Interval categories and their meaning

From Alain Daniélou’s book, “Sémantique Musicale”

The following table lists some notes of the “Daniélou scale”. Each note is given its ratio (both in fractional form and as related to the power of 2, 3 and 5), as well as Alain Daniélou’s description of the character of the interval separating the given note from the tonic.

1. Tonic and octaves (2/1, 4/1, etc.)
Meaning : base, strength, solidity, character

2. Intervals of the cycle of fifths (3 as the numerator)
Meaning : active, sunlike, virile

D 9/8 (32/23) vigorous, confident
E+ 81/64 (34/26) awake, lively
G 3/2 (3/2) sunlight, joyful, strong, active
A+ 27/16 (33/24) restless, playful
B+ 243/128 (35/27) virile, sensuous

Intervals of the cycle of fifths

3. Intervals of the cycle of fourths (3 as the denominator)
Meaning : passive, moonlight

Db 256/243 (28/35) gentle, affectionate, calm
Eb 32/27 (25/33) gentle, loving
F 4/3 (22/3) peaceful, calm, tranquil, passive
Ab 128/81 (27/34) tender, gentle
Bb 16/9 (24/32) beauty, gentle

Intervals of the cycle of fourths

4. Intervals based on major thirds (5 as the numerator)
Meaning : emotive

D- 10/9 (5×2/32) anxious, weak, fearful
E 5/4 (5/22) loving, pleasing
F- 320/243 (5×26/35) doubt, unstable
F# 45/32 (5×32/25) uncertain, emotional
A 5/3 (5/3) soft, calm
B 15/8 (5×3/23) soft, pleasing, loving

Intervals based on major thirds

5. Intervals based on minor thirds (5 as the denominator)
Meaning : sensual, passionate

Db+ 16/15 (24/5×3) erotic, loving, amourous
Eb+ 6/5 (3×2/5) passionate
F+ 27/20 (33/5×22) aggressive, dangerous
F#+ 64/45 (26/5×32) active, vital, passionate
Ab+ 8/5 (23/5) amorous, enterprising
Bb+ 9/5 (32/5) desire, erotic, anxiety

Intervals based on minor thirds

6. Intervals based on double major thirds (5×5 as the numerator)
Meaning : intense emotion, sadness

Db- 25/24 (52/3×23) sad, distressed
Eb- 75/64 (52x3/26) sad, disconsolate
F#- 25/18 (52/32x2) intensely painful
Ab- 25/16 (52/24) deep sorrow
Bb- 225/128 (52x32/27) shopeless, resigned

 

The semantic scale

From Alain Daniélou’s book, “Sémantique Musicale”

For each note of the “Semantic” scale, the following table gives the related ratio (both in fractional form and as related to the power of 2, 3 and 5), as well as the character of the interval separating the given note from the tonic.

Degree Ratio Fraction Character
C * 1/1 – –
C+ 81/80 (34/5×24) –
Db- 25/24 (52/3×23) sad, pathetic
Db * 256/243 (28/35) gentle, affectionate, calm
Db+ * 16/15 (24/5×3) erotic, affectionate, amorous
D- * 10/9 (5×2/32) anxious, weak, fearful
D * 9/8 (32/23) vigorous confident
D+ 256/225 (28/52x32) –
Eb- 75/64 (52x3/26) sad, desolate
Eb * 32/27 (25/33) loving, gentle
Eb+ * 6/5 (3×2/5) passionate
E- 100/81 (52x22/34) –
E * 5/4 (5/22) tender, pleasing
E+ * 81/64 (34/26) awake, lively
F- 675/512 (5×26/35) uncertain, instable
F * 4/3 (22/3) peaceful, calm, tranquil, passive
F+ * 27/20 (33/5×22) aggressive, dangerous
Degree Ratio Fraction Character
F#- 25/18 (52/32x2) intensely painful
F# * 45/32 (5×32/25) uncertain, emotional
F#+ ** 64/45 (26/5×32) active, vital, passionate
G- 40/27 (5×23/33) –
G * 3/2 (3/2) sunlike, joyful, strong, active
G+ 243/160 (35/5×25) –
Ab- 25/16 (52/24) sorrow, despair
Ab * 128/81 (27/34) tender, soft
Ab+ * 8/5 (23/5) loving, enterprising
A- 400/243 (52x24/35) –
A * 5/3 (5/3) soft, pleasing, tender
A+ * 27/16 (33/24) active, playful
Bb- 225/128 (52x32/27) hopeless, resigned
Bb * 16/9 (24/32) beauty, gentle
Bb+ * 9/5 (32/5) desire, erotic anxiety
B- 50/27 (52x2/33) –
B * 15/8 (5×3/23) soft, loving
B+ * 243/128 (35/27) virile, vigorous
C- 160/81 (5×25/34) –
C * 2/1 – –

 

The intervals of the semantic I

From Alain Daniélou’s book, “Sémantique Musicale”

For each of the notes of the scale, the following table gives the ratio (both in fractional form and in cents), corresponding to the interval separating the given note from the previous one.

