User:Holger Stoltenberg/sandbox: Difference between revisions

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<center><small><u>Fig.4</u>: A plane of tonal space in polar projection (up to Mode 16)</small></center>

The center is the location of the fundamental, where Mode n=1 and m=0. This corresponds to the origin of the ~~former~~ Cartesian coordinate system [[#figure03|(Fig.3)]]. The mode axis runs from the center up to the north. A clockwise angle of 2π represents one octave up. Each dot represents a pitch.

The center is the location of the fundamental, where Mode n=1 and m=0. This corresponds to the origin of the Cartesian coordinate system as in [[#figure03|Fig.3]]. The mode axis runs from the center up to the north. A clockwise angle of 2π (in radians) represents one octave up. Each dot represents a pitch.

The labeled spokes reveal a hidden symmetry of tonal space that is not apparent in Cartesian projection.

Note that a line drawn from the center to the south (at an angle of π) represents an irrational interval with a frequency ratio of √2 ≈ 1.414..., a tritone of 600 ¢. Unsurprisingly, no rational interval fits exactly on this line.

==General applicability==

In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:

@@ Line 102: / Line 102: @@
 <center><small><u>Fig.4</u>: A plane of tonal space in polar projection (up to Mode 16)</small></center>
-The center is the location of the fundamental, where Mode n=1 and m=0. This corresponds to the origin of the former Cartesian coordinate system [[#figure03|(Fig.3)]]. The mode axis runs from the center up to the north. A clockwise angle of 2π represents one octave up. Each dot represents a pitch.
+The center is the location of the fundamental, where Mode n=1 and m=0. This corresponds to the origin of the Cartesian coordinate system as in [[#figure03|Fig.3]]. The mode axis runs from the center up to the north. A clockwise angle of 2π (in radians) represents one octave up. Each dot represents a pitch.
+The labeled spokes reveal a hidden symmetry of tonal space that is not apparent in Cartesian projection.
+Note that a line drawn from the center to the south (at an angle of π) represents an irrational interval with a frequency ratio of √2 ≈ 1.414..., a tritone of 600 ¢. Unsurprisingly, no rational interval fits exactly on this line.
 ==General applicability==
 In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different: