The Tonal System, a Base 16 System of Counting

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Swedish-American civil engineer John William Nystrom (1824–1885, born Johan Vilhelm Nyström in Sweden) developed a base 16 system of counting while residing in Philadelphia, Pennsylvania. He called this new system of counting the Tonal System after his new word for 16, ton (after the English word "ten"). He greatly elaborated on his new system in 1862 in a 106-page book, Project of a New System of Arithmetic, Weight, Measure and Coins, Proposed to Be Called the Tonal System, with Sixteen to the Base. Reprints of this book are still available. This system might provide enjoyment for today's computer scientists and others who are used to working in hexadecimal.

He formulated what today would be called a base 16 (or hexadecimal) arithmetic system. The Tonal system introduced several new digits to allow each digit to represent values from 0 through 15. To differentiate them from ordinary decimal numbers, he gave these digits new names. Powers of 16 were also given new names, so as not to be confused with the tens, hundreds, thousands, etc. of decimal arithmetic.

This work was done at a time when a growing interest was developing for an international decimal system coordinated by the International Association for Obtaining a Uniform Decimal System of Weights, Measures, and Coins. John Nystrom presented his ideas to this "International Decimal Association" in England in 1859.

Counting in the Tonal System

John Nystrom gives the following pronunciation guidance for his digit names: "The names of the Tonal figures are contained in the following four words, Andetigo, Subyrame, Nikohuvy, Lapofyton, which should be learned by heart. The vowel y in these names should be pronounced as in the English word cylinder, i as in will, e as in then, a as in all." Those four words encapsulate the Tonal System digits 0x1 ("an") through 0xF ("fy"), and finally 0x10 ("ton").

His book states on p. 17 (decimal!): "The object of employing different consonants to the names of the figures is to render it more difficult to alter a written number from one value to another ; it will also make the expression clearer."

These are the 16 digit names with their glyphs:

Tonal System Digits

Unicode CSURGlyphHexadecimalName
U+E8E0 noll 0x0 Noll
U+E8E1 an 0x1 An
U+E8E2 de 0x2 De
U+E8E3 ti 0x3 Ti
U+E8E4 go 0x4 Go
U+E8E5 su 0x5 Su
U+E8E6 by 0x6 By
U+E8E7 ra 0x7 Ra
U+E8E8 me 0x8 Me
U+E8E9 ni 0x9 Ni
U+E8EA ko 0xA Ko
U+E8EB hu 0xB Hu
U+E8EC vy 0xC Vy
U+E8ED la 0xD La
U+E8EE po 0xE Po
U+E8EF fy 0xF Fy

These digits combine to form larger numbers with regular names following simple rules.

Counting to 100

Unicode CSURGlyphsHexadecimalName
U+E8E1, U+E8E0 an noll 0x10 Ton
U+E8E1, U+E8E1 an an 0x11 Tonan
U+E8E1, U+E8E2 an de 0x12 Tonde
U+E8E1, U+E8E3 an ti 0x13 Tonti
U+E8E1, U+E8E4 an go 0x14 Tongo
U+E8E1, U+E8E5 an su 0x15 Tonsu
U+E8E1, U+E8E6 an by 0x10 Tonby
U+E8E1, U+E8E7 an ra 0x17 Tonra
U+E8E1, U+E8E8 an me 0x18 Tonme
U+E8E1, U+E8E9 an ni 0x19 Tonni
U+E8E1, U+E8EA an ko 0x1A Tonko
U+E8E1, U+E8EB an hu 0x1B Tonhu
U+E8E1, U+E8EC an vy 0x1C Tonvy
U+E8E1, U+E8ED an la 0x1D Tonla
U+E8E1, U+E8EE an po 0x1E Tonpo
U+E8E1, U+E8EF an fy 0x1F Tonfy
U+E8E2, U+E8E0 de noll 0x20 Deton
U+E8E2, U+E8E1 de an 0x21 Detonan
U+E8E2, U+E8E3 de ti 0x23 Detonti
U+E8E2, U+E8EB de hu 0x2B Detonhu
U+E8E2, U+E8EE de po 0x2E Detonpo
U+E8E3, U+E8E0 ti noll 0x30 Titon
U+E8E4, U+E8E0 go noll 0x40 Goton
U+E8E5, U+E8E0 su noll 0x50 Suton
U+E8E6, U+E8E0 an noll 0x60 Byton
U+E8E7, U+E8E0 ra noll 0x70 Raton
U+E8E8, U+E8E0 me noll 0x80 Meton
U+E8E9, U+E8E0 ni noll 0x90 Niton
U+E8EA, U+E8E0 ko noll 0xA0 Koton
U+E8EB, U+E8E0 hu noll 0xB0 Huton
U+E8EC, U+E8E0 vy noll 0xC0 Vyton
U+E8ED, U+E8E0 la noll 0xD0 Laton
U+E8EE, U+E8E0 po noll 0xE0 Poton
U+E8EF, U+E8E0 fy noll 0xF0 Fyton
U+E8E1, U+E8E0, U+E8E0 an noll noll 0x100 San

