The process is called the reciprocal divisor method (RDM) for abbreviated multiplication tables (AMT). See http://www.vincico.com/arqam/RDM2007.pdf for a fuller explanation. I do not know if I discovered this method. Something similar was in play in the middle east long ago, but I am not sure it is exactly this method. If you read the link, I apologize because it can be ponderous.

Note: I am, it will be evident, an "amateur mathematician" woefully bereft of academic training. (Gimme a break!) However, the technique works. Yes there are errata which were corrected but I cannot find the file. This file dates from autumn 2007. I am enhancing it, having learned more between now and then.

Briefly, no, the process does not make sexagesimal more efficient than human scale bases for which one may memorize the entire mutliplication table (the normal method of multiplication I refer to as "direct multiplication" or DM). DM is highly efficient. What RDM does is enable you to wield bases that are mid scale (meaning larger than human scale, about base 16, let's say, to the point when the AMT is comparable in size to the full multiplication table of the largest human scale base). The mid scale bases run between 16 and 120. The size of a base r's multiplication table m is given by m=(r^2 + r)/2. Thus, decimal m = 55, dozenal m = 78, hexadecimal m = 136. The size of the AMT of base 60 is 104. RDM enables sexagesimal to be wielded and used as a tool, when necessary; it does not render sexagesimal more useful than other bases.

Also, the question of symbols, symbols are just tools. Like a mountainclimber, use the tools you desire or trust. I use my tools because, well, it's my idea. You don't have to use my symbols. Use whatcha like.

The process relies on heavily entrained multiplication tables, in other words, highly composite bases. These include the following:
{24, 30, 36, 48} = the lower mid scale.
{60, 72, 84, 90, 96, 108} = the upper mid scale.
{120, 144, 168, 192, 216, 252, 300, 336, 360} = the lower large scale (it is only really feasible with 120).

Two facts contribute to the operation of the RDM technique:
1. Highly composite bases feature multiple pairs of reciprocal divisors (each divisor can be multiplied by another divisor to yield the base: d x d' = r). Each pair of divisors create "avenues" within the multiplication tables of these bases by which we can reach any product in that table by one to three operations. Thus, all we need to know are the avenues in order to wield the base.

2. Every multiplication table is symmetrical and modular. If we can predict the periodicity of the end digits of the products for any factor, and determine the correct first rank digit (i.e. the first power of the base) for each product, we can completely ignore all products in the multiplication table larger than the first period. The period I define as the instance when a divisor's product yields a multiple of the base. In dozenal, the first period for the divisor 3 is reached when we multiply it by 4.

The AMT is produced by truncating the multiplication table to the first period, including no value larger than the base. Additionally, we only need one half of the MT for it to operate.

Don't be annoyed, I am sharing something with you and need multiple posts. Don't like it, don't read it. This post is not meant to offend or put off, to boast or to claim that I discovered anything; it's really a friendly gesture among fellow aficionados. :) enjoy...

TWO ESSENTIAL TOOLS
(I don't have the luxury of sexagesimal symbols so I'll use the convention which delimits digits by semicolons (;), a colon (:) indicating the unit point.)

In order for the technique to work you need to be familiar with two things:

1. the reciprocal divisor pairs of sixty:

{1, 60}
{2, 30}
{3, 20}
{4, 15}
{5, 12}
{6, 10}

The members of the pairs, when multiplied, yield sixty. One acts as a counterweight to the other. This is the crux of the technique; you cannot use the technique if you do not know these pairs.

Next bothersome post: the first and second techniques to use the RDM.

2. The abbreviated multiplication table:

This is the sexagesimal multiplication table truncated to the first "period", or the first instance of a product for any factor equalling the quantity of the base r in play.

For the "1s" facts, we can ignore figures greater than r/2, since the 1s facts simply function as an index. (It should be obvious that 1 x 46: is 46: :) )
Also, we do not need facts "east" of the line of squares 1-4-9-16-25-36-49... since they are repetitions of the facts "south" of the same diagonal line. However, we can retain the facts to 7, since it facilitates use of the table. This way the 1s on the top of the table can function as an index as well.

EASY STUFF

Class A operation: This table functions as a normal multiplication table: to multiply, find the product by selecting the first factor in either the left or the top indices. 04: x 13:, simply locate the larger first, using the left index, and move over till you are directly below the 04: in the top index. The product is 52:.

Example 1: Product in the abbreviated table.

Multiply 12: by 04:.

This appears in the abbreviated table. the result is 48:. :)

Class B operation: For products that are "off the map" you will need to use one of three additional techniques. The simplest of the techniques occurs when you are multiplying using a divisor of sixty. Instead of multiplying by the divisor, you divide by its reciprocal, obtaining a quotient and a remainder. The quotient is the next-highest digit, while the remainder multiplied by the divisor is the digit in play.

Example 2: Product involving a divisor. (the core reciprocal multiplication technique).

Multiply 01;40: by 10:.

Note that 10: is one of the twelve divisors of sixty. Its reciprocal divisor is 06:, since 06: x 10: = 01;00:.

01; x 10; appears in the appreviated multiplication table, the product being less than or equal to 01;00:. Using the abbreviated table, we get 10;00:, adding the 00: in the units position as in traditional multiplication.

40; x 10; will result in a product greater than 01;00:, thus we use the reciprocal divisor of 10;, 06;. 40; divided by 06; is 06; with a remainder of 4. Retain the 06; in the first power of sixty, but multiply the remainder with 10; to obtain 40; in the zeroth power of sixty. For the digit 40;, we have a product 06;40:.

