# Interactions of Quarts and Quints

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Related KMR pages:

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Other related sources of information:

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Some tunes that make use of only quarts and quints

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$\, q_{uart} \cdot q_{uint} = o_{ctave} \,$.

$\, f_a \cdot s_o = 2^{\, \frac{5}{12}} \cdot 2^{ \, \frac{7}{12}} = 2^{\, (\frac{5}{12} + \frac{7}{12})} = 2^{\, \frac{12}{12}} = 2^{\, 1} = 2 \, d_o$.

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Solfege (= key-independent) interaction schema of Quarts and Quints: The Solfege interaction schema of Quarts and Quints instantiated in the key of C: Up-Down frequency-based relationships for both Quarts and Quints:

A slideshow comparing the up- and down motions of the quarts:

Quart up and Quart down

The resulting patterns are symmetric, because:

On a 12 half-tones scale we have
$\, +5n = +6n - n \, (mod \, 12) \,$ and
$\, -5n = -6n + n = +6n + n \, (mod \, 12)$,
and the points
$\, +6n - n \, (mod \, 12) \,$ and
$\, +6n + n \, (mod \, 12) \,$
are symmetrically placed
in relation to the line connecting the points
$\, 6(2n) = 0 \, (mod \, 12) \,$ and
$\, 6(2n+1) = 6 \, (mod \, 12)$.

A slideshow comparing the up- and down motions of the quints:

The resulting patterns are symmetric, because:

On a 12 half-tones scale we have
$\, +7n = +6n + n \, (mod \, 12) \,$ and
$\, -7n = -6n -n = +6n -n \, (mod \, 12)$,
and the points
$\, +6n + n \, (mod \, 12) \,$ and
$\, +6n - n \, (mod \, 12) \,$
are symmetrically placed
in relation to the line connecting the points
$\, 6(2n) = 0 \, (mod \, 12) \,$ and
$\, 6(2n+1) = 6 \, (mod \, 12)$.

Frequency-based relationships between Quarts and Quints: A slideshow comparing the up-motion of the Quints
and the down motion of the Quarts
:

The resulting patterns are identical, because:

On a 12 half-tones scale we have
$\, +7n = -5n \, (mod \, 12) \,$ for all $n \in N$,
which in frequency-based terms translates to
$\, (\frac{3}{2})^n = (\frac{3}{4})^n \,(mod \, 2)\,$ for all $n \in N$.

A slideshow comparing the up-motion of the Quarts
and the down motion of the Quints
:

Quart up and Quint down

The resulting patterns are identical, because:

On a 12 half-tones scale we have
$\, +5n = -7n \, (mod \, 12) \,$ for all $n \in N \,$,
which in frequency-based terms translates to
$\, (\frac{4}{3})^n = (\frac{2}{3})^n \,(mod \, 2)\,$ for all $n \in N$.

A slideshow comparing the up-motion of the Quints
and the up-motion of the Quarts
:

Quint up and Quart up

A slideshow comparing the down-motion of the Quarts
and the down-motion of the Quints
:

A slideshow illustrating the frequencies of the Quart-down motion:

Quart down = 1:2(Quint up) ///////