Mathematical foundations of complex tonality

Figures & data

Table 1. Structures of the three complex tone systems based on three semitones. Each system occurs in a number of variants, one example of which is presented for each system. For each of the three complex systems, the first column gives the aggregation sequence and the second column gives the values taken by the notes of the scale.

Complex Tone Systems based on Three Semitones
Note	System I		System II		System III
C	1	1	1	1	1	1
$C_{\mathrm{♯}}$	α	$\frac{3}{4}(1+\mathrm{i})$	α	$\frac{3}{4}(1+\mathrm{i})$	α	$\frac{3}{4}(1+\mathrm{i})$
D	β	$\frac{1}{2}(1+2\mathrm{i})$	$\overline{\alpha }$	$\frac{9}{8}$	Z	$\frac{4}{5}(1+\mathrm{i})$
$D_{\mathrm{♯}}$	$\overline{\text{β}}$	$\frac{5}{6}(1+\mathrm{i})$	${\alpha }^{\mathrm{\prime }}$	$\frac{5}{6}(1+\mathrm{i})$	ρ	$\frac{5}{6}(1+\mathrm{i})$
E	$\overline{\alpha }$	$\frac{5}{4}$	$\overline{\alpha }$	$\frac{5}{4}$	$\overline{\alpha }$	$\frac{5}{4}$
F	Z	$\frac{4}{3}$	Z	$\frac{4}{3}$	Z	$\frac{4}{3}$
$F_{\mathrm{♯}}$	α	$(1+\mathrm{i})$	α	$(1+\mathrm{i})$	α	$(1+\mathrm{i})$
G	$\overline{\alpha }$	$\frac{3}{2}$	$\overline{\alpha }$	$\frac{3}{2}$	$\overline{\alpha }$	$\frac{3}{2}$
$G_{\mathrm{♯}}$	$\overline{\text{β}}$	$\frac{1}{2}(3-\mathrm{i})$	α	$\frac{9}{8}(1+\mathrm{i})$	Z	$\frac{8}{5}$
A	β	$\frac{5}{3}$	${\overline{\alpha }}^{\mathrm{\prime }}$	$\frac{5}{3}$	ρ	$\frac{5}{3}$
$A_{\mathrm{♯}}$	α	$\frac{5}{4}(1+\mathrm{i})$	α	$\frac{5}{4}(1+\mathrm{i})$	α	$\frac{5}{4}(1+\mathrm{i})$
B	Z	$\frac{4}{3}(1+\mathrm{i})$	Z	$\frac{4}{3}(1+\mathrm{i})$	Z	$\frac{4}{3}(1+\mathrm{i})$
$C^{\mathrm{\prime }}$	$\overline{\alpha }$	2	$\overline{\alpha }$	2	$\overline{\alpha }$	2

Notation: $\alpha =\frac{3}{4}(1+\mathrm{i})$, ${\alpha }^{\mathrm{\prime }}=\frac{80}{81}\alpha$, $\text{β}=\frac{1}{3}(3+\mathrm{i})$, $Z=\frac{16}{15}$, $\rho =\frac{25}{24}$.

Table 2. Four variants of the Complex System II scale based on three semitones. For each variant, the first column gives the aggregation sequence and the second column gives the values of the notes.

