Harmonic Scale Essentials

Views of what the author considered essential properties of a harmonic scale* can be found by combing through the early papers [1,2,3] on what was later dubbed the Bohlen-Pierce scale. In a condensed form these views can be reduced to the necessity to conform with two independent aesthetic principles which are based on the physiology of middle and inner ear:

__1.
Exponential sequence of tone pitches ("Principle of equidistance")
__Based
on the exponential frequency sensitivity characteristic of the
organ of Corti, a musical scale is perceived as aesthetically
pleasing if its tone steps follow or at least closely approximate
the equation

*with
f _{n} meaning the pitch of step n of the scale, f_{0}
the fundamental tone, K the frame interval and N the total number
of steps. I scales for monodic music (one voice only at any time),
K and N can assume quite a range of values. This does not apply
to scales for polyphonic music which at the same time have to
abide by the following principle, too:*

__2.
Gestalt compatibility between intervals ("Principle of consonance")
__The
nonlinearity of the ear, mainly located in middle ear and basilar
membrane, changes the Gestalt impression of intervals and chords
by causing combination tones and altering the harmonics. These
thus modified composite sounds reduce the choice of tones, which
can form a usable scale for polyphonic music, to the members of
a small range of intervals with compatible Gestalt features.

Scales for the creation of harmonic (and finally tonal) music can be developed by a variety of means but will fail to be attractive if the result violates either of the two principles described above. But simultaneously abiding by both of them limits their number to a small pool, even when allowing for some tolerance. The numerically most simple of these scales are the Western 12-tone scale which in its just version is a reasonable approximation to

and the Bohlen-Pierce scale (BP) with the fairly close approximation of its just form to

(*)"Interval gestalt and harmonic scales" provides more details on this issue.

[1]

Original archived at Huygens-Fokker Foundation (Stichting Huygens-Fokker), Amsterdam.

The paper describes the derivation of the 13-step scale (later BP) in both a just and equal-tempered form, in conformance with two basic, independent principles: consonance with combination tones, and approximate equidistance of scale steps. The presentation comprises a 13-step chromatic and four 9-step diatonic versions.

[2] Bohlen,
Heinz: *Die Bildungsgesetze des 12-stufigen Tonsystems und ihre
Anwendung auf einen Sonderfall.
*Manuscript, ink, 50 pages, Hamburg, July 1972.

Original archived at Huygens-Fokker Foundation (Stichting Huygens-Fokker), Amsterdam.

Mainly an expanded and refined version of [1], containing also first considerations of realization essentials (tone names, notation, plans for an electronic organ).

[3]* *Bohlen,
Heinz: *Versuch über den Aufbau eines tonalen Systems auf
der Basis einer 13-stufigen Skala*.

Manuscript, first
version,
typed, 7 pages, Hamburg, Dec. 1972.

Manuscript, second version, typed, 9 pages, Hamburg, July 1974.
(Referenced in [4]).

Originals archived at Huygens-Fokker Foundation (Stichting Huygens-Fokker),
Amsterdam.

These
are mainly abstracts of [2], intended to inform a selected readership
about the 13-step scale.