What are Tonal Bitcoins? What are Tonal Bitcoins? In what application is this system of notation superior to the decimal notation? Are there any clients or other services supporting the tonal system?
After taking a quick look over the Tonal Book (free PDF) and the Wikipedia article it appears to be an early attempt at constructing a number system in base 16.  

Programmers will immediately recognise this as hexadecimal. For non-programmers hexadecimal uses the following representations: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. Note that decimal 10 to 15 are represented by the single letters A to F, such that FF is decimal 255 or (16 * 15)+(1 * 15). To differentiate a decimal number from a hexadecimal number the conventions 0x10 (decimal 16) or 10h are widely recognised.

The Tonal bitcoin representation attempts to offer an alternative representation of BTC known as TBC (Tonal Bitcoin) and uses the various Tonal letters to represent them. The Tonal alphabet is more complex than the hexadecimal system and requires special fonts to be available on the systems using it.

It is unlikely that this archaic approach to base 16 is ever likely to gain traction when much more widespread approaches are available.

Units covering powers of 2 in computing

It should be noted that in 1998 the International Electrotechnical Commission (IEC), Institute of Electrical and Electronics Engineers (IEEE) and the International Committee for Weights and Measures (CIPM) adopted a purely "powers of 2" approach for some use cases. Notably the representation of computer storage and data transmission rates. 

This gives rise to 2 unit systems - one base 10, the other base 2:

The primary power of 10 notation we're all familiar with:


1 kilo byte (kB) = 10 ^ 3 bytes
1 mega byte (MB)= 10 ^ 6 bytes (historically 1000 * 1024 for some storage cases)
1 giga byte (GB) = 10 ^ 9 bytes


The alternative power of 2 notation:


1 kibi byte = 2 ^ 10 bytes
1 mebi byte = 2 ^ 20 bytes
1 gibi byte = 2 ^ 30 bytes


Thus there is even less reason to use the Tonal system since a widespread alternative with well-defined prefixes has already been ratified for some time.