Swar & Shruti question
Paul Barberton
there are 22 swars in an octave and seven are chosen as shrutis but I am
more interested in finding out their frequencies. The western well-tempered
scale is simple to calculate mathematically as each semi-tone is equivalent
to the next, but apparently the swars aren't equally spaced and are based on
a more natural scale. Is there a way to calculate them?
Many thanks
Paul
xxx
raga in which he spends significant time talking about srutis. Check
your local university music library. There also is an article
somewhere about "Bharata's Experiment with the Two Vinas" that goes
into detail about srutis.
There is no simple answer to calculating srutis, and their
intrepretation varies from performer to performer.
Warren Senders
>raga in which he spends significant time talking about srutis. Check
>your local university music library. There also is an article
>somewhere about "Bharata's Experiment with the Two Vinas" that goes
>into detail about srutis.
An important and useful book on the subject is Mark Levy's
"Intonation in North Indian Music," which goes a long way towards
debunking the more outrageous claims of all the shruti theorists.
Levy is a student of Jairazbhoy, and they have both done highly
significant work in clearing the air of nonsensical claims and
specious numerological silliness.
WS
Abhijit Shukla
xxx <nos...@nospam.org> wrote in message
news:38f8853e...@news.mindspring.com...
> Nazir Jairazbhoy, a UCLA ethnomusicology professor, wrote a book on
> raga in which he spends significant time talking about srutis. Check
> your local university music library. There also is an article
> somewhere about "Bharata's Experiment with the Two Vinas" that goes
> into detail about srutis.
>
Chith Eshwaran
Paul Barberton wrote in message <8d90dh$642$1...@ctb-nnrp2.saix.net>...
>Does anyone know where I can get information on swars and shrutis? I know
>there are 22 swars in an octave and seven are chosen as shrutis but I am
>more interested in finding out their frequencies. The western well-tempered
>scale is simple to calculate mathematically as each semi-tone is equivalent
>to the next, but apparently the swars aren't equally spaced and are based
on
>a more natural scale. Is there a way to calculate them?
>
>Many thanks
>
>Paul
>
>
Following is an article I seem to have copied from somewhere some years
ago - hope it helps.
All credits to the author - unknown to me.
=======================================================
RE: question: Hindustani / just intonation tuning
The frequency ratios of Indian classical music are indeed expressed in
ratios of integer numbers with 5 being the highest prime number in
them. Though I don't think that you could exactly call it "just
intonation" since gliding from note to note and shaking of the pitch
is very common.
Below is a short primer that was posted to the newsgroup
rec.music.indian.classical. Recently another more extensive article
was posted to this list which I will mail to you.
Perhaps you already know this: to find the cents value from a ratio,
take the base 2 logarithm and multiply that with 1200.
Example: for 3/2, take 1200 * log2 (1.5) = 701.96 which gives an
offset of 2 cents for the key of G.
-------------------------
Subject: Hindustani Music
In trying to learn more about the theory of Hindustani music I had
made some notes for myself. I'm posting some of these notes to this
newsgroup in the hope that we can discuss about them and in the
process both you and I can learn something new.
Much of the discussion that follows is from a Hindustani point of view,
since I have little or no knowledge of Karnatic traditions. My only
reference for the Karnatic traditions is the monograph published about
a couple years ago by P.S.Sriram from Georgia Tech.
My main references are:
1) "Theory of Indian Music," Rai Bahadur Bishan Swarup, Mittal
Publications, Delhi, India, 1987. [Address: Mittal Publications, B-2
19-B, Lawrence Rd., Delhi - 110 035]. I purchased this book at Stanford
U bookstore and I don't remember how much I paid for it.
2) "The Raags of North Indian Music - Their Structure and Evolution," Nazir
Jairazbhoy, Lok Virsa, Islamabad, Pakistan, 1977. [Address: Lok Virsa
Publishing House, P.O. Box 1184, Islamabad, Pakistan]. I purchased this
book from South Asia Books, Columbia, MO and again I don't remember the
price.
