Interactions
of Quarts and Quints

This page is a sub-page of our page on Music.

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Related KMR pages:

Solfege: An Abstract Key System
Instantiations of Solfege in different Keys
Abstract and Concrete Chord Circles
Generating the Quint Circle from the Mathematical Cogwheels
Chord Ladder
Transposition of Key = Shift of Basis in Music

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Other related sources of information:

• …

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Some tunes that make use of only quarts and quints

Johnny B. Goode {C, F, G}, (Chuck Berry)
Sweet Little Sixteen {E, A, B}, Chuck Berry
Sweet Little Sixteen {E, A, B}, The Rolling Stones
Sweet Little Sixteen {E, A, B}, John Lennon
I Saw Her Standing There {E, A, B} & Twist And Shout {C, F, G},
Paul McCartney & Bruce Springsteen

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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:

Solfege Interaction schema of Qarts and Quints

The Solfege interaction schema of Quarts and Quints instantiated in the key of C:

The Solfege interaction schema of Quarts and Quints instantiated in C

Up-Down frequency-based relationships for both Quarts and Quints:

Up and down frequency 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:

Quint up and Quint down

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:

Frequency relationships between Quarts and Quints

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

Quint up and Quart down

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
:

Quart down and Quint down

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

Quart down = 1:2(Quint up)

Skärmavbild 2018-02-04 kl. 23.50.45

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