Quarter-Tone Harmony/Concepts

The quarter-tone system is a musical system that splits the octave into twenty-four equally-spaced notes, rather than the typical twelve notes in the conventional system.

This means that, between every adjacent pair of notes in the conventional system, there lies another note half-way between the two in the quarter-tone system:

$\fixed c' { \hide Staff.TimeSignature \time 3/2 g2 \once \override NoteHead.transparent = ##t g gs | g gqs gs }$

A quarter-tone is the new smallest interval that is produced, exactly half of a semitone in size.

Notation

The quarter-tone system requires three new accidentals to be devised. Though not standardized for quarter-tones, the most often-used system is the Stein-Zimmerman notation system:

half-sharp or semisharp, raising a note by 1 quarter-tone
three-halves-sharp or sesquisharp, raising a note by 3 quarter-tones
half-flat or semiflat, lowering a note by 1 quarter-tone
three-halves-flat or sesquiflat, lowering a note by 3 quarter-tones

A full quarter-tone chromatic scale ascending and descending is notated like so:

$\fixed c'{ \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \time 6/4 c16[ cqs cs ctqs d dqs ds dtqs e eqs f fqs fs ftqs g gqs gs gtqs a aqs as atqs b bqs] c'[ cqf' b bqf bf btqf a aqf af atqf g gqf gf gtqf f fqf e eqf ef etqf d dqf df dtqf] \time 4/4 c1 }$

This also produces new pairs of enharmonics:

$\fixed c'{ \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \time 2/2 cqs2 dtqf ctqs dqf dqs etqf dtqs eqf eqs fqf fqs gtqf ftqs gqf gqs atqf gtqs aqf aqs btqf atqs bqf bqs cqf' }$

Transposition

Note that the same relationships exist in the set of new quartertonal pitches as in the set of conventional pitches in the same way two identical combs interlock with one another, so passages can be transposed up or down quarter-tones like so:

$\fixed c' { \hide Staff.TimeSignature f8 a bf a f c'4. | fqs8^"up" aqs bqf aqs fqs cqs'4. | fqf8^"down" aqf btqf aqf fqf cqf'4. | }$

Intervals

There are several new intervals for which there are no standardized names:

Size in quarter-tones	System 1	System 2
1	quartertonal second	subminor second, semi-augmented unison
3	neutral second
5	second-third	supermajor second, subminor third
7	neutral third
9	third-fourth, minor fourth	supermajor third, semi-diminished fourth
11	major fourth	semi-augmented fourth
13	minor fifth	semi-diminished fifth
15	major fifth, fifth-sixth	semi-augmented fifth, subminor sixth
17	neutral sixth
19	sixth-seventh	supermajor sixth, subminor seventh
21	neutral seventh
23	quartertonal seventh	supermajor seventh, semi-diminished octave

"System 1" is the system used by Wyschnegradsky in his Manual, whereas "System 2" uses more technical, modern names consistent with the traditional naming system.

This book will hereafter use System 2:

Subminor intervals are quarter-tone smaller than minor.
Supermajor intervals are a quarter-tone larger than major.
Neutral intervals are a quarter-tone smaller than major, or a quarter-tone larger than minor.
Semi-augmented intervals are a quarter-tone larger than perfect, or a quarter-tone smaller than augmented.
Semi-diminished intervals are a quarter-tone smaller than perfect, or a quarter-tone larger than diminished.

$\fixed c' { \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \accidentalStyle modern \cadenzaOn c2^"subminor second" dtqf \bar"|" c2^"neutral second" dqf \bar"|" c2^"subminor third"^"supermajor second," dqs c etqf \bar"|" c2^"neutral third" eqf \bar"|" \break c2^"semi-diminished fourth"^"supermajor third," eqs c fqf \bar"|" c2^"fourth"^"semi-augmented" fqs \bar"|" c2^"fifth"^"semi-diminished" gqf \bar"|" c2^"subminor sixth"^"semi-augmented fifth," gqs c atqf \bar"|"\break c2^"neutral sixth" aqf \bar"|" c2^"subminor seventh"^"supermajor sixth," aqs c btqf \bar"|" c2^"neutral seventh" bqf \bar"|" c2^"supermajor seventh" bqs \bar"|" \break }$

These intervals, like traditional intervals, can be inverted, and doing so produces other quarter-tone intervals: the subminor second (1) inverts into a supermajor seventh (23), the neutral second (3) inverts into a neutral seventh (21), etc.

A rule of thumb for quarter-tone interval inversions:

Subminor intervals invert into supermajor intervals, and vice versa.
Neutral intervals invert into other neutral intervals.
Semi-augmented intervals invert into semi-diminished intervals, and vice versa.