Complex System II
Note	$X_2=\overline{\alpha },X_3={\alpha }^{\mathrm{\prime }}$		$X_2=\alpha ,X_3={\overline{\alpha }}^{\mathrm{\prime }}$		$X_2={\alpha }^{\mathrm{\prime }},X_3=\overline{\alpha }$		$X_2={\overline{\alpha }}^{\mathrm{\prime }},X_3=\alpha$
C	1	1	1	1	1	1	1	1
$C_{\mathrm{♯}}$	α	$\frac{3}{4}(1+\mathrm{i})$	α	$\frac{3}{4}(1+\mathrm{i})$	α	$\frac{3}{4}(1+\mathrm{i})$	α	$\frac{3}{4}(1+\mathrm{i})$
D	$\overline{\alpha }$	$\frac{9}{8}$	α	$\frac{9}{8}\mathrm{i}$	${\alpha }^{\mathrm{\prime }}$	$\frac{10}{9}\mathrm{i}$	${\overline{\alpha }}^{\mathrm{\prime }}$	$\frac{10}{9}$
$D_{\mathrm{♯}}$	${\alpha }^{\mathrm{\prime }}$	$\frac{5}{6}(1+\mathrm{i})$	${\overline{\alpha }}^{\mathrm{\prime }}$	$\frac{5}{6}(1-\mathrm{i})$	$\overline{\alpha }$	$\frac{5}{6}(1-\mathrm{i})$	α	$\frac{5}{6}(1+\mathrm{i})$
E	$\overline{\alpha }$	$\frac{5}{4}$	$\overline{\alpha }$	$\frac{5}{4}$	$\overline{\alpha }$	$\frac{5}{4}$	$\overline{\alpha }$	$\frac{5}{4}$
F	Z	$\frac{4}{3}$	Z	$\frac{4}{3}$	Z	$\frac{4}{3}$	Z	$\frac{4}{3}$
$F_{\mathrm{♯}}$	α	$(1+\mathrm{i})$	α	$(1+\mathrm{i})$	α	$(1+\mathrm{i})$	α	$(1+\mathrm{i})$
G	$\overline{\alpha }$	$\frac{3}{2}$	$\overline{\alpha }$	$\frac{3}{2}$	$\overline{\alpha }$	$\frac{3}{2}$	$\overline{\alpha }$	$\frac{3}{2}$
$G_{\mathrm{♯}}$	α	$\frac{9}{8}(1+\mathrm{i})$	$\overline{\alpha }$	$\frac{9}{8}(1-\mathrm{i})$	${\overline{\alpha }}^{\mathrm{\prime }}$	$\frac{10}{9}(1-\mathrm{i})$	${\alpha }^{\mathrm{\prime }}$	$\frac{10}{9}(1+\mathrm{i})$
A	${\overline{\alpha }}^{\mathrm{\prime }}$	$\frac{5}{3}$	${\alpha }^{\mathrm{\prime }}$	$\frac{5}{3}$	α	$\frac{5}{3}$	$\overline{\alpha }$	$\frac{5}{3}$
$A_{\mathrm{♯}}$	α	$\frac{5}{4}(1+\mathrm{i})$	α	$\frac{5}{4}(1+\mathrm{i})$	α	$\frac{5}{4}(1+\mathrm{i})$	α	$\frac{5}{4}(1+\mathrm{i})$
B	Z	$\frac{4}{3}(1+\mathrm{i})$	Z	$\frac{4}{3}(1+\mathrm{i})$	Z	$\frac{4}{3}(1+\mathrm{i})$	Z	$\frac{4}{3}(1+\mathrm{i})$
$C^{\mathrm{\prime }}$	$\overline{\alpha }$	2	$\overline{\alpha }$	2	$\overline{\alpha }$	2	$\overline{\alpha }$	2

Notation: $\alpha =\frac{3}{4}(1+\mathrm{i})$, ${\alpha }^{\mathrm{\prime }}=\frac{80}{81}\alpha$, $Z=\frac{16}{15}$.

Table 3. Two variants of the Complex System III chromatic scale based on three semitones. For each scale, the first column gives the aggregation sequence and the second column gives the values of the notes of the scale.

