Expanding tonal space: Difference between revisions
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==Polar projection of tonal space== | ==Polar projection of tonal space== | ||
In polar projection | In polar projection the horizontal ''mode'' lines form circles: | ||
[[File:Fig-4 tonal space POLAR 149.png|496px|center]] | [[File:Fig-4 tonal space POLAR 149.png|496px|center]] | ||
<center><small><u>Fig.4</u>: A plane of tonal space in polar projection (up to Mode 16)</small></center> | <center><small><u>Fig.4</u>: A plane of tonal space in polar projection (up to Mode 16)</small></center> | ||
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The labeled spokes reveal a hidden symmetry of tonal space that is not apparent in Cartesian projection. | 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. | 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== | ==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: | 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: | ||