The study of "sangeet" is based on seven broad areas ("adhyAy"s) -
(i) surAdhyAy [study of notes],
(ii) rAgAdhyAy [study of scales (loosely speaking)],
(iii) tAlAdhyAy [study of rhythm and time],
(iv) hastAdhyAy [study of musical instruments],
(v) nrityAdhyAy [study of dance]
(vi) bhAvAdhyAy [study of gesticulating/acting], and
(vii) arthAdhyAy [study of the philosophy of music]
The primary focus of this article is the first area, viz., the study
of notes in Hindustani music. I'll try and post on the other areas
provided I have the time (something that seems to be in short supply
these days :-()
Before setting up the basic notes in an octave, note that there are 5
octaves in Hindustani music - "ati manDra" (low bass), "manDra" (bass),
"madhya" (normal), "tAr" (treble), and "ati tAr" (high treble). The
high treble notes are very rarely used or heard. The low bass notes do
occur in "surbahAr" performances.
The following is a simple way of setting up the notes in an octave (any
one of the above, it really doesn't matter)
Consider a string of length L and locate nodes at the following distances
from the upper support of the string.
0, 1/9, 1/5, 1/4, 1/3, 11/27, 7/15, 1/2 [in terms of L]
The two nodes at 11/27 and 7/15 were added to shorten the interval
between 1/3 and 1/2 and for convenience (as we shall see later).
The vibrating length (from a node to the lower support) is
1, 8/9, 4/5, 3/4, 2/3, 16/27, 8/15, 1/2 [in terms of L]
The frequencies are inversely proportional to the length. Hence, the
frequencies are
1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2 [in terms of string parameters]
The major notes have names - the first is called "krishta" (pulled) since
the other notes are derived from the it. The next four (the harmonics) are,
not surprisingly, "prathama", "dvitIya", "tritIya", and "chaturtha". The
next two are called "panchama" (originally called "manDra") and "atiswarya".
The intervals or the ratios of the successive frequencies are
Kr Pr Dv Tr Ch Pa At Kr2
1 9/8 5/4 4/3 3/2 27/16 15/8 2 Freq.
9/8 10/9 16/15 9/8 9/8 10/9 16/15 Int.
The three major intervals have names too - the 9/8 interval is called
"kAkali" (sweet) the 10/9 interval is called "sAdhAraNa" (ordinary)
(ordinary) and the 16/15 interval is called "antara" (intermediate).
In the Just scale of Western music, the 9/8 interval is called a major
wholetone, 10/9 interval is called a minor wholetone, and the 16/15
interval is called a major semitone. Interestingly, in the Just scale
the fifth and sixth intervals are transposed.
There is a great deal of confusion and controversy about the issue of
"shrutI"s. In the context of present day music this seems to be a
non-issue (or is it?)
The basic question is 'how many shrutIs are there in an octave?' The
answer varies from 'expert' to 'expert' :-) Numbers such as 22, 24,
44, 49, 66, infinity have been offered as answers to this question.
In this article I present Bishan Swarup's argument. His argument is that
if an octave is divided into a certain number of parts then the number of
parts between the first note and any intermediate note varies as the log
of the interval, i.e., n=c*log(i) where c is a constant.
Recall that the intervals are
Kr Pr Dv Tr Ch Pa At Kr2
1 9/8 5/4 4/3 3/2 27/16 15/8 2 Freq.
9/8 10/9 16/15 9/8 9/8 10/9 16/15 Int.
Assuming that the octave is divided into 22 parts he works out the
following
Kr-Kr2: n=22, i=2 => c=22/log(2)
Kr-Pr : n=c*log(9/8 )= 4 (approx.) => "shrutI"s= 4
Kr-Dv : n=c*log(5/4 )= 7 (approx.) => "shrutI"s= 7-4 =3
Kr-Tr : n=c*log(4/3 )= 9 (approx.) => "shrutI"s= 9-7 =2
Kr-Ch : n=c*log(3/2 )=13 (approx.) => "shrutI"s=13-9 =4
Kr-Pa : n=c*log(27/16)=17 (approx.) => "shrutI"s=17-13=4
Kr-At : n=c*log(15/8 )=20 (approx.) => "shrutI"s=20-17=3
Kr-Kr2: n=c*log(2 )=22 => "shrutI"s=22-20=2
You can repeat this exercise with the other numbers and you cannot get
the right distribution. Based on the 22-"shrutI" assumption we see that
the 9/8 interval is represented by 4 "shrutI"s, the 10/9 by 3 "shrutI"s,
and the 16/15 interval by 2 "shrutI"s and the arrangement of "shrutI"s is
Kr Pr Dw Tr Ch Pa At Kr2
1 9/8 5/4 4/3 3/2 27/16 15/8 2 Freq.
