Music As A Distance

What is a Comma?

In terms of music, a Comma is a word used when
referring to the discrepancies of distance between intervals
​inside of any given system.

* There is no single unit that represents a comma but there are
​2 main numeric values in widespread historical use:

The Pythagorean Comma
(also known as the Ditonic comma)
is the discrepancy between twelve pure 5th's
​& seven pure octaves.
​It's the comma used in constructing the Equal Temperament system
& is defined by the ratio 531441:524288
​and the decimal 1.0136433.

​The Syntonic Comma
(also known as the Diatonic comma) 
is the discrepancy between four pure 5th's
& two octaves plus a Major 3rd.
​It's equivalent to the ratio 81:80
​and the decimal 1.0125.

What is a Diesis?

A Diesis is the difference between
a major semitone & a minor semitone.

A notated representation of micro-intervals in between semitones Db & D.

​The distance of a diesis varies depending on
the amount of tempering used in constructing the 5ths.
The greater the tempering of the 5th's, ​the greater the difference between
​major & minor semitones (the diesis).
​
​
​For example:

* Quarter-Comma Meantone tuning has a larger diesis than the 6th-Comma Meantone tuning system.. A quarter (1/4) is larger than a sixth (1/6).