Summary

THIS IS AN ATTEMPT TO DEMYSTIFY “INDIAN CLASSICAL MUSIC AND HOW IT WAS USED”                   THIS IS AN ATTEMPT TO UNDERSTAND WHY SAMA VEDA WAS CALLED SAMA VEDA                              THIS IS AN ATTEMT TO USE SA TO MA  or TONE C TO TONE F  for Cure & Fitness !                               THIS IS AN ATTEMT TO FIND HOW MUSIC (AMUSES &)  CURES SIC MINDS

In summary 22 shrutis have their origin in mathematical value of Pi (π) which is taken as 22/7  so when we have 7 swaras a complete “swaramandal” or cycle will have 22 shruti places !                                                                                                                                                                       Now these 22 places can be taken as equidistant , similar to 12 TET scale which is used in western world. Here A is fixed on 440 hz. Remaining  are  fixed in Twelve Equitempered Tone scale, with a ratio of 2(1/12)  between each note. This scale is represented in terms of equidistant 100 cents where each 100 cents correspond to ratio of  1.0594631.  So 1 cent corresponds to 1.010594631 as it is logarithmic scale !                           The whole scale is comprised of 1200 cents with 1200 cents Being exact 2.0 of any 1.0

In terms of rational numbers (ratios) it is amusing to find out that 1.0594631 which is 100 cents translates to an exact ratio of 31,783,893/30,000,000.  So one cent is frequency ratio of  31783893/3,000,000,000 or 0.010595 cents and distance ratio of 300,000/31783893.

 so 1 per cent or 1 cent is 31,783,893/3,000,000,000 which is 0.010594631 ratio  !                                           so 100 % is 105.94631 which is 100 cent                                    But we have method to convert ratio of frequencies to cents as follows

Formula for converting the interval frequency ratio f2 / f1 to cents (c or ¢).
¢ or c = 1200 × log2 (f2 / f1)
log 2 = 0.301029995

This formula employs a log 2, or logarithm base 2 function. This formula can also be
written using a log 10 function, available on most scientific calculators via the log button:

c = 1200 × 3.322038403 log10 (f2 / f1)
1/log 2 = 1/0.301029995 = 3.322038403
The formula expressed using log10 rather than log 2.

3.322038403 is a conversion factor that converts base 2 logarithms to base 10 logarithms.
1 Cent = 2(1/1200) = 1.0005777895065548592967925757932
One cent is thus the number that multiplied by itself 1200 times results in the number 2.
The cent is an interval which is calculated from the interval frequency ratio as follows:
(In of the interval frequency ratio / ln 2)×1200 = cents value of the interval.
An interval of a halftone is equivalent to: 2(1/12) = 1,0594630943592952645618252949463.
That is: [ln (2(1/12)) / ln (2)]×1200 cent = 100 cent.

 

……. But In TET scale Acual 12 times  multiplication gives 2.000000119 So It appears last tone before octave C is taken on 12/11 ratio which would be on 261.818181  or  11/12 on chord & of Indian value of Upper  Sa or Octave C is 480 frequency so its 11/12 is 440 !        so 440 is taken as A or base tone (& if C is taken then C is taken on 261.6255528  ! ) …. Further C is taken as 16/27 of A as per Indian Ratio then 16/27 of 440 is 260.740740…… So for all practical purpose it comes to 261 any way !                                                                                Here we also note that In Gandhar Gram mentioned in India C was kept on Ni or Nishad or second last tone of Octave …. This appears the reason why A was the name given to first tone of so called western tones with first 7 alphabets names given to tones there by forming ABCDEFG these seven tones and eighth called octave A ……. In reality Indian Sa or Shadja would correspond to Ni or B of western octave with values of Indian Ni or Nishad or B being B flat on 432 & B sharp on 450 on a difference of 18 hz with TET or Equi Tempered scale keeping This B on 440  at approximate being the centre for Equitemparament  keeping 440 to 480 block bigger or 4 shruti  ! leaving rest all slots 3 shruti !                                                                                                                                                              so we get equi tempered scale of 12 tones with A # on 440 frequency                                        & D Being on 12/11 of C or Sa of Indian placed on 240 ,  D on 12/11 ratio is on 261.818181 which is almost the value of  of C of the western scale which is 261.6255528 !                     Reverse if yous see  12/11 of 440 of  A# tone then we get 480 which is value of 4 shruti  tone  called Sa or Shadja    

In Indian Music Shadja has been taught as of frequency 240 from the beginning with three Gram systems  1) Shadja Gram or one group of Tones or scale where C-G ratio 3/2 is the rule for all tones related with this rule which is like pythagorian scale but with tones corrcted for the pythagorian  Comma because of ratio propoertions calculated on chord lengths.                                                         2) Madhyam Gram or one group of Tones or scale where C-F ratio 4/3 is the rule for all tones related with this rule which is again like reverse pythagorian scale but with tones corrcted for the pythagorian Comma because of ratio propoertions calculated on chord lengths.                             3) Gandhar Gram which is again one group of Tones or scale where C-E ratio 6/5 is the rule for all tones related with this rule which is like pythagorian scale but with tones corrcted for the purpose of making it Twelve Equitempered Tone yet related to original positions to make it equidistant  with ratio propoertions calculated on chord lenghts.                         Yet Indians Give importance to 3/2 ratio & 4/3 ratio & 6/5 & 5/4 ratios of chord lenghts.

