power of two times the seventh power of twelve.

wendy.krieger wrote:22 million, (decimal), is 2.2 E7

2.2 is 22 over ten, or eleven fifths, so the number of semitones for 2.2 is the difference between the numbers of semitones of the eleventh and fifth harmonics. The fifth harmonic being nearly four semitones above the fourth harmonic, is at 28 semitones, the last one slightly diminished. The eleventh harmonic is between the tenth and twelfth harmonics, but closer to the twelfth of the 43 semitones, so the eleventh harmonic is nearer to 42 than 41.

wendy.krieger wrote:there's about 13 available.

Subtracting from 42 semitones the 28 of the fifth harmonic gives 14, not 13, as the number of semitones for the ratio 2.2. But 2.2 is a little less than 2.25, the ratio nine quarters, of 14 semitones belonging to a ninth interval, or whole tone second interval compounded with an octave.

wendy.krieger wrote:So it's 8 deficient.

This number 2.2 is multiplied by the pre-exponential factor nine over the fifth power of two. That is the same as dividing 2.2 by (2^5)/9. The division can be done by a subtraction between the numbers of semitones of the ratios, so it is 14 minus 22 equalling minus 8 semitones, the interval of which is a minor sixth, with a ratio of about eight fifths, below the unison, meaning that is inverted to the reciprocal five eighths, which is the number of times of the seventh power of twelve to twice eleven million. Taking out a power of twelve from the seventh gives twelve times five eighths, which is half of three fives, or seven and a half, of sixth powers of twelve to twice eleven million.

Actually, a ratio of eight fifths being a minor musical interval has to be diminished by about a hundredth part to become eight semitones. Ratios for major intervals would be augmented by the same ratio to become tempered to the equal temperament of twelve semitones to the octave. A good ratio to use for a hundredth part augmentation is eight dozen over 95, or 95 over eight dozen for the diminution, but an augmenting ratio of 100/99 can be used where the numerator of the other ratio has a prime number eleven, while 121 twelftieths can be used where the denominator of the other fraction contains the prime number eleven. Multiplying 8/5 by 95/96 gives nineteen twelfths. The nineteenth harmonic is a minor third interval, having a ratio of 19/16 which is the next convergent after six fifths to three equally tempered semitones represented by two to the power of three twelfths or a quarter, compounded with four octaves, so it is a minor sixth interval above the twelfth harmonic. This minor sixth interval ratio of 19/12 is just a perfect fourth above a minor third of 19/16.

The multiplication can also be done using vulgar fractions more directly. 2.2 is two plus a fifth, or eleven fifths as a vulgar fraction. 11/5 by 9/32 gives 99/160. This is more accurate. For quicker mental calculation, one might be more likely to use the simpler ratio without the diminution or augmentation step. In this case, the result would be 11/5 multiplied by 5/18, which is 11/18.

wendy.krieger wrote:The count up to the base is identical in all cases, so 6 is always 31 semitones, and 5 is 28 semitones.

The semitone method replaces memorisation of some ratios of the powers of the two bases decimal and dozenal with memorisation of the numbers of semitones corresponding to a host of numbers. Some smaller numbers, such as seven and eleven, do not align very well with semitones, leading to inaccuracy in these calculations, while it is not so easy to know the number of semitones for large numbers. One could first have to find the nearest power of two. The amount of memorisation for converting between powers of decimal and dozenal to a range without semitones may not be as much as for the memorisation of semitones for the same range. The calculations without semitones involve mainly just changing by factors of ten or twelve and vulgar fractions which is basic arithmetic that may not be more complicated than the use of semitones. Nevertheless, for some years I have been using semitones and justly intoned ratios in a similar way for estimation calculations. Despite that, it was thought that it would be faster to do calculations by learning some correspondences of decimal to dozenal powers. Those learnt facts require no calculations at all, and the speed would be limited only by that of recall.

More accurate results are possible when a finer partition of the octave than twelve notes is used. Partitioning of each semitone into twelve is even more fervently dozenal than the semitone scale, and is more in tune with the true ratios of just intonation. Music played with frequency ratios closer to the justly intoned simple ratios sounds better than when the music is in equal temperament. There is more extreme dozenism in using just intonation rather than an equal temperament of twelve semitones.

The semitone is extensive enough, and exact enough, to handle the 2-3-5 numbers.

When one deals with bases with three or more prime divisors, it is better to find some sort of log scale (like the semi-tones), where numbers can occupy boxes or cells in a table. I have never yet managed by hand to make a regulars table simply by listing the values. It needs something more.

For example, with 70, we put lg2 = 31, lg(5)=72 and lg(7)=87. You still get dual occupancy in a cell (as follows). The formatting has been lost. The stars refer to multiples of 3125v3136, there were originally ten cells 0-9 etc, across the table, and 19 rows.


0 1 2 3 4 5 6 7 8 9
0 1 10130 10310 10828 110
1 11735 **2 12130 12130 128 130 13224
2 142 14420 14702 15235 155 158
3 16714 2 20260 20620 21656 220
4 23135 235 **4 24260 256 260
5 314* 31840 32404 33007 335 340 346
6 36428 4 40550 41240 **7 440
7 463 5 **8 51550 542 550 55926
8 61735 628 * 63710 64804 66014 7 710 722
9 75656 8 81130 82510 85235 865 * 910
10 956 10 **16 103130 1114 1130
11 114152 121735 1235 1256 * 130420 132608 135028 14 1420
12 1444 154342 16 * 162260 165020 171035 1735 1760 *
13 1820 1942 20 **32 206260 2228 *
14 2260 231334 240049 2435 25 2542 * 260840 265216 273056 28
15 2840 * 2918 **49 32 * 324530 333040 3421 35
16 3550 * 3640 3914 40 **64 415530 435235
17 ** 4550 462668 480128 49 50 5114 * 521710 533432 546142
18 56 5710 5836 611735 6235 * 64 * 652060 666110 6842

** 4445 4456 **8 50710 50828 **25 156605 16
**2 11920 11942 **10 62630 628 **28 1760 176428
**4 23840 23914 **14 865 86714 **32 200840 203442
* 31315 314 **16 100420 101656 **35 222235 222800
**7 43235 43342 **20 125260 1256 **40 253550 2542

**49 311735 312514 **98 6235 625028
**50 316210 32 **100 635420 64
**56 3550 355856
**64 405710 406714
**80 510130 5114

If one could decipher this, one might be hireable by an intelligence agency as a decrypter. There is a topic for how to write tables on the forum, https://dozenal.forumotion.com/t31-tables.

An increment about the square dozenth root of the square of five has powers approximating well to the prime numbers two, five, and seven. Its approximation to the third harmonic is less accurate. If powers of the increment approximate to the primes, their multiples to small powers will be approximated as well.