Complex System III
Note	$X_2=\rho ,X_3=Z$		$X_2=Z,X_3=\rho$
C	1	1	1	1
$C_{\mathrm{♯}}$	α	$\frac{3}{4}(1+\mathrm{i})$	α	$\frac{3}{4}(1+\mathrm{i})$
D	ρ	$\frac{25}{32}(1+\mathrm{i})$	Z	$\frac{4}{5}(1+\mathrm{i})$
$D_{\mathrm{♯}}$	Z	$\frac{5}{6}(1+\mathrm{i})$	ρ	$\frac{5}{6}(1+\mathrm{i})$
E	$\overline{\alpha }$	$\frac{5}{4}$	$\overline{\alpha }$	$\frac{5}{4}$
F	Z	$\frac{4}{3}$	Z	$\frac{4}{3}$
$F_{\mathrm{♯}}$	α	$(1+\mathrm{i})$	α	$(1+\mathrm{i})$
G	$\overline{\alpha }$	$\frac{3}{2}$	$\overline{\alpha }$	$\frac{3}{2}$
$G_{\mathrm{♯}}$	ρ	$\frac{25}{16}$	Z	$\frac{8}{5}$
A	Z	$\frac{5}{3}$	ρ	$\frac{5}{3}$
$A_{\mathrm{♯}}$	α	$\frac{5}{4}(1+\mathrm{i})$	α	$\frac{5}{4}(1+\mathrm{i})$
B	Z	$\frac{4}{3}(1+\mathrm{i})$	Z	$\frac{4}{3}(1+\mathrm{i})$
$C^{\mathrm{\prime }}$	$\overline{\alpha }$	2	$\overline{\alpha }$	2

Notation: $\alpha =\frac{3}{4}(1+\mathrm{i})$, $Z=\frac{16}{15}$, $\rho =\frac{25}{24}$.

Table 4. Chromatic scales generated by two semitones for (a) the complex Pythagorean scale, (b) the associated real shadow scales, and (c) a version of the traditional real Pythagorean scale. For each scale the first column gives the aggregation sequence and the second column gives the values of the notes of the scale. The variants that arise in each case are indicated.

Real and Complex Pythagorean Scales based on Two Semitones
Note	Complex Pythagorean		Shadow I		Shadow II		Real Pythagorean
C	1	1	1	1	1	1	1	1
$C_{\mathrm{♯}}$	α	$\frac{3}{4}(1+\mathrm{i})$	θ	$\frac{3}{4}\sqrt{2}$	θ	$\frac{3}{4}\sqrt{2}$	σ	$\frac{256}{243}$
D	$\overline{\alpha }$	$\frac{9}{8}$	θ	$\frac{9}{8}$	θ	$\frac{9}{8}$	τ	$\frac{9}{8}$
$D_{\mathrm{♯}}$	α	$\frac{27}{32}(1+\mathrm{i})$	θ	$\frac{27}{32}\sqrt{2}$	θ	$\frac{27}{32}\sqrt{2}$	σ	$\frac{32}{27}$
E	$\overline{\alpha },\sigma$	$\frac{81}{64},\frac{8}{9}(1+\mathrm{i})$	$\theta ,\sigma$	$\frac{81}{64},\frac{8}{9}\sqrt{2}$	ν	$\frac{3}{4}\sqrt[4]{8}$	τ	$\frac{81}{64}$
F	$\sigma ,\overline{\alpha }$	$\frac{4}{3}$	$\sigma ,\theta$	$\frac{4}{3}$	ν	$\frac{4}{3}$	σ	$\frac{4}{3}$
$F_{\mathrm{♯}}$	α	$1+\mathrm{i}$	θ	$\sqrt{2}$	θ	$\sqrt{2}$	σ	$\frac{1024}{729}$
G	$\overline{\alpha }$	$\frac{3}{2}$	θ	$\frac{3}{2}$	θ	$\frac{3}{2}$	τ	$\frac{3}{2}$
$G_{\mathrm{♯}}$	α	$\frac{9}{8}(1+\mathrm{i})$	θ	$\frac{9}{8}\sqrt{2}$	θ	$\frac{9}{8}\sqrt{2}$	σ	$\frac{128}{81}$
A	$\overline{\alpha }$	$\frac{27}{16}$	θ	$\frac{27}{16}$	θ	$\frac{27}{16}$	τ	$\frac{27}{16}$
$A_{\mathrm{♯}}$	$\alpha ,\sigma$	$\frac{81}{64}(1+\mathrm{i}),\frac{16}{9}$	$\theta ,\sigma$	$\frac{81}{64}\sqrt{2},\frac{16}{9}$	ν	$\frac{3}{2}\sqrt[4]{2}$	σ	$\frac{16}{9}$
B	$\sigma ,\alpha$	$\frac{4}{3}(1+\mathrm{i})$	$\sigma ,\theta$	$\frac{4}{3}\sqrt{2}$	ν	$\frac{4}{3}\sqrt{2}$	τ	$\frac{243}{128}$
$C^{\mathrm{\prime }}$	$\overline{\alpha }$	2	θ	2	θ	2	σ	2