9/8 10/9 16/15 9/8 9/8 10/9 16/15 Int.
4 3 2 4 4 3 2 "ShrutI"s
The arrangement (scale) of "shrutI"s is "shaDja grAma". There are other
possible arrangements.
The twenty two "shrutI"s have names too (surprised? :-)) and they are
"prasUna", "siddhA", "prabhAvatI", "kAntA", "suprabhA", "shikhA",
"dIptimatI", "ugrA", "hlAdI", "nirvIrI", "dirA", "sarpasahA", "kshAnti:"
"vibhUti:', "mAlinI", "chapalA", "bAlA", "sarwaratnA", "shAntA",
"vikalinI", "hrdayonmalinI", and "visAriNI". The note "krishta" was set
on "prasUna". These were the old names from "nArada's sangIta makaranda."
These were replaced later by other names (see next article).
In the table below I have computed the "shrutI" frequencies in terms of
the fundamental frequency, f. The first two columns contain the "shrutI"
# and name, respectively. The third column contains the frequency ratio
(going forwards from a main note). The fourth column contains the freq.
ratio of the main note. The last column contains the freq. ratio (going
backwards from a main note).
S name forward main backward
0 kshobhiNI (Kr) 1
1 tIwrA 81/80
2 kumudvatI 16/15 135/128
3 mandA 10/9
4 chhAndovatI (Pr) 9/8
5 dayaavatI 75/64 32/27 (from 9)
6 ranjanI 6/5
7 raktika (Dw) 5/4
8 raudrI 81/64
9 krodhI (Tr) 4/3
10 vajrikA 25/18 27/20
11 prasAriNI 64/45 45/32
12 prIti: 40/27
13 mArjanI (Ch) 3/2
14 kshiti: 243/160
15 raktA 8/5 405/256
16 sandIpinI 5/3
17 AlApinI (Pa) 27/16
18 madantI 16/9 225/128
19 rohiNI 9/5
20 ramyA (At) 15/8
21 ugrA 243/128
22 kshobhiNI (Kr2) 2
Over the years the names of the primary notes were also changed and the
the rationale behind the new names is really interesting.
"krishta" was replaced by "nishAda" (seated) [Urdu: nikhAd] since the other
notes were derived from it. "prathama" was replaced by "swara" since it was
the first or the chief note. "chathurtha" was replaced by "madhyama" since
it occured in the middle of the octave. "dvitIya" was renamed "rishabha"
[Urdu: rikhab] - the old name of the "shrutI" corresponding to "dvitIya" is
"ugrA" (powerful/also an epithet of Shiva) and this suggested "rishabha"
(bull). "tritIya" was renamed "gAndhAra" - the old name of the "shrutI"
corresponding to "tritIya" is "nirvIrI" ("nirvIra" means a woman whose
husband and children are dead) and this suggested "gAndhArI" (the mother of
the Kauravas in the Mahabharat). "chaturtha" was renamed "dhaivata" - the
old "shrutI" corresponding to this note is "hrdayonmalinI" ("hrdayonmalina"
means black-hearted) and "dhava" means cheat. Somehow this mutates to
"dhaivata". The name "swara" was replaced by "shaDja" (born of six)
[Urdu: khaRaj] and its origin is not adequately explained.
The present day names for the notes are
Ni, Sa , Re, Ga, Ma, Pa, Dha
It is interesting to note that in the old system the octave was reckoned
from Ni and not from Sa. Current Hindustani music reckons the octave from
Sa and so some of the notes of the present day music are sharper than the
corresponding old notes.
Note: Many moons ago Ramesh Mahadevan (of Ajay Palvayanteeswaran fame)
had sent me a table of "shrutI"s and the corresponding intervals.
My calculations match almost all of his values except for a couple
and I cannot explain the discrepancies. Maybe Ramesh can tell me
where I've made the errors.
