Summary

Table of the intervals and kleismic variations of the Semantic-53 scale

This table contains all the intervals, within an octave, that can be found between two notes of the full Semantic-53 scale.
They are classified in the 1st column by degrees, that is the number of keys of the instrument to cover the whole range between the notes of the interval, in other words, the sum of the numbers of commas and disjunctions separating them.
Except for the unison (or the octave), for any interval of a given number of degrees, according to its position two variants are encountered : the most common option, identical to the 53 intervals stemming from 1/1 ( C ), is written in bold face, and its variation, deriving from other notes of the scale, is written in italics.
Thus, the Semantic scale contains 53 main intervals and 52 variants, totalizing 105 different intervals.

The 2nd column indicates the size of all these intervals, in integers of kleismas (that are 171 per octave).
This measure is the simplest we can find to refer to the notes or intervals of the Semantic scale.
Because of the perfectly-balanced division of the octave performed by the Semantic-53 scale, for a given number of degrees (other than 0 or 53) there exists only two possibilities of intervals and always differing from one kleisma. Whatever the number of degrees and its position in the scale, an interval indeed can have only two possible numbers of disjunctions, differing from one unit.
For example, the 1st degree (= separating two any consecutive keys of the keyboard) can be only a syntonic comma (Pramana shruti, of 3 kleismas), or one disjunction (comma septimal, of 4 kleismas) ;
Other example, the 5th degree can be only an apotome (16/15) composed of 4 commas + 1 disjunction (that sums up to 16 kleismas), or a septimal semitone of the 15/14 type, composed of 3 commas + 2 disjunctions (that sums up to 17 kleismas) ;
The 11th degree (semifourth) idem can be of two types: 144/125 or 125/108, etc.

The 3rd column names each one of the 105 intervals, these more or less common names arising depending on the circumstances from several sources : names generally used by microtonalists (for example borrowed from the Scala software, in which these tunings were written), Indian names, size of the intervals or relevant combinations of other intervals from the scale, names of linear temperaments, or even fractal algorithms and other remarkable generators of the semantic system being part of these Semantic Daniélou-53 tunings, etc.

The 4th column indicates the intervals belonging to the 22 shrutis, in the traditional numbering used in Indian music.

The 5th column refers to the chromatic writing of notes, increased or diminished by syntonic commas used by Alain Daniélou in his book “Semantique musicale”.

The 6th column gives the ratios of all intervals in 5-harmonic limit, in their least deviant schismatic variant with the whole system. If the ratios remain the most precise way to define an interval in just intonation, it is necessary not to loose sight of the fact that any interval of the Semantic scale is potentially double and therefore also shows at least one relevant schismatic variation. For example a limma, given for a ratio of 135/128 in this table, can according to its position in the scale be found somewhere else with a value of 256/243, etc. In practice, two notes differing from a schisma are considered as equivalent to the Semantic system.

Expressed alternately in 7-harmonic limit in the 7th column, are reported the ratios of the kleismic variations of the most complex intervals in limit 5, the difference with these later ones being generally of a ragisma (4375 / 4374), approximately one fifth of a schisma.
We observe that for all the kleismic variants of the main intervals, exists a simpler ratio in 7-limit than in 5-limit, and it is also the case for six of the main intervals.

The 8th column gives the values in cents of the 5-limit versions for all the bold intervals, and of the 7-limit versions for all the intervals in italics. These values in cents are necessary to tune the drones, mastertunings, or pitchbends of the instrument.
To tune a drone to one of the 53 notes of the Semantic-53 default scale, it is necessary to use the bold values exclusively. However, it is appropriate to use the italic values whenever their ratios belong to other Semantic Daniélou-53 tunings.

The last column indicates the numbers of ascending fifths necessary to generate every interval.
When negative, their absolute value corresponds to the number of downward fifths (= or of ascending fourths).
We observe that the sum of the absolute values of the number of fifths of an interval and of its kleismic variation is always equal to 53.
The absolute value of the number of fifths necessary for the generation of an interval is also always the number of occurrences of its kleismic variation within the scale. Therefore, from this 9th column we know that the Semantic-53 scale contains 52 fourths or fifths, 51 major tones (9/8), 46 diatonic semitones (16/15), 45 major thirds (5/4), 44 minor thirds (6/5), 34 semitones of Zarlino (27/25), 22 neutral seconds (35/32), 21 Thai thirds (close to 128/105), 20 neutral thirds (close to 16/13), 16 supermajor thirds (9/7), 15 minor septimal thirds (7/6), 14 septimal sevenths (7/4), 8 Turkish major thirds (56/45), etc. The scale being symmetrical, an interval and its complement (ex. 7/5 and 10/7) always have the same number of occurrences.

