# Tonal system

Nystrom's "Tonal Watch, or Clock-dial". Note the use of his invented digits for hex values 9-F. Also note that midnight (hour 0) is at the bottom, rather than the top of the clockface.

The Tonal system is a base 16 system of notation (predating hexadecimal), arithmetic, and metrology proposed in 1859 by John W. Nystrom. In addition to new weights and measures, his proposal included a new calendar with sixteen months, a new system of coinage, and a clock with sixteen major divisions of the day (called tims).

"I am not afraid, or do not hesitate, to advocate a binary system of arithmetic and metrology. I know I have nature on my side; if I do not succeed to impress upon you its utility and great importance to mankind, it will reflect that much less credit upon our generation, upon scientific men and philosophers."
(Quotation: John W. Nystrom, ca. 1863)[1]

## Names for the numbers

He proposed names for the digits, calling zero "noll" and counting (from one to sixteen):

"An,  de,  ti,  go,  su,  by,  ra,  me,  ni,  ko,  hu,  vy,  la,  po,  fy,  ton." (Therefore tonal system.)

Because hexadecimal requires sixteen digits, Nystrom supplemented the existing decimal digits 0 through 8 with his own invented characters. (These can be seen on his clockface at right.) Later, the hexadecimal notation overcome this same obstacle by using the digits 0 through 9 followed by the letters A through F.

The numbers 1116 and 1216 would be said "tonan", "tonde", etc. The table below shows Nystrom's names for successive powers of 1016.

Base 16 Number Tonal Name Base 10 Equivalent
10 ton 16
100 san 256
1000 mill 4,096
1,0000 bong 65,536
10,0000 tonbong 1,048,576
100,0000 sanbong 16,777,216
1000,0000 millbong 268,435,456
1,0000,0000 tam 4,294,967,296

Thus, the hexadecimal number 1510,0000 would be "mill-susanton-bong". This first hexadecimal system, proposed in the 19th century, has thus far not achieved widespread usage.

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## Geography

Nystrom's hexadecimal compass

For latitudes he put 0 at the North Pole, 4 at the equator and 8 at the South Pole. The units were called tims. They are the same as the colatitudes measured in turns times 16.

Tonal (in tims) ISO 6709 Colatitude (in degrees) Colatitude (in turns)
0 090 0
1 67.5
2 045 45° 0.125
3 022.5
4 000 90° 0.25
5 -22.5
6 -045 135° 0.375
7 -67.5
8 -090 180° 0.5

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## Music

A page in the tonal system book

In his book he made a reference to music notation, where binary division is already in use.

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## References

1. ^ The Art of Computer Programming section 4.1, Donald Knuth.
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Last modified on 21 April 2013, at 21:04