Having discussed the old basis for Hindustani music we come to the present
day structure. Music today recognizes 7 "shuddha svara"s (pure notes) and
5 "vikrita svara"s (modified notes). Of the "shuddh" notes "sa" and "pa" are
considered "achal" (invariant). Of the five modified notes, 4 are "komal"
(soft) taken at two "shrutI" intervals from the main notes sa, re,
pa, and dha and one "tIvra" (sharp) taken at two "shrutI" intervals from
ma. This works out to the following
# Note Freq. ratio
0 shuddh sa 1 ["achal svar"]
1
2 komal re 16/15
3
4 shuddh re 9/8
5
6 komal ga 6/5
7 shuddh ga 5/4
8
9 shuddh ma 4/3
10
11 tIvra ma 45/32
12
13 shuddh pa 3/2 ["achal svar"]
14
15 komal dha 8/5
16
17 shuddh dha 27/16
18
19 komal ni 9/5
20 shuddh ni 15/8
21
22 shuddh sa2 2
Chith Eshwaran
Paul Barberton wrote in message <8d90dh$642$1...@ctb-nnrp2.saix.net>...
>Does anyone know where I can get information on swars and shrutis? I know
>there are 22 swars in an octave and seven are chosen as shrutis but I am
>more interested in finding out their frequencies. The western well-tempered
>scale is simple to calculate mathematically as each semi-tone is equivalent
>to the next, but apparently the swars aren't equally spaced and are based
on
>a more natural scale. Is there a way to calculate them?
>
>Many thanks
>
>Paul
>
>
Another one I seem to have ... credits are obvious.
- Chith Eshwaran
======================================================
From: parr...@mimicad.Colorado.EDU (Rajan Parrikar)
Subject: Shrutis: Still some more!
Organization: University of Colorado, Boulder
Date: Thu, 11 Feb 1993 04:01:13 GMT
The following essay was originally published in the Journal of
Music Academy, Madras, 1966. This reproduction is from Chapter 8 of
The MUSIC OF INDIA: A SCIENTIFIC STUDY by B. CHAITANYA DEVA (Munshiram
Manoharlal Publishers Pvt. Ltd.) first published in 1981.
Rajan
=====
ps: I had to do a bit of massaging on the text since the scanner
didn't recognise some parts due to the poor state of the book. Hence
some typos might have crept in.
**********************************************************************
pp94-103
The Problem of Continuity in Music and Sruti Studies in Indian Musical
Scales -2 by B. Chaitanya Deva
......................I strongly suspect that this problem of
continuity-discontinuity, which is a fundamental one in human
perception, is at the foundation of tbe theory of sruti-s. That is
why, inspite of the arithmetical problem (to which we will turn
later), sruti-s are said to be infinite in number. The reduction of
the infinite to the perceivable and measurable finite, introduces
approximations and errors. These are inherent in the process and I
again suspect that the sruti-s have this occultic background: for,
notice that there are 22 sruti-s and 7 notes, with a telling pointer
to 22/7 (pi) - a number of great mathematical and occultic
implications. The sruti is, therefore, the primordial perceptive gap
between time and non-time. Only for arithmetical and mathematical
purposes is a significant number, 22, attached to it. Even then sruti
is only a pointer and not a measure.
SRUTI:
The word sruti has been used with slightly different meanings by
various writers. In general we may say they are additive measures of
pitch relations in music.
Matanga (3) has defined it as the sound which can be grasped
by the ear. If we interpret this very broadly we may say that any
audible sound is sruti. As Kohala(4) says, some took the sruti-s
to be infinite. Thus we see that all the sounds we use in music are
sruti-s, and it is obvious that we do use really uncountable number of
pitches in music. It is practically impossible to measure all the
pitch variations of steady tones, gamaka-s, glides, etc.
The sruti-s are convenient steps of measurement of pitch
analogous to notes. We say that Bhup has five notes. It does not mean
that we use only five perfectly steady frequencies. It means we
recognize five points where we can conveniently stand and measure the
pitch. If we try to measure all the " mind-s", gamaka-s used in
singing or playing Bhup, the pitches would be infinite.
To define these steps or srutis Sarngadeva gives a method (5).
A string is fixed on a vina(harp) in such a way that it can produce
its lowest pitch. Now, tune another string at a slightly higher pitch.