To browse in this table a series of notes formed by the repetition of the same interval, it is necessary to add the value in kleismas of the generative interval in a continuous way, and to deduct 171 kleismas every time the sum exceeds 171, as to bring back the interval in the initial octave. So to browse a series of fifths it is necessary to constantly add 100 kleismas to the initial note ; or to browse a series of fourths (or downward fifths) it is necessary to constantly add 71 kleismas.
Ex : series of fifths from C back in the initial octave :
0 ; 100 ; 29 (= 200 – 171) ; 129 ; 58 (= 229 – 171) ; 158 ; 87 (= 258 – 171), etc.

Semantic-53 interval table

n° Kleismas Interval Shruti Note L.5 ratio L.7 ratio Cents Fifths
0 0 Unison 0 C 1/1 1/1 0 0
1 3
4		 Pramana shruti, syntonic c.
Septimal comma		 C+
Db – – –		 81/80
20000/19683		 875/864
64/63		 21,506
27,264		 12
-41
2 6
7		 Diesis
Septimal diesis / quartertone		 C++
Db – –		 128/125
250/243		 36/35 41,059
48,770		 24
-29
3 9
10		 Archytas’ 1/3 tone, septim. lagu
Lagu (5th limit)		 C+++
Db –		 648/625
25/24		 28/27
729/700		 62,961
70,672		 36
-17
4 12
13		 Minor semitone, Damlas
Major limma, first shruti		 1 C++++
Db		 6561/6250
135/128		 21/20 84,467
92,179		 48
-5
5 16
17		 Apotome, diatonic semitone
Reverse Zira’at, 1/10 octave		 2 Db+
D – – – –		 16/15
3125/2916		 15/14 111,731
119,443		 7
-46
6 19
20		 Zarlino semitone, 1/9 octave
Tsaharuk, 1/5 fifth, “13/12”		 Db++
D – – –		 27/25
625/576		 175/162
243/224		 133,238
140,949		 19
-34
7 22
23		 Neutral second, Totem
Dlotkot		 Db+++
D – –		 2187/2000
800/729		 35/32
192/175		 155,140
160,897		 31
-22
8 25
26		 Olzal, quarter fifth
Minor whole tone		 3 Db++++
D –		 3456/3125
10/9		 448/405 174,692
182,404		 43
-10
9 29
30		 Major whole tone
Double 17 th harmonic, 96/85		 4 D
Eb – – – –		 9/8
15625/13824		 2025/1792 203,910
211,622		 2
-51
10 32
33		 Double apotome
Septimal whole tone		 D+
Eb – – –		 256/225
2500/2187		 8/7 223,463
231,174		 14
-39
11 35
36		 Low semifourth, Semka
High semifourth, 37/32		 D++
Eb – –		 144/125
125/108		 280/243
81/70		 244,969
252,680		 26
-27
12 38
39		 Septimal minor third
Augmented second		 D+++
Eb –		 729/625
75/64		 7/6 266,871
274,582		 38
-15
13 41
42		 Basepbis, 85/72
3rd limit minor third		 5 D+++
Eb		 18432/15625
32/27		 189/160 288,377
294,135		 50
-3
14 45
46		 5th limit minor third
Superkleismic, double Dlotkot		 6 Eb+
E – – –		 6/5
3125/2592		 135/112 315,641
323,353		 9
-44
15 48
49		 Double Zalzal (54/49)^2
Thaï third, “39/32”		 Eb++
E – –		 243/200
625/512		 175/144
128/105		 337,148
342,905		 21
-32
16 51
52		 Rast third, Mogar, “16/13”
Dble minor tone, 79/64		 Eb+++
E –		 19683/16000
100/81		 315/256
216/175		 359,050
364,807		 33
-20
17 54
55		 Turkish major third
5th limit major third		 7 Eb++++
E		 3888/3125
5/4		 56/45 378,602
386,314		 45
-8
18 58
59		 3rd limit major third
Riham, 10 steps of 29-edo		 8 E+
F – – – –		 81/64
25000/19683		 80/63 407,820
413,578		 4
-49
19 61
62		 Daghboc, dim. 4th, 41/32
Supermajor septimal third		 E++
F – – –		 32/25
625/486		 9/7 427,373
435,084		 16
-37
20 64
65		 Augm. 3rd, quadruple apotome
High augmented third		 E+++
F – –		 162/125
125/96		 35/27
729/560		 449,275
456,986		 28
-25
21 67
68		 Septimal fourth
Biseptimal Slendro fourth		 E++++
F –		 6561/5000
320/243		 21/16
1152/875		 470,781
476,539		 40
-13
22 70
71		 Persian fourth, 85/64
3rd limit fourth		 9 E+++++
F		 20736/15625
4/3		 896/675 490,333
498,045		 52
-1
23 74
75		 Fourth plus pramana s.
Fourth + septimal comma		 10 F+
F# – – –		 27/20
80000/59049		 256/189 519,551
525,309		 11
-42