Degree Ratio intervals cents raga intervals
C * 1/1 – –
C+ 81/80 21.51 –
Db- 25/24 49.17 –
Db * 256/243 19.55 90.22
Db+ * 16/15 21.51 21.51
D * – 10/9 70.67 (21.51 + 49.17) 70.67
D * 9/8 21.51 21.51
D+ 256/225 19.55 –
Eb- 75/64 51.12 –
Eb * 32/27 19.55 90.22
Eb+ * 6/5 21.51 21.51
E- 100/81 49.17 –
E * 5/4 21.51 70.67
E+ * 81/64 21.51 21.51
F- 675/512 68.72 (19.55 + 49.17) –
F * 4/3 21.51 90.22
F+ * 27/20 21.51 21.51
F#- 25/18 49.17 –
F# * 45/32 21.51 90.22
F#+ ** 64/45 19.55 92.18
Degree Ratio intervals cents raga intervals
G- 40/27 70.67 (21.51 + 49.17) –
G * 3/2 21.51 90.22
G+ 243/160 21.51 –
Ab- 25/16 49.17 –
Ab * 128/81 19.55 21.51
Ab+ * 8/5 21.51 70.67
A- 400/243 49.17 –
A * 5/3 21.51 21.51
A+ * 27/16 21.51 90.22
Bb- 225/128 70.67 (19.55 + 49.17) –
Bb * 16/9 19.55 21.51
Vb+ * 9/5 21.51 70.67
B- 50/27 49.17 –
B * 15/8 21.51 21.51
B+ * 243/128 21.51 –
C- 160/81 68.72 (21.51 + 49.17) –
C * 2/1 21.51 90.22

 

The intervals of the semantic II

From Alain Daniélou’s book, “Sémantique Musicale”

For each of the notes of the Semantic scale, the following table gives the related ratio (both in fractional form and in decimals), as well as the entire number of cents corresponding to the interval separating the given note from the tonic. Also indicated are the nearest note on the tempered scale, the number of cents corresponding to the interval separating the said note from the tonic, as well as the number of cents to be added or substracted to obtain the Semantic pitch.

Scale Note Ratio Decimal Interval equivalent note Interval in cents Nearest tempered note Transposition in cents
C 1/1 – – C (0) 0
C+ 81/80 1,0125 22 C (0) + 22
Db- 25/24 1,0417 71 C# (100) – 29
Db 256/243 1,0535 90 C# (100) – 10
Db+ 16/15 1,0667 112 C# (100) + 12
D- 10/9 1,1111 182 D (200) – 18
D 9/8 1,125 204 D (200) + 4
D+ 256/225 1,1378 223 D (200) + 23
Eb- 75/64 1,1719 275 D# (300) – 25
Eb 32/37 1,1852 294 D# (300) – 6
Eb+ 6/5 1,2 316 D# (300) + 16
E- 100/81 1,2346 365 E (400) – 35
E 5/4 1,25 386 E (400) – 14
E+ 81/64 1,2656 408 E (400) + 8
Scale Note Ratio Decimal Interval equivalent note Interval in cents Nearest tempered note Transposition in cents
F- 320/243 1,3169 477 F (500) – 23
F 4/3 1,3333 498 F (500) – 2
F+ 27/20 1,35 520 F (500) + 20
F#- 25/18 1,3889 569 F# (600) – 31
F# 45/32 1,4062 590 F# (600) – 10
F#+ 64/45 1,4222 610 F# (600) + 10
G- 40/27 1,4815 680 G (700) – 20
G 3/2 1,5 702 G (700) + 2
G+ 243/160 1,5187 723 G (700) + 23
Ab- 25/16 1,5625 773 G# (800) – 27
Ab 128/81 1,5802 792 G# (800) – 8
Ab+ 8/5 1,6 814 G# (800) + 14
A- 400/243 1,6461 863 A (900) – 37
A 5/3 1,6667 884 A (900) – 16
A+ 27/16 1,6875 906 A (900) + 6
Bb- 225/128 1,7578 977 A# (1000) – 23
Bb 16/9 1,7778 996 A# (1000) – 4
Bb+ 9/5 1,8 1018 A# (1000) + 18
B- 50/27 1,8519 1067 B (1100) – 33
B 15/8 1,875 1088 B (1100) – 12
B+ 243/128 1,8984 1110 B (1100) + 10
C- 160/81 1,9753 1178 C (1200) – 22