Add the 10;00: and the 06;40: to get 16;40:, the answer.

Proof: 01;40: decimally is 100, while digit 10; is 10. 100 x 10 = 1000.
16;40: = (16 x 60^1) + (40 * 60^0) = 16 x 60 + 40 = 960 + 40 = 1000.

Class C operation: For multiples of the divisor which are themselves divisors, e.g. 24 = 2 x 12, or 32 = 2 x 2 x 2 x 2 x 2, you may divide the problem into several parts and apply class A to each of the parts. Note that the goal isn't to produce burden: the class C operation on the factor 32 = five repetitions of multiplication by five may be more easily handled by 4*4*2 or a class D operation. (multiplicative breaking of the problem)

Example 3: Product involving a multiple of a divisor which is also a divisor.

Multiply 14;24: by 24:.

Note that 24: is double the divisor 12;. So we'll break the problem into two segments. We can either treat it as:
2(24;14: x 12;) or 12:(24;14: x 02:). Let's use the former.

the reciprocal divisor of 12: is 05:, since 12: x 05: = 01;00:.

14;00: x 12: : divide 14: by 05: to get 02: remainder 4. Multiply the remainder by 12:. We get 02;48;00: as the result for 14;00: x 12:.
24: x 12: : divide 24: by 05: to get 04: remainder 4. Multiply the remainder by 12:. We get 04;48: as the sub-product for the digit 24:.
Add 02;48;00: and 04;48: to get 02;52;48.

Now we need to reapply the factor 2 to the result, since we halved the multiplicand.
02;52;48: x 02: :
02;00;00: x 02; appears in the AMT - it's 04;00;00:.
52;00: x 02: : divide 52: by 30: to get 01: remainder 22: ; multiply 22; by 02; to get 44;. We get 01;22;00: as the result for 52;00: x 02:.
48: x 02: : divide 48: by 30: to get 01: remainder 18:. Multiply 18; by 02; to get 36;. We get 01;36: as the result for 48: x 02:
Add the results to get 05;45;36:.

14;24: decimally is 864, while 24: is 24. Thus 864 x 24 = 20736.
05;45;36: = (5 x 60^2) + (45 x 60^1) + (36 x 60^0) = 18000 + 2700 + 36 = 20736.

Class D: Product involving totatives or any multiple of totatives. (Additive breaking of the problem)
The totatives of sixty, in their pairs, are:
{01, 59} {07, 53} {11, 49} {13, 47} {17, 43} {19, 41} {23, 37} {29, 31}
Totatives are digits which are relatively prime to the base r. They are either "unrepresented primes" or UPs which do not appear in the unique prime factorization (UPF) of the base r, e.g. 7 in base 60: 7 doesn't appear among {2, 3, 5}, or they are "products of unrepresented primes" or PUPs, all the factors of which do not appear in the UPF of r, e.g. 49 in base 60: neither 7 nor 7 appear among {2, 3, 5}. All bases greater than 30 will possess at least one PUP. (12 does not possess a PUP digit, but 10, 16, etc. do). You don't need to be familiar with totatives but it could help. The class D operation is a general catch-all. I'd use it to handle 32; times x, since 32;x can be broken into 30;x + 02;x.

Example 4: Product involving totatives or any multiple of totatives.

Multiply 05;43: by 49:

The problem involves totative digits.

Let's break the problem into two parts. 05;43: x 49: = 05;43: x 48: + 05;43:.
Now we have a class C operation to which we need to add 05;43: at the end of the process.
05;43: x 48: = 4(05;43: x 12:).
05;00: x 12: = 01;00;00:.
43: x 12; = [43:/05: = 08; (r 3 x 12: = 36:)] 08;36:
01;00;00: x 08;36: = 01;08;36:.
Now reapply the factor 4:
4 x 01;08;36: = 04;34;24:
Now reapply the 05;43: we subtracted from the class D problem:
04;34;24: + 05;43: = 04;40;07:.

Proof: 05;43: is 343 decimally, the cube of 7, while 49 is the square of 7. The answer is the fifth power of seven, 16807.
04;40;07: = (4 x 60^2) + (40 x 60^1) + 7 = 14400 + 2400 + 7 = 16807.

Oh cute! my 144th post! that's 02;24: in base 60, when we convert lol (sorry ruthe!) BTW this is the end. Enjoy.

More stuff to think about:
Here are two basic base-60 studies developed in 2007.

The first is a test of the original sexagesimal font, a sexagesimal multiplication table at:

http://www.vincico.com/arqam/112-DMT.pdf.

The second is a study of factors, which was produced using the Reciprocal Divisor Method on several flights, waiting for planes, etc. This was then entered into excel. This can be found at:

http://www.vincico.com/arqam/112-FactorStudy.pdf.

The document multiplies the three prime factors of sixty, considering 2 up to the 24th, 3 up to the 18th, and 5 up to the 12th power, as well as the products of these powers, 2 x 3, 2 x 5, and 3 x 5. This study was inspired by a similar dozenal study which appeared recently in the Dozenal Journal, but considered the products of the powers of 2 and 3 to about the 12th power. The chart took 8-10 months to produce. (I guess my version of a crossword puzzle. Once it's done though what else to do on planes?). This is the reason for my previous thread about patterns, have you ever seen these patterns, especially in the end digits, as the products are continually multiplied by one another? how the end digits of the products of the powers of 3 and 5 end in digit 15 and digit 45, alternating along the 5-axis?