24 77
78		 Fourth +diesis, Zinith
Septimal neutral fourth		 F++
F# – –		 512/375
1000/729		 175/128
48/35		 539,104
546,815		 23
-30
25 80
81		 7th + 3rd limit minor thirds
Major third + minor tone		 F+++
F# –		 864/625
25/18		 112/81
243/175		 561,006
568,717		 35
-18
26 83
84		 Septimal tritone
Diatonic tritone		 11 F++++
F#		 4374/3125
45/32		 7/5 582,512
590,224		 47
-6
27 87
88		 Reverse tritone
Euler’s septimal tritone		 12 F#+
G – – – –		 64/45
3125/2187		 10/7 609,776
617,488		 6
-47
28 90
91		 Double minor third
9/7 + 9/8, “13/9”		 F#++
G – – –		 36/25
625/432		 350/243
81/56		 631,283
638,994		 18
-35
29 93
94		 Double Aksaka
Narayana, rev. Zinith		 Gb+++
G – –		 729/500
375/256		 35/24
256/175		 653,185
660,896		 30
-23
30 96
97		 Fifth minus septimal com.
Fifth minus pramana		 Gb++++
G –		 3456/3125
40/27		 189/128 674,691
680,449		 42
-11
31 100
101		 3rd limit perfect fifth
Persian fifth, 128/85		 13 G
Ab – – – –		 3/2
15625/10368		 675/448 701,955
709,667		 1
-52
32 103
104		 Fifth plus pramana s.
Septimal extended fifth		 G+
Ab – – –		 243/160
10000/6561		 875/576
32/21		 723,461
729,219		 13
-40
33 106
107		 Low trisemifourth
High trisemifourth		 G++
Ab – –		 192/125
125/81		 54/35 743,014
750,725		 25
-28
34 109
110		 Septimal minor sixth
Low minor 6th, dble 5/4		 G+++
Ab –		 972/625
25/16		 14/9 764,916
772,627		 37
-16
35 112
113		 Mahir, 19 steps of 29-edo
3rd limit minor sixth		 14 G++++
Ab		 24576/15625
128/81		 63/40 786,422
792,180		 49
-4
36 116
117		 5th limit minor sixth
Turkish minor sixth		 15 Ab+
A – – –		 8/5
3125/1944		 45/28 813,686
821,398		 8
-45
37 119
120		 Double Zalzal
Bayati sixth, “13/8”		 Ab++
A – –		 81/50
625/384		 175/108
512/315		 835,193
840,950		 20
-33
38 122
123		 Thaï sixth, “64/39”
Double Daghboc		 Ab+++
A –		 1024/625
400/243		 105/64
288/175		 857,095
862,852		 32
-21
39 125
126		 Turkish 6th, rev. Superkleismic
Major sixth		 16 Ab++++
A		 5184/3125
5/3		 224/135 876,647
884,359		 44
-9
40 129
130		 3rd limit major sixth
Sybis, reverse Basepbis		 17 A+
Bb – – – –		 27/16
15625/9216		 320/189 905,865
911,623		 3
-50
41 132
133		 Diminished seventh
Supermajor septimal sixth		 A++
Bb – – –		 128/75
1250/729		 12/7 925,418
933,129		 15
-38
42 135
136		 Triple minor third
Rev. semifourth, 111/64		 A+++
Bb – –		 216/125
125/72		 140/81
243/140		 947,320
955,031		 27
-26
43 138
139		 Harmonic seventh
Rev. double apotome		 A++++
Bb –		 2187/1250
225/128		 7/4 968,826
976,537		 39
-14
44 141
142		 Persian minor 7th, 85/48
Minor 7th, dble fourth		 18 A+++++
Bb		 27648/15625
16/9		 567/320 990,332
996,090		 51
-2
45 145
146		 High minor seventh
Reverse Olzal		 19 Bb+
B – – –		 9/5
3125/1728		 405/224 1017,596
1025,308		 10
-43
46 148
149		 Reverse Dlotkot
Neutral seventh, “117/64”		 Bb++
B – –		 729/400
4000/2187		 175/96
64/35		 1039,103
1044,860		 22
-31
47 151
152		 Reverse Tsaharuk, 59/32
Rev. Zarlino semitone		 Bb+++
B –		 1152/625
50/27		 448/243 1059,051
1066,762		 34
-19
48 154
155		 Zira’at
Major seventh – 15th h.		 20 Bb++++
B		 5832/3125
15/8		 28/15 1080,557
1088,269		 46
-7
49 158
159		 Reverse limma
Major 6th + septimal tone		 21 B+
C – – – –		 256/135
12500/ 6561		 40/21 1107,821
1115,533		 5
-48
50 161
162		 Reverse lagu, 123/64
Supermajor septimal 7th		 B++
C – – –		 48/25
625/324		 27/14 1129,328
1137,039		 17
-36
51 164
165		 Rev. septimal quartertone
Triple major third		 B+++
C – –		 243/125
125/64		 35/18 1151,230
160,897		 29
-24
52 167
168		 Octave – septimal comma
Octave minus pramana		 B++++
C –		 19683/10000
160/81		 63/32 1172,736
1178,494		 41
-12
53 171 Octave 22 C 2/1 2/1 1200 0