Now it will be interesting to know the importance of 440 frequency as per Indian concept Shadja or Sa or C is placed on 240 with corresponding G or Pa or Pancham on 360 on 3/2 ratio   Here we seen This is a scale between 240 to 480 frequency with 12 tones Equitempered or Equidistance Shruti the word used for frequency (called shruti for ability to differentiate by ear) we will have 20 hz frequecy difference for Equitempered scale as 240 was C naturally previous to octave C on 480 would be on 440 to maintains C as 4 shruti or larger compartment with scale discribe as 4 – 3 -3 – 3 – 3 – 3- 3 shruti swaras with 220 being the fequency of A from where scale begins  for making it Equitempered …. Gandhar Gram … which was supposed to be played in heaven only  !  THIS IS THE IMPORTANCE OF 440 FREQUENCY WITH LABLE ” A ”  & C on 480 ! There are descriptions where Ni or Nishad or B is supposed to merge into C or Sa with appearence that scale starts on Ni or Nishad or B or Gandhar Gram Sa or C is put on Ni or B  !         So this 440 frequency being used as base for preparing TET has Indian … Universal as word Indian has been in use when Columbus & Wasco D’gama started search for India  !  It is UNIVERSAL MUSIC with starting scale from A on 440 !

Indian Classical Music is given as follows on Wikipedia in shruti section table                         Though it is wrong as 1st shruti of C is  Tivra  &                                                                           Kshobhini is 22nd shruti or Last Shruti of B  !  It is shown & there after corrected in next. 

Shrutis 12-TET Notes 53-TET Notes Perfect FIFTHs
Name Ratio Cents Frequency
(Hz)
Name Frequency
(Hz)
Note
No.
Frequency
(Hz)
FIFTH
No.
Frequency
(Hz)
Kṣobhinī 1 0 261.6256 C 261.6256 0 261.6256 0 261.6256
Tīvrā 256/243 90 275.6220 C♯ 277.1826 4 275.6763 -5 275.622
Kumudvatī 16/15 112 279.0673 5 279.3053 7 279.3824
Mandā 10/9 182 290.6951 D 293.6648 8 290.4816 -10 290.3672
Chandovatī 9/8 203 294.3288 9 294.3056 2 294.3288
Dayāvatī 32/27 294 310.0747 D♯ 311.1270 13 310.1114 -3 310.0747
Ranjanī 6/5 316 313.9507 14 314.1937 9 314.3052
Raktikā 5/4 386 327.0319 E 329.6275 17 326.7661 -8 326.6631
Raudrī 81/64 407 331.1198 18 331.0677 4 331.1199
Krodhā 4/3 498 348.8341 F 349.2282 22 348.8478 -1 348.8341
Vajrikā 27/20 519 353.1945 23 353.4401 11 353.5933
Prasāriṇī 45/32 590 367.9109 F♯ 369.9944 26 367.5829 -6 367.496
Prīti 729/512 612 372.5098 27 372.4218 6 372.5098
Mārjanī 3/2 702 392.4383 G 391.9954 31 392.4229 1 392.4384
Kṣiti 128/81 792 413.4330 G♯ 415.3047 35 413.4982 -4 413.433
Raktā 8/5 814 418.6009 36 418.9415 8 419.0736
Sandīpanī 5/3 884 436.0426 A 440.0000 39 435.7053 -9 435.5508
Ālāpinī 27/16 906 441.4931 40 441.441 3 441.4932
Madantī 16/9 996 465.1121 A♯ 466.1638 44 465.1488 -2 465.1121
Rohiṇī 9/5 1017 470.9260 45 471.2721 10 471.4578
Ramyā 15/8 1088 490.5479 B 493.8833 48 490.1298 -7 489.9947
Ugrā 243/128 1110 496.6798 49 496.582 5 496.6798
Kṣobhinī 2 1200 523.2511 C 523.2511 53 523.2512 0 523.2511

 

Before comming to corrected table certain things need explainations ….                                     The intervals of 12-TET closely approximate some intervals in just intonation. The fifths and fourths are almost indistinguishably close to just.

In the following table the sizes of various just intervals are compared against their equal-tempered counterparts, given as a ratio as well as cents.

In the following table the sizes of various just intervals are compared against their equal-tempered counterparts, given as a ratio as well as cents.The intervals of 12-TET closely approximate some intervals in just intonation. The fifths and fourths are almost indistinguishably close to just.

Western Pyathgorian  Scale Here he has tried to get tones with C-G(3/2) or C-F (3/4) ratios             Here table-pythagoras

 

In the following table formation of Each Tone & its value in Llogarithmic scale  & decimal calue of the square root on calculators are given with each defined as 100 cents … Further values of Ratio Proportions in just intonation which is same for Indian natural scale except F #  or Teevra Madhyam tone of Indian natural scale Based on Shadja Gram or Ratio of 3/2                           which would be  3/2 of C # so 64/45 or ……                                                                                                More correct wound be 45/32 which is 3/2 of 15/8 B #   of precious Octave B as 3/2 of 4/3 F or Ma or Madhyam is 2/1  !   

Here we See that 16/15 is 1.06666 & 1.0594631 is  100 cents  111.73 as

c = 1200 × 3.322038403 log10 (f2 / f1)
1/log 2 = 1/0.301029995 = 3.322038403

Here we See that 16/15 is 1.06666 is  111.74 cents                                                                                   Here we See that 135/128 is 1.0546875 is 92.18 cents                                                                       Here we See that 25/24 is 1.0416666 is  70.67 cents                                                                         Here we See that 81/80 is 1.0125 is  21.51 cents