Cycles

In the conventional twelve-tone system, different intervals (exclusing any larger than the octave) behave differently when repeatedly stacked in series. For example, only four intervals produce a cycle that goes through all twelve notes in the conventional system:

the minor second (1 semitone)
the perfect fourth (5 semitones)
the perfect fifth (7 semitones)
the major seventh (11 semitones)

Let us now see the behavior of the new intervals:

The subminor second (1 quarter-tone) produces the aforementioned quarter-tone chromatic scale.
The neutral second (3 quarter-tones) splits the minor third into equal halves, producing a scale with eight equal tones in one octave:
$\fixed c' { \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \accidentalStyle modern \cadenzaOn c2 dqf4 ef fqf fs gqs a bqf c'2 }$

Notably, this scale consists of two interlocking diminished sevenths chords a neutral second apart.
The supermajor second (5 quarter-tones) splits the perfect fourth into equal halves, producing a scale with all 24 tones in five octaves:
$\fixed c' { \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \accidentalStyle modern \cadenzaOn \clef "bass_8" c,,,2 dqs,,,4 f,,, gqs,,, bf,,, cqs,, ef,, fqs,, af,, \clef bass bqf,, df, eqf, gf, aqf, b, \clef treble ctqs e ftqs a bqs d' etqs' g' aqs' c''2 }$
The neutral third (7 quarter-tones) splits the perfect fifth into equal halves, producing a scale with all 24 tones in seven octaves:
$\fixed c' { \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \accidentalStyle modern \cadenzaOn \clef "bass_15" c,,,,2 eqf,,,,4 g,,,, bqf,,,, d,,, fqs,,, \clef "bass_8" a,,, cqs,, e,, gqs,, b,, \clef bass dqs, fs, aqs, \clef treble cs eqs gs bqs ef' gqf' \clef "treble^8" bf' dqf'' f'' aqf'' c'''2 }$
The supermajor third (9 quarter-tones) splits the major sixth into equal halves, producing a scale with the same eight tones as in the neutral second cycle but in three octaves:
$\fixed c' { \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \accidentalStyle modern \cadenzaOn \clef bass c,,2 eqs,,4 a,, dqf, fs, bqf, \clef treble ef gqs c'2 }$
The semi-augmented fourth (11 quarter-tones) splits the major seventh into equal halves, producing a unique scale with all 24 tones in eleven octaves, not unlike a warped circle of fourths:
$\fixed c' { \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \accidentalStyle modern \cadenzaOn \clef "bass_22" c,,,,,2 fqs,,,,,4 b,,,,, eqs,,,, bf,,,, eqf,,, \clef bass a,,, dqs,, gs,, ctqs, g, \clef treble cqs fs bqs f' bqf' e'' \clef "treble^22" aqs'' ef''' aqf''' d'''' gqs'''' cs''''' gqf''''' c''''''2 }$
The semi-diminished fifth (13 quarter-tones) is an inversion of the semi-augmented fourth, producing the cycle of semi-augmented fourths in reverse order.
The subminor sixth (15 quarter-tones) is an inversion of the supermajor third, producing the cycle of supermajor thirds in reverse order.
The neutral sixth (17 quarter-tones) is an inversion of the neutral third, producing the cycle of neutral thirds in reverse order.
The supermajor sixth (19 quarter-tones) is an inversion of the subminor third, producing the cycle of subminor thirds in reverse order.
The neutral seventh (21 quarter-tones) is an inversion of the semi-augmented fourth, producing the cycle of neutral seconds in reverse order.
The supermajor seventh (23 quarter-tones) is an inversion of the subminor second, producing the quarter-tone chromatic scale in reverse order.

Contrary Motion

Interestingly, contrary motion in quarter tones starting on the unison produces all conventional intervals (labeled here with their quartertonal sizes):

$\fixed c' { \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \accidentalStyle modern \cadenzaOn << {c'4^"0" cqs'^"2" cs'^"4" dqf'^"6" d'^"8" dqs'^"10" ds'^"12" eqf'^"14" e'^"16" eqs'^"18" f'^"20" fqs'^"22" fs'^"24"} \\ {c'4 bqs b bqf bf aqs a aqf gs gqs g gqf fs} >> }$

And contrary motion on the semi-augmented unison likewise produces all of the new quartertonal intervals:

$\fixed c' { \hide Staff.TimeSignature \hide Staff.Stem \hide Staff.Beam \hide Score.BarNumber \accidentalStyle modern \cadenzaOn << {c'4^"1" cqs'^"3" cs'^"5" dqf'^"7" d'^"9" dqs'^"11" ef'^"13" eqf'^"15" e'^"17" eqs'^"19" f'^"21" fqs'^"23"} \\ {bqs b bqf bf btqf a aqf gs gqs g gqf gf} >> }$