User:Holger Stoltenberg/sandbox: Difference between revisions

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Like ''frequency'', ''musical pitch'' is one-dimensional. Fig.1 illustrates the horizontal axis of tonal space, which we call the ''interval axis''. It points in keyboard direction from low keys to high keys. The first harmonic ''(h1)'' is known as the common ''fundamental'' of all upcoming overtone scales.

[[File:Fig-1_tonal_space_0846c.png|~~520px~~|center]]

[[File:Fig-1_tonal_space_0846c.png|480px|center]]

:: Fig.1: One octave (from first harmonic h1 to second harmonic h2)

:::: Fig.1: One octave (from first harmonic h1 to second harmonic h2)

==The mode axis==

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Since each mode (row) begins with a member of the harmonic series, we can space adjacent modes by corresponding rational intervals for plotting on the logarithmic vertical axis.

[[File:Fig-2_tonal_space_0851f.png|~~520px~~|center]]

[[File:Fig-2_tonal_space_0851f.png|480px|center]]

:::: Fig. 2: Overtone scales from Mode 1 (''h1'') to Mode 4 ''(h4)'' on a 2D-plane of tonal space

:: Fig. 2: Overtone scales from Mode 1 (''h1'') to Mode 4 ''(h4)'' on a 2D-plane of tonal space

The result of visualizing more modes of the overtone scale (up to Mode 16) is shown in Fig. 3:

[[File:Fig-3_tonal_space_0844g.png|458px|center]]

::Fig. 3: The Horizon Chart: Relations of overtone scales (up to Mode 16) on a plane of tonal space

:::: Fig. 3: The Horizon Chart: Relations of overtone scales (up to Mode 16) on a plane of tonal space

Each pitch is labeled with the size of an interval in cents, measured from the tonic (0 ¢) to the corresponding pitch marker ('''+'''). Each pitch marker is connected to the next vertical 12edo-line by a ''delta'' ''indicator''. We define the direction and length of this indicator as the ''signed intonation'' interval of the respective pitch.

@@ Line 22: / Line 22: @@
 Like ''frequency'', ''musical pitch'' is one-dimensional. Fig.1 illustrates the horizontal axis of tonal space, which we call the ''interval axis''. It points in keyboard direction from low keys to high keys. The first harmonic ''(h1)'' is known as the common ''fundamental'' of all upcoming overtone scales.
-[[File:Fig-1_tonal_space_0846c.png|520px|center]]
+[[File:Fig-1_tonal_space_0846c.png|480px|center]]
-:: <small><u>Fig.1</u>: One octave (from first harmonic h1 to second harmonic h2)</small>
+:::: <small><u>Fig.1</u>: One octave (from first harmonic h1 to second harmonic h2)</small>
 ==The mode axis==
@@ Line 31: / Line 31: @@
 Since each mode (row) begins with a member of the harmonic series, we can space adjacent modes by corresponding rational intervals for plotting on the logarithmic vertical axis.
-[[File:Fig-2_tonal_space_0851f.png|520px|center]]
+[[File:Fig-2_tonal_space_0851f.png|480px|center]]
+:::: <small><u>Fig. 2</u>: Overtone scales from Mode 1 (''h1'') to Mode 4 ''(h4)'' on a 2D-plane of tonal space</small>
-:: <small>Fig. 2: Overtone scales from Mode 1 (''h1'') to Mode 4 ''(h4)'' on a 2D-plane of tonal space</small>
 The result of visualizing more modes of the overtone scale (up to Mode 16) is shown in Fig. 3:
 [[File:Fig-3_tonal_space_0844g.png|458px|center]]
-::<small>Fig. 3: The Horizon Chart: Relations of overtone scales (up to Mode 16) on a plane of tonal space</small>
+:::: <small><u>Fig. 3</u>: The Horizon Chart: Relations of overtone scales (up to Mode 16) on a plane of tonal space</small>
 Each pitch is labeled with the size of an interval in cents, measured from the tonic (0 ¢) to the corresponding pitch marker ('''+'''). Each pitch marker is connected to the next vertical 12edo-line by a ''delta'' ''indicator''. We define the direction and length of this indicator as the ''signed intonation'' interval of the respective pitch.