Notation: $\alpha =\frac{3}{4}(1+\mathrm{i})$, $\theta =\frac{3}{4}\sqrt{2}$, $\nu =\frac{8}{9}\sqrt[4]{2}$, $\sigma =\frac{256}{243}$, $\tau =\frac{2187}{2048}$.

Table

$\pm 1$, $\pm i$	unit (ϵ)
$\frac{3+4i}{5}$	five-limit phase factor (ζ)
$\frac{81}{80}$	syntonic comma (κ)
$\frac{25}{24}$	minor chromatic semitone, small half tone (ρ)
$\frac{20}{27}(1+i)$	grave complex semitone (${\alpha }^{\mathrm{\prime }}$)
$\frac{256}{243}$	minor Pythagorean semitone, diesis (σ)
$\frac{1}{3}(3+i)$	minor complex semitone (β)
$\frac{135}{128}$	major chromatic semitone, major limma
$\frac{3}{4}(1+i)$	major complex semitone $(\alpha$)
$\frac{16}{15}$	minor diatonic semitone (Z)
$\frac{2187}{2048}$	major Pythagorean semitone, apotome (τ)
$\frac{27}{25}$	major diatonic semitone
$\frac{10}{9}$	minor whole tone (Y)
$\frac{1}{2}(1+2i)$	minor complex whole tone (γ)
$\frac{9}{8}$	major whole tone (X)
$\frac{4}{5}(1+i)$	major complex whole tone (δ)
$\frac{5}{6}(1+i)$	complex minor third
$\frac{32}{27}$	minor Pythagorean third
$\frac{27}{32}(1+i)$	complex minor Pythagorean third
$\frac{6}{5}$	minor third
$\frac{5}{4}$	major third
$\frac{81}{64}$	major Pythagorean third
$\frac{4}{3}$	perfect fourth
$\frac{27}{20}$	acute fourth
$\frac{25}{18}$	alternate tritone
$\frac{1024}{729}$	Pythagorean tritone
$\frac{45}{32}$	diatonic tritone
1 + i	complex tritone (ξ)
$\frac{64}{45}$	complementary diatonic tritone
$\frac{36}{25}$	complementary alternate tritone
$\frac{40}{27}$	grave fifth
$\frac{3}{2}$	perfect fifth
$\frac{25}{16}$	augmented fifth
$\frac{128}{81}$	minor Pythagorean sixth
$\frac{1}{2}(3-i)$	complex minor sixth
$\frac{9}{8}(1+i)$	complex minor Pythagorean sixth
$\frac{8}{5}$	minor sixth
$\frac{5}{3}$	major sixth
$\frac{27}{16}$	major Pythagorean sixth
$\frac{5}{4}(1+i)$	complex minor seventh
$\frac{16}{9}$	minor Pythagorean seventh
$\frac{81}{64}(1+i)$	complex minor Pythagorean seventh
$\frac{9}{5}$	minor seventh
$\frac{15}{8}$	major seventh
$\frac{4}{3}(1+i)$	complex major seventh
$\frac{243}{128}$	major Pythagorean seventh
2	octave