User:Holger Stoltenberg/sandbox: Difference between revisions
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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 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== | |||
== 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.<br>[Fig. 1] | 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.<br>[Fig. 1] | ||
<small>Fig. 1: One octave (from first harmonic h1 to second harmonic h2)</small> | <small>Fig. 1: One octave (from first harmonic h1 to second harmonic h2)</small> | ||
== The mode axis == | ==The mode axis== | ||
The vertical axis of our model of tonal space indicates ''modes'' of the harmonic series. All modes start at the same normalized frequency (or pitch) of 0 cents. This means that the ''tonics'' of all of these overtone scales share exactly the same pitch (Fig. 2), which can be chosen arbitrarily. | The vertical axis of our model of tonal space indicates ''modes'' of the harmonic series. All modes start at the same normalized frequency (or pitch) of 0 cents. This means that the ''tonics'' of all of these overtone scales share exactly the same pitch (Fig. 2), which can be chosen arbitrarily. | ||
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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: | The AFDO-page can help to reproduce this plot: | ||
'' | |||
''Within each period of any n-afdo system, the [[frequency ratio]] r of the m-th degree is'' | |||
<math>\displaystyle | <math>\displaystyle | ||
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where | where | ||
* r is a rational frequency ratio which – after conversion to cents –<br> is displayed against the horizontal interval axis of tonal space | *r is a rational frequency ratio which – after conversion to cents –<br> is displayed against the horizontal interval axis of tonal space | ||
* n is the mode of an overtone scale, plotted on the vertical axis | *n is the mode of an overtone scale, plotted on the vertical axis | ||
* m addresses (indexes, counts) the elements of each overtone scale in horizontal direction from the tonic (left, starting at 0) to the right. | *m addresses (indexes, counts) the elements of each overtone scale in horizontal direction from the tonic (left, starting at 0) to the right. | ||
In particular ... | In particular ... | ||
* if m = 0, then r = 1 (0 ¢) and | *if m = 0, then r = 1 (0 ¢) and | ||
* if m = n, then r = 2 (1200 ¢) | *if m = n, then r = 2 (1200 ¢) | ||
<math>\displaystyle r = (n + m)/n</math> | <math>\displaystyle r = (n + m)/n</math> | ||