Topic 59: Difference between the Semantic Danielo-53 System and 22 shrutis

Alain Daniélou (1907-1994), an expert of European musical intervals, had studied the Indian Classical music too (22 shrutis) for a large part of his life.

Danielou wanted to evolve a system that comes closer to many of the world music systems. Hence, he developed a musical scale of 53 notes, only using the ratios of the prime factors 2, 3 and 5, which is similar to the 22-Shruti system.

The attempt was of course laudable, but the Semantic Daniélou-53 system differs from 22 shrutis as shown below. Only 19 out of the 22 shrutis (shown in Red) are present in the Semantic Daniélou-53 system as shown in the following table.

The details of the 3 missed shrutis are given at the bottom of the table.

Number Note Ratio Cents Interval (Shruti)
0 C 1-Jan 0 Unison
1 C+ 81/80 21,506 Pramana shruti, syntonic comma
2 C++ 128/125 41,059 Diesis, small quartertone
3 Db- 25/24 70,672 5-limit Lagu
4 Db 135/128 92,179 Major limma, 1st shruti
5 Db+ 16/15 111,731 Diatonic semitone, apotome (r2)
6 Db++ 27/25 133,238 Zarlino semitone
7 D-- 800/729 160,897 High neutral 2nd, Dlotkot
8 D- 10-Sep 182,404 Minor whole tone (R1)
9 D 9-Aug 203,910 Major whole tone (R2)
10 D+ 256/225 223,463 Double apotome
11 D++ 144/125 244,969 Low semifourth
12 Eb- 75/64 274,582 Low minor third
13 Eb 32/27 294,135 3-limit minor third (g1)
14 Eb+ 6-May 315,641 5-limit minor third (g2)
15 Eb++ 243/200 337,148 Double Zalzal (54/49)^2
16 E- 100/81 364,807 Double minor tone
17 E 5-Apr 386,314 5th harmonic major third (G1)
18 E+ 81/64 407,820 3-limit major third (G2)
19 E++ 32/25 427,373 Supermajor third, Daghboc
20 F-- 125/96 456,986 Hypermajor third
21 F- 320/243 476,539 Biseptimal slendroic fourth
22 F 4-Mar 498,045 3-limit natural fourth (M1)
23 F+ 27/20 519,551 Fourth + pramana shruti (M2)
24 F++ 512/375 539,104 Fourth + diesis, Zinith
25 F#- 25/18 568,717 Major third + minor tone
26 F# 45/32 590,224 Diatonic tritone, 11th shruti (m1)
27 F#+ 64/45 609,776 High tritone, 12th shruti
28 F#++ 36/25 631,283 Double minor third
29 G-- 375/256 660,896 Narayana, reverse Zinith
30 G- 40/27 680,449 Fifth minus pramana
31 G 3-Feb 701,955 3rd harmonic perfect fifth (P)
32 G+ 243/160 723,461 Fifth plus pramana
33 G++ 192/125 743,014 Low trisemifourth
34 Ab- 25/16 772,627 Low minor sixth, double 5/4
35 Ab 128/81 792,180 3-limit minor sixth (d1)
36 Ab+ 8-May 813,686 5-limit minor sixth (d2)
37 Ab++ 81/50 835,193 Double Zalzal
38 A- 400/243 862,852 Double Daghboc
39 A 5-Mar 884,359 5-limit major sixth (D1)
40 A+ 27/16 905,865 3-limit major sixth (D2)
41 A++ 128/75 925,418 Supermajor sixth
42 Bb-- 125/72 955,031 Reverse semifourth
43 Bb- 225/128 976,537 Low minor seventh
44 Bb 16-Sep 996,090 3-limit minor seventh (n1)
45 Bb+ 9-May 1,017,596 5-limit minor seventh (n2)
46 Bb++ 729/400 1,039,103 Low neutral seventh
47 B- 50/27 1,066,762 Reverse Zarlino semitone
48 B 15-Aug 1,088,269 Major seventh, 15th harmonic (N1)
49 B+ 256/135 1,107,821 High major seventh, 21th shruti
50 B++ 48/25 1,129,328 Reverse 5-limit Lagu
51 C-- 125/64 1,158,941 Triple major third
52 C- 160/81 1,178,494 Octave minus pramana
53 C 2-Jan 1,200,000 Octave (S’)

It can be appreciated that 3 out of 22 shrutis are missing in the above table, namely, r1 (256/243), m2 (729/512) and N2 (243/128).

This is because the system does not ‘strictly’ follow the ‘Poorna-Praman-Nyuna’(mathematical ratios in a series) of the 22 shruti-system (See Diagram on our homepage), e.g., Danielou gives 135/128 (1.0546875) which is close to r1 (256/243 or 1.053497942), 64/45 (1.42222222) which is close to m2 (729/512 or 1.423828125),and 256/135 (1.8962962962) which is close to N2 (243/128 or 1.8984375). 

The differences are small but there is no reason to have imperfect numbers for the 22 shrutis, when perfect numbers can be easily selected.

In short, the above Danielou’s 53-note system is close to 22 shrutis, but not perfect.



Leave a Reply

Your email address will not be published. Required fields are marked

  1. You are wrong about your statement regarding Danielou

    The 22 shruti scheme published by Danielou in Music and the Power of Sound (1996) is identical to your scheme apart from the assignment of the fractional value for just one shruti 12, which Danielou assigns as the just tritone (64/45) and you the Pythagorean augmented fourth (729/512). These two notes differ by just 1.954 cents. Look at Fig 11 (p 80) and Fig 17 (p 142-145).

    For shruti 1 he gives it (in that publication) as 256/243 – the Pythagorean limma just as you do , but with an alternative of 135/128. The same goes for shruti 21 243/128 with alternative 256/135. So these shrutis are not missing in that scheme published by Danielou. Also. these notes differ by just 1.953 cents, which are effectively indistinguishable by any normal mortal being.

    I find it impossible to believe that Indian musicians can reproducibly produce these exact relative frequencies when singing or playing an instrument. In fact they can't and they don't as shown by Wim van der Meer. Every swara sung has a significant range of pitch, which depends on the artist, the raga, and the specific performance. So your efforts and everyone else's on this matter are merely academic and mathematical exercises.

{"email":"Email address invalid","url":"Website address invalid","required":"Required field missing"}