Once the 13 tone barrier is broken, the additional tones are related by Pythagorean (fifths) commas to the smaller subset. The ratios become quite large, and the notation reflects the comma relations with the awkward result that flat notes in higher positions specify lower pitches than adjacent sharp notes in lower positions, as shown below in the chain of 19 tones.

Doubly Augmented Third

The number 19 is not arbitrarily chosen. The tones at either end of this system, C and E, are related to the end tones of the diatonic naturals, B and F, by commas, as shown by the grey and blue color coding of these tones in the diagrams below.

The end tones C and E create a new interval, the doubly augmented third.

This interval sounds similar to the diminished fifth, the interval between the end tones of the diatonic naturals, B and F.

Comparing these intervals side by side again shows that the notation is misleading, as the interval spanning the smaller staff distance specifies a wider interval. The qualities of the intervals do however suggest the correct relationship between their sizes, as a doubly augmentated third is larger than a once diminished fifth.

Doubly Diminished Thirds

There are also other new intervals, doubly diminished thirds, occuring between end tones E up to G and A up to C.

These new intervals sound very much like the halfsteps of the diatonic naturals, which occur in the analagous positions of the subset.

Side by side comparison of these intervals shows the interval having the larger staff distance is the one which is smaller in actual size, although the qualities of the intervals do suggest correct relative sizes, a minor second being a bit larger than a doubly diminished third.

Summary: 19 Tones, Letters, Notes & Intervals

The placement of these doubly diminished thirds within the scale show that the 19 tone system has special significance, with these two new intervals recalling the two diatonic halfsteps of the diatonic naturals.

The 19 tone system has 7 Pythagorean commas, whereas 13 tones only has one. Still larger systems give greater numbers of commas and other new and interesting interval qualities.

NEXT: 31 Tones