User:Holger Stoltenberg/sandbox: Difference between revisions

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= Expanding tonal space =

This article describes how to visually arrange a set of overtone scales in order to expand a particular plane of tonal space.

This article describes how to visually arrange a set of overtone scales in order to expand a particular ''plane of tonal space''.

== Why expand tonal space in this way? ==

The concept is a step towards building an electronic keyboard instrument that allows the player to interactively map ''sequences of rational intervals'' to consecutive keys – in real time, even live on stage. Therefore, the player needs convenient and intuitive access to the mapping process, which becomes an integral part of the musical performance.

The concept is a step towards building an electronic keyboard instrument that allows the player to interactively map ''sequences of rational intervals'' to consecutive keys – in real time, even live on stage. Therefore, the player needs convenient and intuitive access to the mapping process, which becomes an integral part of the musical performance.

With a prototypical instrument in place, the search for ''intermediary chords'' in tonal music begins. We can try to identify (and play) supplementary consonant chords that fit seamlessly into the gaps of familiar chord progressions.

The proposed model of tonal space utilizes the uniform structure and simple mathematical description of ''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.

== 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. [Fig. 1]

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

== 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.

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

[Fig. 2]

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:

[Fig. 3]

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.

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

r=\frac{n+m}{n}</math>

where

* r is a rational frequency ratio which – after conversion to cents – is displayed against the horizontal interval axis of tonal space

* 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.

In particular ...

* if m = 0, then r = 1 (0 ¢) and

* if m = n, then r = 2 (1200 ¢)

<math>\displaystyle r = (n + m)/n</math>

@@ Line 1: / Line 1: @@
 = Expanding tonal space =
-This article describes how to visually arrange a set of overtone scales in order to expand a particular plane of tonal space.
+This article describes how to visually arrange a set of overtone scales in order to expand a particular ''plane of tonal space''.
 == Why expand tonal space in this way? ==
-The concept is a step towards building an electronic keyboard instrument that allows the player to interactively map ''sequences of rational intervals'' to consecutive keys – in real time, even live on stage. Therefore, the player needs convenient and intuitive access to the mapping process, which becomes an integral part of the musical performance.<br><br>
+The concept is a step towards building an electronic keyboard instrument that allows the player to interactively map ''sequences of rational intervals'' to consecutive keys – in real time, even live on stage. Therefore, the player needs convenient and intuitive access to the mapping process, which becomes an integral part of the musical performance.
+With a prototypical instrument in place, the search for ''intermediary chords'' in tonal music begins. We can try to identify (and play) supplementary consonant chords that fit seamlessly into the gaps of familiar chord progressions.
+The proposed model of tonal space utilizes the uniform structure and simple mathematical description of ''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.
+== 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]
+<small>Fig. 1: One octave (from first harmonic h1 to second harmonic h2)</small>
+== 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.
+Since each mode (row) begins with an element of the harmonic series, we can space adjacent modes by corresponding rational intervals for plotting on the logarithmic vertical axis.
+[Fig. 2]
+<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:
+[Fig. 3]
+<small>Fig. 3: 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.
+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
 r=\frac{n+m}{n}</math>
+where
+* 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
+* 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 ...
+* if m = 0, then  r = 1 (0 ¢) and
+* if m = n, then  r = 2 (1200 ¢)
 <math>\displaystyle r = (n + m)/n</math>