But it must be so close to the first in pitch that a third tone cannot
be introduced between them. Similarly tune a third string just above
in pitch to the second, so that there cannot be introduced another
tone between the second and third strings and so on. The strings so
tuned are said to be one sruti apart. It is clear, therefore,
according to Sarngadeva, that sruti is the just noticeable difference
in pitch. In modern language we may say that sruti is the difference
limen for pitch. However, if we carefully try the experiment on a
svaramandal (psaltery) we can get many more than 22 sruti-s in an
octave. In our preliminary experiments we could get nearly 40 tones
between Sa and Ga!
Also, by modern experiments with pure tones, it has been found
that a normal ear can discern a difference of nearly three cps to five
cps in pitch. That is, if there is a tone of 240 cps and another of
243 cps or 237 cps, the latter will be heard as different in pitch.
But if the other tone is, say, 241 cps or 239 cps, the ear cannot
distinguish between this and 240 cps. Of course, this is under
experimental conditions with very accurate instruments in the
laboratory with pure tones! But under ordinary conditions with complex
tones the differentiation will be definitely less. Again, even if the
number of different pitches within an octave which the ear can make
out may be more than 22, they may not be 'musically' different or
significant.
In this connection we may refer to an experiment by Ellis
(England, 19th Cent.)(6). He took a stretched string with moveable
bridges under it. By moving the bridges, the length of the vibrating
string could be altered, thus changing the pitch of its sound. He
found that to produce a just noticeable difference in the pitch of the
string he had to shorten the length of the string by 1/32 of its
*previous* length. For instance, let the wire be 1024 mm. long. Let
this be Sa. To get the next just noticeable pitch reduce the length
by 1/32 of this, that is by 1024/32. The new length is 996 mm. The
next length to produce a just noticeable difference in pitch will be
31/32 of this new length, i.e., 996 x 31/32 =964 mm. The next note
will have a length 31/32 of this, i.e., 964 x 31/32=932 mm. and so
on, till we get Sa' with length 512 mm.
We know that string length is inversely proportional to
frequency. So, every time we decrease the length by 31/32 of the
previous value, we are increasing the frequency by 32/31 of its
previous value. If we actually calculate the number of such steps
possible from Sa to Sa' we will find that there are nearly 22!
(For those who want to calculate these, here is the method. Let Sa be
l. The next audible 'note' will be 1x32/31. The third audible note
wi11 be (lx 32/31)x 32/31. The fourth audible note will be (1x 32/31 x
32/31)x 32/31, and so on. Now, Sa'=2. How many steps of 32/31 will it
require to get Sa'? Let this number be n. Then 1 x (31/32)**n = 2 and
n=21.98 or very nearly 22.)
That is, by Ellis' experiment, the number of rough-steps of
pitch which can be distinguished in an octave are about 22. This is
indeed a surprising agreement with the 22 sruti-s mentioned by our ancients!
The above experiment has an important point in it. That is
this: the difference in any sensation which can be perceived depends
on the sensation already present in a person. It is common experience
that a cup of tea is insipid after a few sweets have been eaten.
Similarly, even a bright lamp does not have an appreciable effect on
the eye in sun light. This dependence of differential perception on
the prior condition of an organism is expressed in Weber's law which
states that "in any given kind of perception, equal relative (not
absolute) differences are equally perceptible." It has, however, been
found that this law is only approximately correct.(Mathematically,
this is put as k=df/f and known as Weber-Fechner law).
The same thing happened in Ellis' experiment described above.
Everytime the increase in pitch was by 1/32 of the immediately
preceding pitch. It was not 1/32 of the original 1024 mm. That is, the
sensation of pitch difference depends on the previous sound just
heard. This is a familiar physiological process and is expressed
mathematically as a logarithmic scale.
Now we shall examine a musical fact familiar to us. Let Sa
=240 cps. Its Pa will be 360 cps. That is, we have ascended by 120
cps. Now take the upper Sa'=480 cps. If we now ascend by 120 cps from
it, will we get Pa'?
We certainly do not get Pa'. Actually we have to ascend by 240
x 3/2 and reach 720 cps to get Pa'. This is the meaning of our
discussion above. From Sa=240 cps we ascended by 120 cps. But since 480 cps
is a higher pitch the same degree of ascent will not do. Our sensation
is now higher; therefore, our ascent also must be by a higher degree.
