Expanding tonal space/projections: Difference between revisions

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If we bend the projection plane around the vertical axis and glue the lower end of the octave to its upper end, we can observe the match of the slanted mode-lines at the octave boundary (Fig.2).

On the surface of this cylinder, a multi-start thread is created. All modes of the overtone scale that can be strung together in octaves form a common continuous thread on the surface. Modes 1, 3, 5, 7, 9... and all other odd-numbered modes each have their own thread.

==== Sequences of intervals with common numerator ====

In ''Part II: Planar extensions'', we mentioned in the [[Expanding tonal space/planar extensions#Extending tonal space to the right|explanation of Fig.4]] that there the

:''“…slanted fine blue lines connect intervals that share a common numerator, since Harry Partch also known as utonalities”''.

Here, '''in a projection with slanted mode-lines, interval sequences with a common numerator appear horizontally'''. [[#Slanted mode-lines|Fig.1]] shows an example at mode <math>n_0 </math>= 6,

<math>(\frac{6}{6}, \frac{6}{5}, \frac{6}{4}, \frac{6}{3})</math>,

printed in blue.

== Polar projection using slanted mode-lines ==

In polar projection the slanted mode-lines form a separate spiral for each odd mode. Odd modes can never be an octave of a lower mode. Fig.3 shows an artistic interpretation of the polar projection of tonal space with slanted mode-lines.

[[File:Fig-3_POLAR_3174_Tilted_Mode-lines.png|thumb|482px|center|Fig.3: Polar projection of tonal space applying slanted mode-lines]]

:The center is the location of the fundamental, where mode <math>n_0</math>=1 and <math>m</math>=0. This corresponds to the origin of the Cartesian coordinate system as in Fig.1. The mode axis runs from the center upward. A clockwise angle of 2π (in radians) represents one octave up. Each yellow ball represents a pitch.

== Logarithmic mode-lines ==

For the sake of completeness, another projection is shown here. This view may be of less practical importance to the musician, but from an aesthetic point of view it has a particularly well-balanced shape, especially in polar projection.

We can find a continuous function that binds the right boundary of each mode <math>n</math> of the overtone scale to the left side of the adjacent mode <math>n</math>+1-overtone scale. Measured vertically we recognize a set of lines of constant frequency difference (Fig.4).

[[File:Fig-4_Projections_1248x.png|thumb|470px|center|Fig.4: Projection of tonal space with lines of constant frequency difference]]

Projecting our familiar pitch markers vertically onto the graph, this logarithmic function continuously connects all markers.

== Polar projection with logarithmic spiral ==

When drawn in polar coordinates, the projection in Fig.4 shows a beautiful logarithmic spiral (Fig.5):

[[File:Fig-5_wt_POLAR_159_Spiral_Lin+Spokes.png|thumb|470px|center|Fig.5: Polar projection with logarithmic spiral]]

== Find out more about tonal space… ==

==== [[Expanding tonal space|Part I: Expanding tonal space]] ====

==== [[Expanding tonal space/planar extensions|Part II: Planar extensions]] ====

@@ Line 30: / Line 30: @@
 If we bend the projection plane around the vertical axis and glue the lower end of the octave to its upper end, we can observe the match of the slanted mode-lines at the octave boundary (Fig.2).<br>
 On the surface of this cylinder, a multi-start thread is created. All modes of the overtone scale that can be strung together in octaves form a common continuous thread on the surface. Modes 1, 3, 5, 7, 9... and all other odd-numbered modes each have their own thread.
+==== Sequences of intervals with common numerator ====
+In ''Part II: Planar extensions'', we mentioned in the [[Expanding tonal space/planar extensions#Extending tonal space to the right|explanation of Fig.4]] that there the
+:''“…slanted fine blue lines connect intervals that share a common numerator, since Harry Partch also known as utonalities”''.
+Here, '''in a projection with slanted mode-lines, interval sequences with a common numerator appear horizontally'''. [[#Slanted mode-lines|Fig.1]] shows an example at mode <math>n_0 </math>= 6,
+<math>(\frac{6}{6}, \frac{6}{5}, \frac{6}{4}, \frac{6}{3})</math>,
+printed in blue.
+== Polar projection using slanted mode-lines ==
+In polar projection the slanted mode-lines form a separate spiral for each odd mode. Odd modes can never be an octave of a lower mode. Fig.3 shows an artistic interpretation of the polar projection of tonal space with slanted mode-lines.
+[[File:Fig-3_POLAR_3174_Tilted_Mode-lines.png|thumb|482px|center|<u>Fig.3</u>: Polar projection of tonal space applying slanted mode-lines]]
+:The center is the location of the fundamental, where mode <math>n_0</math>=1 and <math>m</math>=0. This corresponds to the origin of the Cartesian coordinate system as in Fig.1. The mode axis runs from the center upward. A clockwise angle of 2π (in radians) represents one octave up. Each yellow ball represents a pitch.
+== Logarithmic mode-lines ==
+For the sake of completeness, another projection is shown here. This view may be of less practical importance to the musician, but from an aesthetic point of view it has a particularly well-balanced shape, especially in polar projection.<br>
+We can find a continuous function that binds the right boundary of each mode <math>n</math> of the overtone scale to the left side of the adjacent mode <math>n</math>+1-overtone scale. Measured vertically we recognize a set of lines of constant frequency difference (Fig.4).
+[[File:Fig-4_Projections_1248x.png|thumb|470px|center|<u>Fig.4</u>: Projection of tonal space with lines of constant frequency difference]]
+Projecting our familiar pitch markers vertically onto the graph, this logarithmic function continuously connects all markers.
+== Polar projection with logarithmic spiral ==
+When drawn in polar coordinates, the projection in Fig.4 shows a beautiful logarithmic spiral (Fig.5):
+[[File:Fig-5_wt_POLAR_159_Spiral_Lin+Spokes.png|thumb|470px|center|<u>Fig.5</u>: Polar projection with logarithmic spiral]]
+== Find out more about tonal space… ==
+==== [[Expanding tonal space|Part I: <span style="font-weight:normal">Expanding tonal space</span>]] ====
+==== [[Expanding tonal space/planar extensions|Part II: <span style="font-weight:normal">Planar extensions</span>]] ====