Lessons · Harmony

Intervals

Distances, quality, consonance

% An INTERVAL is a distance, measured in semitones. Taken one note
% after another it is MELODIC; the brace field {a, b} runs its fields
% at once over one span, and the same distance comes back HARMONIC.
% This bugle call is only 1 3 5 '1 — up a major third of 4, a minor
% third of 3, a fourth of 5, and those three add to the octave's 12.
% Over a drone on .1 the call measures 12, 16, 19, 24 instead: an
% octave plus each simple size, which is what a COMPOUND interval is.
 
bpm 40
key g maj
 
Fanfare ::= 1 _2 3 _2 5 _4 '1 _2 5 _2 3 _2 1 _2
 
assert 1 3 5 '1 == 1 i4 i3 i5
 
Fanfare
{Fanfare, .1 _16}
 
 
% Doubling a line at a fixed distance is the oldest harmony there is.
% '@-7' moves the copy down exactly 7 semitones, so every dyad is a
% perfect fifth — parallel ORGANUM — but the fifth under degree 4 is an
% f natural, outside the key. '@~-4' moves four SCALE STEPS instead and
% never leaves G major, at the price of one dyad of 6 semitones, the
% TRITONE, because the ladder's rungs are not evenly spaced.
 
Chant ::= 1 _2 2 _2 3 _2 4 _2 3 _2 2 _2 1 _4
 
assert 1 s-4 == 1 i-7
assert 4 s-4 == 4 i-6
 
{Chant, Chant@-7}
{Chant, Chant@~-4}
 
 
% Sizes divide by CHARACTER. Thirds and sixths are the sweet ones and
% a chain of sixths can run all day; sevenths and seconds are the sharp
% ones, and a seventh has to fall. Sixths and Chain walk down the same
% .6 .5 .4 .3, but Chain holds its first note two units longer, so
% every move lands late and each standing sixth turns into a seventh
% that steps down into the next sixth — a 7-6 chain.
 
Bass   ::= .1 _4 ..7 _4 ..6 _4 ..5 _4 .1 _4
Sixths ::= .6 _4 .5 _4 .4 _4 .3 _4 .3 _4
Chain  ::= .6 _6 .5 _4 .4 _4 .3 _6
 
{Bass, Sixths}
{Bass, Chain}
 
 
% Turn an interval upside down — the lower note up an octave — and
% the two sizes add to 12: a fifth of 7 becomes a fourth of 5, a sixth
% of 9 becomes a third of 3, a seventh of 10 a second of 2. So the
% organum above, its lower voice lifted an octave, is parallel FOURTHS;
% and the chain with its voices swapped is a 2-3 chain, the dissonance
% still stepping down into the consonance.
 
assert 1 5 '1 == 1 i7 i5
assert 1 3 '1 == 1 i4 i8
assert Chant@5 == Chant@-7@12
 
{Chant, Chant@5}
{Bass@12, Sixths}
{Bass@12, Chain}
 
 
% A duet is usually one tune and its own shadow. '@~-2' drops a copy
% two scale steps below, so every dyad is a third — 4 semitones or 3,
% whichever the key holds there, and that alternation is what makes it
% sound like G major and not like organum. Five steps down inverts each
% third into a sixth. Nothing here is spelled as a pitch, so the whole
% duet travels: in Eb the sizes come out in the same order.
 
Tune ::= 5 '1 7 6 5 _2 6 5 4 3 _2 2 1 _4
 
{Tune, Tune@~-2}
{Tune, Tune@~-5}
 
key eb maj
{Tune, Tune@~-2}
 
 
% All of it in one pass, home in G: the call over its drone, the
% chant doubled four steps down, the 7-6 chain, the duet in thirds, and
% a last octave with the fifth inside it.
 
key g maj
 
{Fanfare, .1 _16}
{Chant, Chant@~-4}
{Bass, Chain}
{Tune, Tune@~-2}
{.1, .5, 1} _8

% An INTERVAL is a distance, measured in semitones. Taken one note
% after another it is MELODIC; the brace field {a, b} runs its fields
% at once over one span, and the same distance comes back HARMONIC.
% This bugle call is only 1 3 5 '1 — up a major third of 4, a minor
% third of 3, a fourth of 5, and those three add to the octave's 12.
% Over a drone on .1 the call measures 12, 16, 19, 24 instead: an
% octave plus each simple size, which is what a COMPOUND interval is.

bpm 40
key g maj

Fanfare ::= 1 _2 3 _2 5 _4 '1 _2 5 _2 3 _2 1 _2

assert 1 3 5 '1 == 1 i4 i3 i5

Fanfare
{Fanfare, .1 _16}

% Doubling a line at a fixed distance is the oldest harmony there is.
% '@-7' moves the copy down exactly 7 semitones, so every dyad is a
% perfect fifth — parallel ORGANUM — but the fifth under degree 4 is an
% f natural, outside the key. '@~-4' moves four SCALE STEPS instead and
% never leaves G major, at the price of one dyad of 6 semitones, the
% TRITONE, because the ladder's rungs are not evenly spaced.

Chant ::= 1 _2 2 _2 3 _2 4 _2 3 _2 2 _2 1 _4

assert 1 s-4 == 1 i-7
assert 4 s-4 == 4 i-6

{Chant, Chant@-7}
{Chant, Chant@~-4}

% Sizes divide by CHARACTER. Thirds and sixths are the sweet ones and
% a chain of sixths can run all day; sevenths and seconds are the sharp
% ones, and a seventh has to fall. Sixths and Chain walk down the same
% .6 .5 .4 .3, but Chain holds its first note two units longer, so
% every move lands late and each standing sixth turns into a seventh
% that steps down into the next sixth — a 7-6 chain.

Bass   ::= .1 _4 ..7 _4 ..6 _4 ..5 _4 .1 _4
Sixths ::= .6 _4 .5 _4 .4 _4 .3 _4 .3 _4
Chain  ::= .6 _6 .5 _4 .4 _4 .3 _6

{Bass, Sixths}
{Bass, Chain}

% Turn an interval upside down — the lower note up an octave — and
% the two sizes add to 12: a fifth of 7 becomes a fourth of 5, a sixth
% of 9 becomes a third of 3, a seventh of 10 a second of 2. So the
% organum above, its lower voice lifted an octave, is parallel FOURTHS;
% and the chain with its voices swapped is a 2-3 chain, the dissonance
% still stepping down into the consonance.

assert 1 5 '1 == 1 i7 i5
assert 1 3 '1 == 1 i4 i8
assert Chant@5 == Chant@-7@12

{Chant, Chant@5}
{Bass@12, Sixths}
{Bass@12, Chain}

% A duet is usually one tune and its own shadow. '@~-2' drops a copy
% two scale steps below, so every dyad is a third — 4 semitones or 3,
% whichever the key holds there, and that alternation is what makes it
% sound like G major and not like organum. Five steps down inverts each
% third into a sixth. Nothing here is spelled as a pitch, so the whole
% duet travels: in Eb the sizes come out in the same order.

Tune ::= 5 '1 7 6 5 _2 6 5 4 3 _2 2 1 _4

{Tune, Tune@~-2}
{Tune, Tune@~-5}

key eb maj
{Tune, Tune@~-2}

% All of it in one pass, home in G: the call over its drone, the
% chant doubled four steps down, the 7-6 chain, the duet in thirds, and
% a last octave with the fifth inside it.

key g maj

{Fanfare, .1 _16}
{Chant, Chant@~-4}
{Bass, Chain}
{Tune, Tune@~-2}
{.1, .5, 1} _8

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Intervals · Pattern Engine