Long number names use hyphens to separate components, just as in English we say, for example, "forty-two". Long numbers can be written with commas every four digits, which is every 16 bits. These commas extend into the written names of larger numbers, as shown in the table below. Numbers 0x200 and above are the examples that appear in Nystrom's book.

Large Numbers

an noll noll 0x100 San
an noll an 0x101 Sanan
an noll by 0x106 Sanby
an an noll 0x110 Santon
an an ti 0x113 Santonti
an de noll 0x120 Sandeton
an de ko 0x12A Sandetonko
de noll noll 0x200 Desan
de me fy 0x28F Desan-metonfy
ti la hu 0x3DB Tisan-latonhu
ra noll noll 0x700 Rasan
an noll noll noll 0x1000 Mill
de noll noll noll 0x2000 Demill
me hu vy su 0x8BC5 Memill-husan-vytonsu
an, noll noll noll noll 0x10000 Bong
vy, noll by an noll 0xC0610 Vybong, bysanton
an noll, noll noll noll noll 0x10 0000 Tonbong
an noll noll, noll noll noll noll 0x100 0000 Sanbong
an su an noll, noll noll noll noll 0x1510 0000 Mill-susanton-bong
an, noll noll noll noll, noll noll noll noll 0x1 0000 0000 Tam
an, noll noll noll noll, noll noll noll noll, noll noll noll noll 0x1 0000 0000 0000 Song
an, noll noll noll noll, noll noll noll noll, noll noll noll noll, noll noll noll noll 0x1 0000 0000 0000 0000 Tran
de, me la su hu, ra po noll fy 0x2 8D5B 7E0F Detam,


Any fraction that is a multiple of a reciprocal of a power of 2 (1/2, 3/4, 7/8, 5/16, etc.) has an exact representation in the Tonal System. Any fraction that is not is a repeating fraction. Here are some examples.

Decimal and Tonal Fractions

Decimal FractionTonal Fraction
1/64 = 0.015625 an/gonoll = noll.nollgo
1/32 = 0.03125 an/denoll = noll.nollme
1/16 = 0.0625 an/annoll = noll.an
1/8 = 0.125 an/me = noll.de
1/4 = 0.25 an/de = noll.go
1/3 = 0.333… an/ti = noll.sususu
25/64 = 0.380625 anni/gonoll = noll.bygo
1/2 = 0.5 an/de = noll.me
3/4 = 0.75 ti/go = noll.vy

Tonal Musical Notation

The second part of the Tonal System is a new marking for music clefs. The following table gives a summary of these new clef notations.

Tonal Music Clefs

U+E8F4canto clef CantoSopranoTwo octaves above treble pitch.
U+E8F3alto clef AltoContraltoOne octave above treble pitch.
U+E8F2treble clef TrebleDescantNormal female voice.
U+E8F1tenor clef TenorTenorOne octave below treble clef; natural voice of a man.
U+E8F0bass clef BassBassTwo octaves below treble clef; ordinary bass clef.

Tonal Time, Angles, and Length

To understand the note pitch indicated by the Tonal System musical notation, it is necessary to understand Tonal time. The Tonal System divides a circle into 16 parts. It applies this division to time (16 decimal hours in the day), and to longitude. This unit of division is the "tim". A circle is divided into annoll tims (16 tims decimal). The tim is further subdivided as follows:

Tonal Time Divisions

Circleannoll Tims
Timannoll Timtons
Timtonannoll Timsans
Timsanannoll Timmills
TimmillanMillimeter =
1.31836 seconds of time
or 19.77″ of a circle

The Tonal System defines the millimeter as the length of one tinmill of time of rotation at the Earth's equator.