Our sensation of Pa depends on the previous sensation.
This is perhaps the basic scientific principle in measuring
musical scales by sruti-s. In the above examples, the interval was
Sa-Pa. But the increase in frequency in both cases was not the same.
In the first case we ascended by 120 cps and in the second by 210 cps.
We therefore, require a scale which has equal units for equal musical
intervals. In the sruti scale both are equal to 13 sruti-s, as in the
above example.
Of course, if we do not subtract the frequencies but express
their relation by ratios both will be found to be the same: for
Pa=360/240= 720/480=3/2. So, equal musical intervals are expressible
by equal ratios.
However, it is not easy to work with ratios. We require a
simple scale which can be easily used. And sruti scale is such a
scale. It can be easily worked, for, (I) sruti-s are small in number,
(2) they can be added or subtracted and (3) equal musical intervals
are equal in sruti-s.
There have been great many discussions on the nature and
number of sruti-s. We need not enter into the merits and demerits of
all these. We may note here the following:
(I) Our ancients do not talk of frequency ratios. They talk of numbers
which can be added or subtracted.
(2) The ratios given by various authors for the 22 sruti-s are so
close to one another that they may not make any practical difference.
For example, take the case of one-sruti. We can get one-sruti interval
as follows:
_______________________________________________________________________
Subtraction of srutis Ratio Savarts Cents
_______________________________________________________________________
4 - 3=1 9/8 / 10/9 = 81/80 5 22
3 - 2=1 10/9 / 16/15=25/24 18 70
3 - 2=1 10/9 / 135/128=256/243 23 90
2 - 1=1 16/15 / 81/80=256/243 23 90
2 - 1=1 16/15 / 25/24=128/125 10 52
2 - 1=1 135/128 / 81/80=25/24 18 70
2 - 1=1 135/128 / 256/243
= 32805/32768 2 6
_______________________________________________________________________
It is not necessary to remember all these numbers to
understand the basic idea! We need only note that one-sruti interval
is equal to many ratios some of which have been given above. It is
possible to obtain similarly differing ratios also for two-sruti values.
Thus we have the important idea that sruti does not correctly measure
a tonal interval. For when we say one-sruti, it may mean any of the
above ratios! It indicates a position in the octave. When we say 21st
sruti, it means the 21st position from Sa, where the octave is
measured by 22 positions from Sa.
For instance, when some one asks, "How far is the post office
from here ?," we may say "Oh! just pass by five lamp-posts and you
will reach it." We definitely do not mean that the lamp-posts are at
equal distances from one another. The lamp-post is only a numeral
indicative.
The sruti then may be considered an ordinal number. It shows
the position of a sound on a scale of 22. The actual ratio is a
cardinal number which shows the correct relation between two pitches.
We may compare the sruti-s to the position of a student in the class.
This is his sruti number. The actual marks he gets in the examination
is like the pitch ratios.
Again, imagine that we have to group students in a school
according to their heights. Let us assume that we want them in 12
groups, starting with boys 5' tall to boys 6' tall. Then we can have
groups like this:
(I) 5' to 5' 1" (2) 5' l" to 5' 2" and so on till the 12th group of
5' ll" to 6'. Now all boys who have heights 5'1.25" 5' 1.5", 5' 1.75"
and less than 5' 2" will come into group (2). We will call them second
group boys, though their heights are not the same.
Similarly we divide the octave into 22 parts. In each sruti
come all intervals which are very close, according to this scale.
Thus sruti-s are equal when they indicate a position in the octave but
unequal when we express them as ratios.
In the discussion on sruti-s by various authors the following
points will be found important:
1. Are the sruti-s intervals or notes?
Perhaps this is not so complicated as it seems. Let us consider them
as notes. We require 22 in an octave. That is, starting from Sa, the
last note must be the 22nd, as in this figure:
/ / / / / / / / / / / / / / / / / / / / / / /
Sa Sa'
The 23rd note is Sa' which we do not count, for it is a
'repetition' of Sa.
{\b If the sruti-s are intervals, we must have 22 gaps in our figure,
as below:
/ / / / / / / / / / / / / / / / / / / / / / /
Sa Sa'
The Sa' comes after 22 intervals. Here we take into consideration the
last note (Sa'), omitting Sa. For, the first interval is from Sa to
ri (one-sruti) and the last interval (22nd) is from Ni to Sa'.
