User:Holger Stoltenberg/sandbox

Expanding tonal space/planar extensions

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Expanding tonal space/projections

Part I: Expanding tonal space

Don't forget about the function

Part I: Expanding tonal space

Navigating tonal space

Number of distinct intervals

The first five octaves of Tonal Space contain a fairly large number of intervals footed on a common tonic of 0 ¢. The intervals are well structured in rows, with each row corresponding to a mode of the overtone scale. It may be of interest to the reader to know how many different intervals are present, since some obviously occur more than once.
To find out, we will scan the Horizon Chart line by line (mode by mode), from the bottom up. Mode 1 has no intervals between the fundamental and the next octave. In Mode 2 we find a pure fifth, the third harmonic. This is the first time the pure fifth appears, and - like any other interval - it is only considered once as we scan. Table 1 summarizes the scanning results from Mode 1 through Mode 16.

Table 1: Count of distinct intervals depending on the highest implemented mode

Mode 1 up to Mode 16 Mode	Number of new intervals found at this mode	Aggregated number of different rational intervals	Total number of intervals scanned	Mode found in octave number
16	8	79	120	5
15	8	71	105	4
14	6	63	91	4
13	12	57	78	4
12	4	45	66	4
11	10	41	55	4
10	4	31	45	4
9	6	27	36	4
8	4	21	28	4
7	6	17	21	3
6	2	11	15	3
5	4 (...)	9	10	3
4	2 (3rd, b7th)	5	6	3
3	2 (4th, 6th)	3	3	2
2	1 (5th)	1	1	2
1	0	0	0	1

Rob Ickes performing with his band, Blue Highway, on June 21, 2010.

Heather Leigh-0981

Dobro guitar - Bluegrass Band, Kentucky (2011-10-16 by Navin75)

See also…

Sethares, William A. Tuning Timbre Spectrum Scale. London: Springer Verlag , 1999. [p65, 3.7. Overtone Scales]