 Here we See that 16/15 is 1.06666 is  111.73  so
Name Exact value in 12-TET Decimal value in 12-TET Cents Just intonation interval Cents in just intonation Error
Unison (C)  2 raised to  0/12 1.000000 0 1/1 = 1.0000000 0.00 0
Minor second (C♯/D♭)  2 raised to  1/12 1.059463 100 16/15 = 1.06666 111.73 −11.73
Major second (D)  2 raised to  2/12 1.122462 200  9/8= 1.1250000 203.91 −3.91
Minor third (D♯/E♭)  2 raised to  3/12 1.189207 300 6/5 = 1.2000000 315.64 −15.64
Major third (E)  2 raised to  4/12 1.259921 400  5/4= 1.2500000 386.31 +13.69
Perfect fourth (F)  2 raised to  5/12 1.334840 500  4/3= 1.33333… 498.04 +1.96
Tritone (F♯/G♭)  2 raised to 6/12 1.414214 600  7/5= 1.4000000 582.51 +17.49
Perfect fifth (G)  2 raised to 7/12 1.498307 700  3/2= 1.5000000 701.96 −1.96
Minor sixth (G♯/A♭)  2 raised to  8/12 1.587401 800  8/5= 1.6000000 813.69 −13.69
Major sixth (A)  2 raised to  9/12 1.681793 900  5/3= 1.66666… 884.36 +15.64
Minor seventh (A♯/B♭)  2 raised to  10/12 1.781797 1000  16/9= 1.77777… 996.09 +3.91
Major seventh (B)  2 raised to  11/12 1.887749 1100  15/8= 1.87500 1088.27 +11.73
Octave (C)  2 raised to  12/12 2.000000 1200  = 2.0000000 1200.00 0

c = 1200 × 3.322038403 log10 (f2 / f1)
1/log 2 = 1/0.301029995 = 3.322038403

Here we See that 16/15 is 1.06666 is  111.74 cents                                                                               Here we See that 135/128 is 1.0546875 is 92.18 cents                                                                       Here we See that 25/24 is 1.0416666 is  70.67 cents                                                                         Here we See that 81/80 is 1.0125 is  21.51 cents

But This needs to be modified as per Indian concepts of Shruti which are not equidistant !  Also In Indian Shruti tables 1st shruti of C or Sa or Shadja is Tivra & NOT Kshobhini !            where as 22nd shruti is last shruti of B or Ni or Nishad is called Kshobhini !

Shrutis 12-TET Notes 53-TET Notes Perfect FIFTHs
Name Indian Shruti Ratio Cents Frequency
(Hz)
Name Frequency
(Hz)
Note
No.
Frequency
(Hz)
FIFTH
No.
Frequency
(Hz)
Tīvrā 1 0 261.6256 C 261.6256 0 261.6256 0 261.6256
Kumudvatī 81/80 21.51 264.8959 C♯ 277.1826 4 275.6763 -5 275.622
Mandā 25/24 70.67 272.5267 5 279.3053 7 279.3824
Chandovatī 135/128 92.18 275.9333 D 293.6648 8 290.4816 -10 290.3672
Dayāvatī 16/15 111.74 279.0673 9 294.3056 2 294.3288
Ranjanī 10/9 186.10 290.6951 D♯ 311.1270 13 310.1114 -3 310.0747
Raktikā 9/8 203.91 294.3288 14 314.1937 9 314.3052
Raudrī 6/5 315.65 315.64 E 329.6275 17 326.7661 -8 326.6631
Krodhā 5/4 386.31 327.032 18 331.0677 4 331.1199
Vajrikā 4/3 498.04 348.8341 F 349.2282 22 348.8478 -1 348.8341
Prasāriṇī 27/20 519.57 353.1946 23 353.4401 11 353.5933
Prīti 25/18 568.74 363.3689 F♯ 369.9944 26 367.5829 -6 367.496
Mārjanī 45/32 590.24 367.911 27 372.4218 6 372.5098
Kṣiti 64/45 609.81 372.0897 G 391.9954 31 392.4229 1 392.4384
Raktā 36/25 631.30 376.7409 G♯ 415.3047 35 413.4982 -4 413.433
Sandīpanī 40/27 680.47 387.5935 36 418.9415 8 419.0736
Ālāpinī 3/2 701.98 392.4384 A 440.0000 39 435.7053 -9 435.5508
Madantī 8/5 813.71 418.6011 40 441.441 3 441.4932
Rohiṇī 5/3 884.39 436.0427 A♯ 466.1638 44 465.1488 -2 465.1121
Ramyā 27/16 905.89 441.4932 45 471.2721 10 471.4578
Ugrā 9/5 1017.63 470.9261 B 493.8833 48 490.1298 -7 489.9947
Kṣobhinī 15/8 1088.30 490.548 49 496.582 5 496.6798
Tīvrā 2 1200 523.2512 C 523.2511 53 523.2512 0 523.2511

Basic Concept is as follows

When you take any swara or frequancy as Sa or Shadja , Madhyam , or Pancham ( 4 shruti swaras)  in these if you know Pancham is called Pratishadja as like other Sa or C or Shadja it has 6 shrutis with “swe swe sware api Madhyatvam” In Indian Music One can keep Shadja on any swara as per scale of your voice Still They are achal swaras as as soon as C or Sa or Shadja is fixed its G or Pa or Pancham or fifth get automatic position on 3/2 ratio  ! Its Madhyam or F tone comes on 4/3 ratio !  Here you can see Underlined tones are 1st shruti of next swar  Even if you lower Sa from 135/128 further one shruti or 81/80 You get  2187/2048 = 1.067810938 which crosses 16/15 = 1.066666 ! so becomes  in Next swar area !  Here comes the “VIVADI” concept if one is able to differentiate in so much detail which is not possible except experts They feel something has gone wrong !   Here we must understand Underlined are 15/8 Last shruti of Ni or Nishad or B sharp !  Further 16/15 is 1st shruti of Re or Rishabh or D flat !  And 8/5 is 1st shruti of Dha or Dhaivat or A flat !   In the following chart we can see what would be the ratio

Here we see Any 4 shruti swara say 1/1 will have four shrutis 81/80 , 25/24 , 135/128 & 16/15       Further its 3 shruti which would be 7 th shruti ( 4 + 3) will be on 9/8 shruti ratio with 5th shruti can be put on 27/25 with ati-komal very flat on 16/15  & 6th shruti on 10/9 ratio Here whether to use Very flat tone on 16/15 or a little higher 27/25 depends on the choice of the master and the listeners moods and what mood is to be created…. this is the buty of Indian Classical Music

In the following table one would seen how Shadja , Madhyam , & Pancham have three shruti ratios before & after their own ratios which are 1/1 , 4/3 & 3/2  respectively …. Where as Re or Rishabha or D Tone and Dha or Dhaivat or A Tone have only 3 shruti

 Fequancy  Ratios of tones                                                                  No of  Shrutis distance

15/8            256/135             48/25            160/81                1/1                     

1/1              81/80                   25/24           135/128             16/15                          4

16/15          27/25                   10/9                9/8                  256/225                       3

9/8              729/640                75/64            32/27                   6/5

6/5              243/200                   5/4              81/64                32/25                           2

5/4              512/405               96/75            320/243             4/3

4/3              27/20                    25/18            45/32                64/45                           4

64/45           36/25                  40/27               3/2          1024/675                             4

3/2              243/160                25/16         405/256               8/5                                4

8/5              81/50                      5/3               27/16           128/75                               3

27/16          2187/1280            225/128      3645/2048         9/5

9/5                729/400                 15/8           243/128          144/75                             2

 

→→→→→→                            →→→→→→                                     This is 1 shruti 81/80 distance        →→→→→→→→→→→→→                                                            This is 2 shruti 25/24  distance                               →→→→→→→→→→→→→→                                 This is 2 shruti 25/24  distance        →→→→→→→→→→→→→ →→→→→→→                            This is 3 shruti 135/128 distance  →→→→→→→→→→→→→ →→→→→→→→→→→→→        This is 4 shruti 16/15 distance             This 4 shruti distance is from 16/15 ratio from 1 st shruti of a 4 shruti swar to 1st shruti of next  3 shruti or 4 shruti swara or even 2 shruti swara.  Further in case of gram system when they say Pancham is brought one shruti down one has to resember that these are swaras of Sama Veda & its rule is   that in Sama gayan or singing of Sama first one is Ma or F Tone , Second is Ga or E tone  further third is Re or D tone , & fourth is Shadja Sa oer C tone  ( Meaning Ma,Ga,Re,Sa or F,E,D,C which is avrohi kram or descending order is followed )  Further if you seen the underlined shloka or part of stanza underlined in the Sansrit rule given bellow –                    Further it says Fifth is Dha or A  ! & here people get confused that when pancham is placed 1 shruti bellow or fith is place 1 shruti bellow but here Old masters say Fifth is Dha or A  & it is put one shruti down Meaning A Sharp is to be played on ration 5/3 instead of  27/16                     & accordingly Re or D is to be placed on 10/9 instead of 9/8 which maintains C-G Ratio or But then These become definitions – of Shruti system !                                                                                                                         when you say 9/8 Ratio this means Sa-Re or Shadja Rishabha or Central shruti C or Shadja (1/1) & Last shruti of D Ratio !

  य सामगायनाः प्रथम वेण्णो मध्यम स्वर                                                                                                         यो द्वितीय गांधार तृतीय ऋषभ स्मृत ।।                                                                                                     चतुर्थः षड्ज इति आहुः पंचमो धैवतो भवेत्  ।                                                                                                           षड्जे निषाद विज्ञेय सफ्तम पंचम स्मृत ।।                                                                                                                Means sequnce would be    Ma , Ga ,Re , Sa ,Dha ,Ni, Pa                                                                        In western tone system this becomes  ( F ,E ,D , C, A ,B, G )                                                    This is supposed to have some simmilarity world wide spiritual toning ! 

 

These are linked to each other in following way according to original Indian Musical System –

1. If First shruti can be labelled as 1/1 this is the shadja (C Tone) of  any person or swar-mandal.

2. Automatically 2/1 becomes23 rd shruti of upper scale shadja (C Tone) (situated on 1/2 on             chord)

3. Further 3/2 becomes shruti of Pancham (G Tone 17th shruti ) (situated on 2/3 on chord                 making 3 parts of cord ).

4. Further 4/3 becomes shruti of Madhyam (F Tone 10th shruti)  (situated on 3/4 on chord               making 4 parts of cord)

5. Further 6/5 becomes first shruti of Gandhar (D# Tone 8th shruti) (situated on 5/6 on chord           1/2 of 2/3 rd Pancham)

6. Further 5/4 becomes second shruti of Gandhar (E Tone 9th shruti)  (situated on 4/5 on chord       making 5 parts of chord)

7. Further 9/8 becomes third shruti of Rishabh ( D Tone 7th shruti) (situated on 8/9 on chord            making further 3 parts of 2/3 Pancham of C-G distance )

8. Further 16/15 becomes first shruti of  Rishabh ( C# tone situated on 5th shruti ) (situated on       15/16 on chord making 4 parts of further 4 parts of 3/4 Ma or C – F distance on chord )

9. Further 10/9 becomes second shruti of Rishabh ( D flat tone on 6th shruti (situated on                   9/10 on chord making 2 parts of 4/5 or Ga or C – E distance )

10. Further 8/5 becomes first shruti of Dhaivat ( G # tone sistuated on 18th shruti  (situated on          5/8 on chord “centre of 4/8 upper scale Sa or C  & 6/8 positon of Pa or G tone” )

11. Further 5/3 becomes second shruti of Dhaivat or A flat tone situated on 19th shruti (situated       on 3/5 on chord centre of lower scale Gandhar (E tone) & upper scale Gandhar (E tone) as           seen in ratio distances lower scale Gandhar (E)  4/5 3/5  2/5 upper scale Gandhar (E) as this         is madhyama gram shruti & is situated one shruti above third shruti of Gandhar on 27/16             which is at a ratio distance of 81/80

12. Further 27/16 becomes third shruti of Dhaivat ( A # tone ) on 20th shruti  (situated on                    chord on  16/27 on chord 8/9 from 2/3 further divided into 27 parts from Pancham or G                tone. Beuty of this is ratio distance from 27/16 A# tone & 5/3 A tone is 81/80 which is                    Praman shruti or Pythagorian coma indicating Univerasal music was based on Sama or                same ! ) Here we can see Ratio distance between D on 10/9 & D# on 9/8 is also 81/80 ! Here        we would also note that A flat would be term for Tone on 8/5, A would be 5/3 & A# would            be tone on 27/16 parallel to three shruti Dhaivat or A Tone ! In simmilar way D flat would              term for tone on 16/15 , with D on 10/9 & D # would be on 9/8 called as three shruti Re or            Rishabh in universal music !

13. Further 9/5 becomes first shruti of Nishad situate on 21st shruti  (situated on 5/9 on chord         situated in centre of 2/3 Pancham or G tone & 1/3 Pancham of upper scale so on previous           one third of  6/9 5/9 4/9 3/9  or    3/2 ratio  of 6/5 Gandhar or 2/3rd of 5/6 Gandhar (E )

14. Further 15/8 becomes second shruti of Nishad ( B Tone)  on 22nd shruti position  (situated         on 8/15 on chord at a ratio distance of 15/16 from 2/1 Sa or Shadja or upper scale C                       position ! Beuty is 0n 16/15 is the 1st shruti of 1st tone of  Re or Rishabh or D tone on the             lower side of scale where as last shruti of Ni or Nishad or B tone is on 15/16 of C or Sa of           upper scale showing how balanced was the scale of Universal ! It can be seen also as  2/3 rd        of 4/5 Gandhar or E tone or 3/2 ratio of 5/4 Gandhar or E tone ! )

Now we come to “Atar-shruti of Shadja” swaras which have 4 shruti ( 3 above & 3 bellow the            self position taken as 1/1 with passover to next swara ratio being 16/15 above or 15/16                bellow ! this being true for any swara which is to be placed on Shadja these were labelled  “Madhya Jati” shruti with rule “Swe swe sureshu Madhyatvam” !                          Here we would see –

15. Further 81/80 becomes first shruti of Shadja ! (situated on 80/81 on chord ratio this                  being evolved from praman shruti ratio of 16/27 & 3/5 Dha or Dhaivat or A tone or                    9/10 & 8/9 of Re or Rishabha or D tone  ) “81/80 or 80/81 being one shruti distance”

16. Further 25/24 becomes second shruti of shadja (situated on 24/25 on chord this being evolved   from ratio distance between  5/4 & 5/6 of Ga or Gandhar or E tone & also ratio distance   between 15/8 & 5/9 of Ni or Nishad or B tones  which are described as dwishrutic swaras !  So “25/24 or 24/25 becomes 2 shruti distance”

17. Further 135/128 becomes third shruti of Shadja (situated on 128/135 on chord ratio of 8/9       & 15/16 Re or Rishabha or D swaras & also ratio distances between 5/8 & 16/27 of Dha or             Dhaivat or A tone which are described as trishrutic swaras ! So “135/128 or 128/125 becomes three shruti distance”

18) Further 16/15 becomes fourth shruti of Shadja ( Situated on 16/15 on chord ratio of 128/135 from Re or D Sharp tone situated on 9/8 which is called three shruti swara ! Here we would note that As per Indian Musical frequencies described with Sa or C on 240 the fourth shruti of Sa or C would come on 240 multiplied by 16/15 which is 256 But third shruti of Sa which is on 135/128 would come on 240 multiplied by 135/128 which is 253.125  or 253 1/8 by ratio. But from this third shruti if we move the swara or tone by 1 shruti which is 81/80 then we get ratio of  135/128 multiplied by 81/80 which is  2187/2048  & from Sa or tone C on 240 we get 256.2890625 but not 256 but for Human ear this is not possible to differentiate so 256 is taken as 1st shruti of Re komal or D flat ! But Theoretically it is shruti of Sa or C Sharp !

19. Further 27/20 becomes first shruti of Madhyam (situated on 20/27 on chord 81/80 of 4/3        Ma or F tone)

20. Further 25/18 becomes second shruti of Madhyam  ( situated on 18/25 on chord 25/24 of 4/3  Ma or F tone )

21. Further 45/32 becomes third shruti of Madhyam (situated on 32/45 on chord 135/128 of            4/3 Ma or F tone )

22. Further 64/45 becomes fourth shruti of Madhyam (situated on 45/64 on chord 128/135 of 3/2  Pa or G) Here we would note that As per Indian Musical frequencies described with Ma or F tone on 320 the fourth shruti of Ma or F would come on 320 multiplied by 16/15 which is        341 1/3 or  341.3333 But third shruti of Ma or F  which is on 135/128 would come on 320 multiplied by 135/128 which is 337.5 or 337 1/2  by ratio. But from this third shruti if we move the swara or tone by 1 shruti which is 81/80 then we get ratio of  135/128 multiplied by 81/80 which is  2187/2048  & from Ma or F tone on 320 we get 341.71875 but not 341.3333 but for Human ear this is not possible to differentiate so 341.3333 is taken as 1st shruti of Pa komal or G flat ! But Theoretically it is shruti of Ma or F Sharp !

23. Further 36/25 becomes third shruti of Pancham (situated on 25/36 on chord 24/25 of 3/2          Pa)

24. Further 40/27 becomes second shruti of pancham (situated on 27/40 on chord 80/81 of 3/2 Pa)

In this 2/1 the upper sa is the same shruti as first swara of upper scale so we have 22 shrutis        of 7 swaras  !  Here No 1 is Sa or C Tone & No 2 is uppper scale of Sa C Tone & So we have 22 shruti of 7 swaras or tones  3rd is Pa or G tone these are achala swaras or Tone

A) Shrutis of chatuhshruti swaras are linked by ratios 81/80, 25/24, 135/128 & 16/15.

B) Shrutis of trishruti swaras are linked by ratios 2087/2048 ( Taken as 16/15) 25/24, 135/128.

C) Shrutis of dwishruti swaras are linked by ratios 25/24.

D) Lowest shruti of upper swara or uppermost shruti of lower swara have ratio of 16/15 or        15/16 respectively.

E) Uppermost shruti of chatuhshruti swara is situated on 135/128 or 128/135 ratio respectively.

Shadja is called shadja because it gives rise to shad-shrutis as well as shad-swaras.

Shadja is any shruti which is taken as 1/1 & has three shrutis above & three shrutis
bellow. Madhayam & Pancham are also called chatuhshruti swar as they have own shrutis defined by ratios 4/3 for Madhyam (F) & 3/2 for Pancham (G) & in addition              
3 shrutis above & bellow like Shadja swara. 

Shruti Jaati mentioned in shloka on shruti jaatis clearly mention Dweepta , Ayata to Karuna naam Mrudu-Madhyamyoh tatha indicating 1st shruti as Dweepta , Last or fourth shruti as Ayata & Karuna being 1st shruti of next swara with mrudu as 2nd shruti & Madhyama as 3 rd shruti ( Dweepta – shrutis being 1/1 , 4/3 , 6/5, 9/5                                                                                                 Karuna – shrutis being 16/15 , 8/5 , 3/2                                                                                                      Ayata – shrutis being 135/128, 9/8, 45/32 , 3/2 , 27/16 ,                                                                             Madhya shrutis being 25/24 , 10/9 , 25/18 ,

How Do we get 25/24 , 135/128 , 16/15 , 81/80 etc ratios ?

1) 25/24 is the ratio distance between two Gandhar Shrutis  6/5 & 5/4  this is one shruti distance for Gandhar swar meaning any swaras which are on 25/24 ratio distance have Shuddha & Teevra Gandhars relationship this is dwishruti shrutyantar of Gandhar swaras.

2) 135/128 is ratio distance between “1nd shruti of chatuhshruti & 4th shruti of same chatuhshruti swar” it is placed on single shrutyantar ratio (same of Gandhar ) 25/24 ratio from 81/80 shruti so also 1st Shruti of chatuhshruti swara Shadja on 1/1 & 3rd shruti shadja on 25/24 this gives 1/1 , 81/80 , 25/24 , & 135/128 shadja shrutis but Shadja has three shrutis bellow 1/1 , 160/81 , 48/25 , & 256/135 which are shrutis of Shadja & Not of Nishad. Nishad is Dwishruti swar with 1st shruti on 9/5 & 2nd shruti on 15/8 which are on also on single shruti distance or shrutiantar of 25/24 and also on Shadja – Pancham. Ratio 3/2 from Gandhar on 6/5 & 5/4

3) Ratio of 16/15 is ratio distance between 2nd shruti of Gandhar 5/4 & 4/3 , 1st shruti of Madhyam The word “Antarshruti” as against “Shrutiantar” is seen here 16/15 is antarshruti distance between 1st shruti of any swar & last shruti of previous swara. This Range 16/15 above to 15/16 bellow is the range of a chatuhshruti swar.  where as 135/128 bellow & above becomes range of three shruti swaras 25/24 above or bellow becomes range of dwi shruti swaras & 81/80 above or 250/243 bellow is the range of one shruti swar movement of dwi shruti swara as total difference in dwishruti swar is 25/24  with two parts 81/80 & 250/243 because this make 25/24 & 81/80 of this becomes 135/128 the three shruti distance & from this on 2048/2025 is 16/15 lowest of upper swara of a four shruti swar distance ! on 16/15 is lower tone of Re(D) Lower tone of E from 9/8 tone of D then lower tone of  Ma (F) from upper tone of Ga (E ) on 5/4 !

“This appears Gamak kriya” “which is same as finding Sa (C) from Teevra Ni (B)” or Finding Ma shruti from Ga shruti or otherway with ratio of  16/15 from Teevra Ga to Shuddha Ma, &  ratio of 15/16 from Shuddha Ma to  Teevra Ga. as this is situated on 15/16 distance of wire cord Indian music has described 15 gamak kriyas !

4) Ratio of 81/80 is the shruti distance between Dhaivat swaras obtained by noting two Dhaivat Shrutis obtained by using Shadja Pancham Bhav (Ratio of 3/2)  &  shadja Madhayam Bhav to get Dhaivat swara from  Rishab on 9/8   by Shadja Pancham Bhav so 3/2 ratio we get 27/16 the last 3rd shruti of Dhaivat where as from Gandhar on 5/4 by Shadja Madhyam Bhav so 4/3 ratio we get 5/3 which is 2nd shruti of Dhaivat.  This also seen in derivation of positions from meaning of shlokas. Now one can note Ratio Distance between 5/3 & 27/16 the two shrutis of Dha is 81/80 This is one shrutianar of Dhaivat & Rishab swaras where as on Gandhar shruti distance of 25/24 from Rishabh 0n 16/15 is seen 2nd shruti Rishabh 10/9 , and also on Gandhar shruti distance of 25/24 from  Dhaivat on 8/5 , is seen 2nd shruti Dhaivat 5/3 One can see how perfect the shruti ratios were placed in grand old Indian classical music !

ITS USE IN MUSIC IN SHORT IS CALLED “RIYAJ”     –   It is Combination of  Ri & YAJ                              RI of Rigved which have Richa which are stanzas of Mantra to be sung in SAMA Ved style for        YAJ is of Yajurved that is POOJA of GOD

So Singing Songs of GOD in different ways at different times one can balance your own body !

Hindu Religion gives so much freedom that one can sing “Radha-Krishna Songs” or                          “Hanuman songs”   or  “Songs of Any Goddess”  Important is your accurate “Swar & Bhav” !

In Western Music because of TET system we have logarithmically equidistant swaras or Tones which are ruling over world because lack of understanding of sankrit stanzas & equidistant tones become easier to play as one can start at any key & further 12 keys are all equitempered so for manufacurers as well as for beginners it becomes easy …… & purpose is to rejoice but….

when one wants medical effects then the exact shruti ratios or exact ratios between tones become important

Twelve-tone equal temperament

In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio of the interval between two adjacent notes, is the twelfth root of two:

\sqrt[12]{2}=2^{\frac{1}{12}}\approx 1.0594631

This interval is divided into 100 cents.

SOME REAEARCHERS HAVE USED same ratio 1.0594631 to get 12 tones & then Each one was moved 81/80 or Pythagorian coma or praman shruti distance up & down with 5 swaras to add 10 shruti more to make it 22 shruti  !  This effort reduces exact frequecy ratio undeerstanding & Shruti jaati system unexplained. 

53 equal temperament is for all practical purposes equivalent to an extended Pythagorean tuning. as mentioned bellow  Pythagorean table is given bellow.

53  TET system

Theoretical interest in this division goes back to antiquity. Ching Fang (78–37 BC), a Chinese music theorist, observed that a series of 53 just fifths ((3/2)^{53}) is very nearly equal to 31 octaves ((2/1)^{31}). He calculated this difference with six-digit accuracy to be 177147 / 176776.[2] Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c. 1620–1687), who calculated this value precisely as ( 3^{53} / 2^{84} = 19383245667680019896796723/19342813113834066795298816), which is known asMercator’s comma.[3] Mercator’s comma is of such small value to begin with (≈ 3.615 cents), but 53 equal temperament flattens each fifth by only 1/53 of that comma (≈ 0.0682 cent ≈ 1/315 syntonic comma). Thus, 53 equal temperament is for all practical purposes equivalent to an extended Pythagorean tuning.

After Mercator, William Holder published a treatise in 1694 which pointed out that 53 equal temperament also very closely approximates the just major third (to within 1.4 cents), and consequently 53 equal temperament accommodates the intervals of 5-limit just intonation very well.[4][5] This property of 53-TET may have been known earlier; Isaac Newton‘s unpublished manuscripts suggest that he had been aware of it as early as 1664–65.[6]

In the following Pythagorean Table also one can see                                                                             Perfect fifth is on 3/2 this is the fifth tone  called Pancham in Sanskit meaning fifth  !                   Minor second is 256/243 the second tone Re or Rishabh in its komal or flat version  !                     Major second is 9/8 the second tone Re or Rishabh in its Shudhha or Sharp version  !                     Minor third  is  32/27 third tone or Ga or Gandhar in its Komal or flat version  !                             Major third  is   81/64 the third tone or Ga or Gandhar in its Teevra or Sharp version  !                   Perfect fourth is 4/3 the fourth tone or Ma or Madhyam in its Shudhha or flat version  !                 Augmentated fourth is 729/ 512 the fourth tone Ma or Madhyam in its Teevra or Sharp version ! Unison is 1/1 the tone which is in unison with perfect fifth or Pa or Pancham is Sa or Shadja  !     Here we see Unison & ferfect fifth are called Sa or Shadja & Pa or Pancham & called achal swara  or tones which can not be changed or moved or need perfection  !                                                     Though there is diminished fifth on 1024/729 which is Pa or Pancham on its 1st shruti but used as augmented fourth only !  This creating arguments !                                                                           Major Sixth is 27/16 the sixth tone or Dha or Dhaivat in its Shuddha or Sharp form  !                       Minor sixth is 128/81 the sixth tone or Dha or Dhaivat in its Komal or flat form  !                             Minor seventh is 16/9 the seventh tone or Ni or Nishad in its komal or flat form  !                           Major seventh is 243/128 the seventh tone or Ni or Nishad in its Teevra or Sharp form  !

Western Pyathgorian  Scale Here he has tried to get tones with C-G(3/2) or C-F (3/4) ratios             Here table-pythagoras

 

But Indian Classical Music is as follows on Wikipedia shruti table

 

Shrutis 12-TET Notes 53-TET Notes Perfect FIFTHs
Name Ratio Cents Frequency
(Hz)
Name Frequency
(Hz)
Note
No.
Frequency
(Hz)
FIFTH
No.
Frequency
(Hz)
Kṣobhinī 1 0 261.6256 C 261.6256 0 261.6256 0 261.6256
Tīvrā 256/243 90 275.6220 C♯ 277.1826 4 275.6763 -5 275.622
Kumudvatī 16/15 112 279.0673 5 279.3053 7 279.3824
Mandā 10/9 182 290.6951 D 293.6648 8 290.4816 -10 290.3672
Chandovatī 9/8 203 294.3288 9 294.3056 2 294.3288
Dayāvatī 32/27 294 310.0747 D♯ 311.1270 13 310.1114 -3 310.0747
Ranjanī 6/5 316 313.9507 14 314.1937 9 314.3052
Raktikā 5/4 386 327.0319 E 329.6275 17 326.7661 -8 326.6631
Raudrī 81/64 407 331.1198 18 331.0677 4 331.1199
Krodhā 4/3 498 348.8341 F 349.2282 22 348.8478 -1 348.8341
Vajrikā 27/20 519 353.1945 23 353.4401 11 353.5933
Prasāriṇī 45/32 590 367.9109 F♯ 369.9944 26 367.5829 -6 367.496
Prīti 729/512 612 372.5098 27 372.4218 6 372.5098
Mārjanī 3/2 702 392.4383 G 391.9954 31 392.4229 1 392.4384
Kṣiti 128/81 792 413.4330 G♯ 415.3047 35 413.4982 -4 413.433
Raktā 8/5 814 418.6009 36 418.9415 8 419.0736
Sandīpanī 5/3 884 436.0426 A 440.0000 39 435.7053 -9 435.5508
Ālāpinī 27/16 906 441.4931 40 441.441 3 441.4932
Madantī 16/9 996 465.1121 A♯ 466.1638 44 465.1488 -2 465.1121
Rohiṇī 9/5 1017 470.9260 45 471.2721 10 471.4578
Ramyā 15/8 1088 490.5479 B 493.8833 48 490.1298 -7 489.9947
Ugrā 243/128 1110 496.6798 49 496.582 5 496.6798
Kṣobhinī 2 1200 523.2511 C 523.2511 53 523.2512 0 523.2511

 

But This needs to be modified as per Indian concepts of Shruti which are not equidistant !  Also In Indian Shruti tables 1st shruti of C or Sa or Shadja is Tivra & NOT Kshobhini !            where as 22nd shruti is last shruti of B or Ni or Nishad is called Kshobhini !

Shrutis 12-TET Notes 53-TET Notes Perfect FIFTHs
Name Ratio Cents Frequency
(Hz)
Name Frequency
(Hz)
Note
No.
Frequency
(Hz)
FIFTH
No.
Frequency
(Hz)
Tīvrā 1 0 261.6256 C 261.6256 0 261.6256 0 261.6256
Kumudvatī 81/80 90 275.6220 C♯ 277.1826 4 275.6763 -5 275.622
Mandā 25/24 112 279.0673 5 279.3053 7 279.3824
Chandovatī 135/128 182 290.6951 D 293.6648 8 290.4816 -10 290.3672
Dayāvatī 16/15 203 294.3288 9 294.3056 2 294.3288
Ranjanī 10/9 294 310.0747 D♯ 311.1270 13 310.1114 -3 310.0747
Raktikā 9/8 316 313.9507 14 314.1937 9 314.3052
Raudrī 6/5 386 327.0319 E 329.6275 17 326.7661 -8 326.6631
Krodhā 5/4 407 331.1198 18 331.0677 4 331.1199
Vajrikā 4/3 498 348.8341 F 349.2282 22 348.8478 -1 348.8341
Prasāriṇī 27/20 519 353.1945 23 353.4401 11 353.5933
Prīti 25/18 590 367.9109 F♯ 369.9944 26 367.5829 -6 367.496
Mārjanī 45/32 612 372.5098 27 372.4218 6 372.5098
Kṣiti 64/45 702 392.4383 G 391.9954 31 392.4229 1 392.4384
Raktā 36/25 792 413.4330 G♯ 415.3047 35 413.4982 -4 413.433
Sandīpanī 40/27 814 418.6009 36 418.9415 8 419.0736
Ālāpinī 3/2 884 436.0426 A 440.0000 39 435.7053 -9 435.5508
Madantī 8/5 906 441.4931 40 441.441 3 441.4932
Rohiṇī 5/3 996 465.1121 A♯ 466.1638 44 465.1488 -2 465.1121
Ramyā 27/16 1017 470.9260 45 471.2721 10 471.4578
Ugrā 9/5 1088 490.5479 B 493.8833 48 490.1298 -7 489.9947
Kṣobhinī 15/8 1110 496.6798 49 496.582 5 496.6798
Tīvrā 2 1200 523.2511 C 523.2511 53 523.2512 0 523.2511

 

(Indian )            ( International )

Chandovatī            9/8              203                       294.3288 9              294.3056 2              294.3288

Dayāvatī 32/27       294          310.0747 D♯           311.1270 13            310.1114 -3             310.0747

Ranjanī   6/5           316          313.9507 14            314.1937 9              314.3052

Raktikā   5/4           386          327.0319 E              329.6275 17            326.7661 -8             326.6631

Raudrī    81/64       407          331.1198 18            331.0677 4              331.1199

Krodhā   4/3           498          348.8341 F              349.2282 22            348.8478 -1             348.8341

Vajrikā    27/20       519          353.1945 23            353.4401 11            353.5933

Prasāriṇī 45/32       590          367.9109 F♯            369.9944 26            367.5829 -6             367.496

Prīti         729/512   612          372.5098 27            372.4218 6              372.5098

Mārjanī   3/2           702          392.4383 G             391.9954 31            392.4229 1              392.4384

Kṣiti        128/81     792          413.4330 G♯           415.3047 35            413.4982 -4             413.433

Raktā      8/5           814          418.6009 36            418.9415 8              419.0736

Sandīpanī              5/3           884          436.0426 A             440.0000 39            435.7053 -9             435.5508

Ālāpinī    27/16       906          441.4931 40            441.441   3              441.4932

Madantī  16/9         996          465.1121 A♯           466.1638 44            465.1488 -2             465.1121

Rohiṇī    9/5           1017        470.9260 45            471.2721 10            471.4578

Ramyā    15/8         1088        490.5479 B             493.8833 48            490.1298 -7             489.9947

Ugrā        243/128   1110        496.6798 49            496.582   5              496.6798

Kṣobhinī                2              1200        523.2511 C             523.2511 53            523.2512 0              523.2511