Angles are given as an integer followed by two Tonal digits. For example, the angle su.meby is described as "sutim and metonby". [John Nystrom gives this example in his book on p. 33, but accidentally gives part of the number in decimal names.]

Tonal time is given as a whole number of tims, followed by a two-digit timton value. For example, the time ko.tihu is "kotim and titonhu". A third hand on the dial would measure timbongs (i.e., an/annollnollnollnoll of a day (or 0.082 of our seconds).

The illustration below shows a Tonal clock. Midnight is at the bottom, noon is at the top.

Tonal Clock

Tonal Clock

The "natural" pitch that John Nystrom describes is annoll (16 decimal) vibrations per tinmill on p. 33, but as. annollnoll (256 decimal) vibrations per tinmill on p. 45. Of these two, the latter value is likely what he meant. With 86,400 seconds in a day, this gives 86,400 / 166 ≈ 0.00514984 vibrations per second, which is a frequency of 194.18 Hz. This is approximately two semitones below 218 Hz, which is very close to concert pitch of 220 Hz for the A below middle A. (The ratio between semitones is the 12th root (decimal) of 2. On p. 47, the Tonal System gives the base pitch as 195 Hz [though 194 Hz is more accurate], approximately the pitch of F sharp.)

The Tonal System begins the natural music scale with the A note.

John Nystrom extolls the virtues of the Boehm flute as "the only wind instrument constructed on purely scientific principles, and which has attained perfection..." He mentions meeting Mr. Boehm in 1858. John Nystrom appears to have travelled far and wide in search of metrological perfection.

Tonal Money

The Tonal System also extends itself to monetary values. John Nystrom recommended dividing the dollar into 16 decimal shillings, and dividing each shilling into 16 decimal cents. Thus there would be annollnoll tonal cents to the dollar (256 decimal cents) and annoll tonal shillings to the dollar (16 decimal shillings).

Tonal coin denominations were recommended as follows:

Tonal Arithmetic Devices

The Tonal Counting Machine, shown below, is modeled after the abacus. It can expedite performing addition and subtraction using Tonal arithmetic, and even multiplication and division. It is constructed using brass wires in a square frame with tonal "balls" for counting. Line c is a "void line", meant to separate a fractional part in rows a and b from the whole part. Line h lies between mill and bong; in writing Tonal numbers, this line represents where one would place a comma.

Tonal Counting Machine (Abacus)

Tonal Abacus

To use the Tonal Counting Machine, begin with all the balls on the left-hand side, and move balls to the right as necessary. Numbers are read from bottom to top. Thus the number displayed above (reading the balls on the right-hand side from bottom to top) is ragohu.deni. If used in the shop or market to tally money, this could represent ragohu dollars, de shillings, and ni cents, with dollars on rows f through d, shillings on row b, and cents on row a.

Multiplication, Division, and Trigonometric Operations

Nystrom's calculator, shown below, facilitates multiplication and division of tonal numbers. He was awarded Patent Number 7961 (720 kB) for this device.

Nystrom's Calculator

A picture of the patent model appears below (©National Museum of American History):

Nystrom's Calculator

Quoting from Nystrom's book, p. 105: "This calculating machine consists of a silvered brass plate of about nine inches in diameter, on which are fixed two movable arms, extending from the centre to the periphery. On the plate are engraved a number of curved lines in such form and divisions that with their intersection with the arms, the most complicated calculations can be performed almost instantly.

"The arrangement for trigonometrical calculations is such that it is not necessary to notice the functions sine, cosine, tangent, &c., operating only by the angle expressed in degrees and minutes, and without any tables, which makes it so easy that any one who can read figures, will be able to solve trigonometrical questions. Any kind of calculation can be performed on this instrument, no matter how complicated it may be, whilst there is nothing intricate in its use."

To learn more on its operation, you can download the manual (20 Megabytes).

But wait! There's more!

The above descriptions of applications of the Tonal System are not exhaustive. For example, the book also goes into systems of weights and measures, and a year with 16 months. Those whose interest this page has piqued can search for the book; several sources publish it using print-on-demand.

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