User:Holger Stoltenberg/sandbox: Difference between revisions

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The proposed model of tonal space utilizes the uniform structure and simple mathematical description of ''[[Overtone scale|overtone scales]]'' as well as the closely related ''arithmetic frequency division of the octave'' ([[AFDO]]).

For each note rendered, the model should effectively determine the current deviation from 12-tone equal temperament ([[12edo]]) in order to control the intonation of a commercial sound module or software plug-in on the fly.

For each note to be rendered, the model should effectively determine the current deviation from 12-tone equal temperament ([[12edo]]) in order to control the intonation of a commercial sound module or software plug-in on the fly.

==How to expand tonal space – the interval axis==

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.

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[[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 the first harmonic h1 to the second harmonic h2)

==The mode axis==

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[[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.

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.

The [[AFDO]]-page can help to reproduce this plot:

@@ Line 18: / Line 18: @@
 The proposed model of tonal space utilizes the uniform structure and simple mathematical description of ''[[Overtone scale|overtone scales]]'' as well as the closely related ''arithmetic frequency division of the octave'' ([[AFDO]]).
-For each note rendered, the model should effectively determine the current deviation from 12-tone equal temperament ([[12edo]]) in order to control the intonation of a commercial sound module or software plug-in on the fly.
+For each note to be rendered, the model should effectively determine the current deviation from 12-tone equal temperament ([[12edo]]) in order to control the intonation of a commercial sound module or software plug-in on the fly.
 ==How to expand tonal space – the interval axis==
 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.
@@ Line 24: / Line 24: @@
 [[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 the first harmonic h1 to the second harmonic h2)</small>
 ==The mode axis==
@@ Line 38: / Line 38: @@
 [[File:Fig-3_tonal_space_0844g.png|458px|center]]
-:::: <small><u>Fig. 3</u>: 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.
+Each pitch is labeled with the size of an interval in cents, measured from the tonic (0 ¢) to the corresponding pitch marker (<small>'''+'''</small>). 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.
 The [[AFDO]]-page can help to reproduce this plot: