User:Holger Stoltenberg/sandbox: Difference between revisions

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Section "Number of distinct intervals" moved to main page: Expanding tonal space
Line 16: Line 16:
'''Navigating 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. <br>
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.
: <u>Table 1</u>: Count of distinct intervals depending on the highest implemented mode
: {| class="wikitable" style="text-align:center;"
|- style="background-color:#6b82c6; color:#484848;"
! <span style="font-weight:normal"> Mode 1 <br /> up to <br /> Mode 16 </span> <br /> Mode
! <span style="font-weight:normal"> Number of <br />new intervals <br />found at <br />this mode </span>
! Aggregated <br />number <br />of different  <br />rational intervals
! <span style="font-weight:normal"> Total number<br />of intervals<br />scanned </span>
! <span style="font-weight:normal"> Mode<br />found in<br />octave<br />number </span>
|-
| style="background-color:#e2e2e2; color:#484848;" | '''16'''
| 8
| style="background-color:#e2e2e2;" | 79
| 120
| 5
|-
| style="background-color:#e2e2e2;" | '''15'''
| 8
| style="background-color:#e2e2e2;" | 71
| 105
| 4
|-
| style="background-color:#e2e2e2;" | '''14'''
| 6
| style="background-color:#e2e2e2;" | 63
| 91
| 4
|-
| style="background-color:#e2e2e2;" | '''13'''
| 12
| style="background-color:#e2e2e2;" | 57
| 78
| 4
|-
| style="background-color:#e2e2e2;" | '''12'''
| 4
| style="background-color:#e2e2e2;" | 45
| 66
| 4
|-
| style="background-color:#e2e2e2;" | '''11'''
| 10
| style="background-color:#e2e2e2;" | 41
| 55
| 4
|-
| style="background-color:#e2e2e2;" | '''10'''
| 4
| style="background-color:#e2e2e2;" | 31
| 45
| 4
|-
| style="background-color:#e2e2e2;" | '''9'''
| 6
| style="background-color:#e2e2e2;" | 27
| 36
| 4
|-
| style="background-color:#e2e2e2;" | '''8'''
| 4
| style="background-color:#e2e2e2;" | 21
| 28
| 4
|-
| style="background-color:#e2e2e2;" | '''7'''
| 6
| style="background-color:#e2e2e2;" | 17
| 21
| 3
|-
| style="background-color:#e2e2e2;" | '''6'''
| 2
| style="background-color:#e2e2e2;" | 11
| 15
| 3
|-
| style="background-color:#e2e2e2;" | '''5'''
| 4 <small>''(...)''</small>
| style="background-color:#e2e2e2;" | 9
| 10
| 3
|-
| style="background-color:#e2e2e2;" | '''4'''
| 2  <small>''(3rd, b7th)''</small>
| style="background-color:#e2e2e2;" | 5
| 6
| 3
|-
| style="background-color:#e2e2e2;" | '''3'''
| 2  <small>''(4th, 6th)''</small>
| style="background-color:#e2e2e2;" | 3
| 3
| 2
|-
| style="background-color:#e2e2e2;" | '''2'''
| 1 <small>''(5th)''</small>
| style="background-color:#e2e2e2;" | 1
| 1
| 2
|-
| style="background-color:#e2e2e2;" | '''1'''
| 0
| style="background-color:#e2e2e2;" | 0
| 0
| 1
|}


[[File:Rob Ickes performing with Blue Highway California USA June 2010.jpg|thumb|180px|Rob Ickes performing with his band, Blue Highway, on June 21, 2010.]]
[[File:Rob Ickes performing with Blue Highway California USA June 2010.jpg|thumb|180px|Rob Ickes performing with his band, Blue Highway, on June 21, 2010.]]

Revision as of 16:37, 25 February 2025

Expanding tonal space

Expanding tonal space/planar extensions


< User:Holger Stoltenberg

Expanding tonal space/projections


Part I: Expanding tonal space

Don't forget about the function

Part I: Expanding tonal space

Navigating tonal space


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]

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