That is, in the first case Sa is counted but not Sa'. In the
second case Sa is not counted but Sa' is taken into account.
2. Are the sruti-s equal or unequal?
Expressed as ratios they are unequal. We have seen this particularly
in the case of one-sruti intervals. If they are unequal what are their
mathematical values? Much discussion and difference of opinion obtain
on this problem. However, most of these opinions are theoretical with
very little practical evidence.
But we have also seen that when ratios are expressed as additive
numbers like sruti-s, they are logarithmic and equal.
In other words, sruti-s are both equal and unequal.
In so far as they indicate a position in an octave (as ordinal
numbers), they are equal. In this capacity they do not measure but
show only a position in a series of pitches. But each position or sruti
may have many close ratios which we measure. In this capacity they are
unequal. Much confusion can be avoided by following this idea.
3. Why are there only 22 sruti-s?
This is a mathematical question for which many solutions have been
offered. Some say that 22 sruti-s and 7 notes are closely related to
the ratio of the circumference and radius of a circle (22/7). Some are
of opinion that this is a small number which does not introduce much
error when we change ratios into additive numbers(7). We have already
noted Sarngadeva's and Ellis' experiments, where it was seen that
there are roughly 22 distinguishable pitches in an octave.
The concept of sruti has gone through many vicissitudes and
dimensions. Setting aside, for the present, the philosophical and the
epistemological implications, the important fact to be noted is that
the theory of sruti-s and their practice were based on polychords and
hence were conceived as discrete units and levels. Add to this the
static scalic concept from the West and we get the current craze for
frequency ratios in Indian musicology. Little is it realized that we
have to develop a calculus of continuous pitch movements and also
that the sruti phenomenon is an infinite series suffering
approximations for adjustments of octave relations.
References:
1) B.C. Deva, Transitive Elememnts in Music, Nada Rupa, pp 44ff(Jan
1963)
2) C.R. Sankaran, Process of Speech, Deccan College Monograph Series,
27, pp23 ff, Poona, 1963.
3) Matanga, Brhaddesi, p.4(Travancore, 1928)
4) Kohala. Quoted by Matanga in Brhaddesi. See commentary on 1, 28
(Travancore).
5) Sarngadeva, Sangitratnakara, Sect, 1, Ch. 3, verse 12 (Adyar
Library)
6) H. Helmholtz (Tr. A.J. Ellis), On the sensations of tone, p 523
(New York 1954)
7) H.V. Modak, Propriety of dividing an octave into 22 shrutis,
Physics Sect., Indian Science Congress, 1961
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Keith Erskine
: Does anyone know where I can get information on swars and shrutis? I know
: there are 22 swars in an octave and seven are chosen as shrutis but I am
: more interested in finding out their frequencies.
You have it somewhat backwards. There are 12 swaras, not shrutis, in an octave,
just as in western music (C, C#, D, D#, E, F, F#, G, G#, A, A#, B)
(Sa re Re ga Ga ma Ma Pa da Da ni Ni)
For each of the swaras OTHER THAN Sa AND Pa, there are (to a first approximation)
2 intonational values possible - these alternative intonations are the SHRUTIS.
Thus 2 (Sa, Pa) + 2*10 (all else) = 2 + 20 = 22 SHRUTIS.
While physics can suggest logical choices for exact shruti intonation, in
practice there is quite a bit of flexibility.
Going further, sometimes the shruti concept is applied to how an individual
note is treated, not just it's intonation. In Mian ki Malhar, for example,
komal ga is always approached via a meend from shuddha ma, in effect rendering
it a "sharpish, smeared" komal ga. In Bhairav, re and da are andolan, or slowly
oscillated. In Darbari Kanada the komal ga is very flat and oscillated.
: The western well-tempered
: scale is simple to calculate mathematically as each semi-tone is equivalent
: to the next, but apparently the swars aren't equally spaced and are based on
: a more natural scale. Is there a way to calculate them?
The differences in pitch are minute, 0.8% is the maximum deviation (8/1000),
for well tempered major 3rd which is sharper than a just/pythagorean major 3rd.
These differences, while real, are only discernible to a well trained ear.
Pt. Buddhadev dasGupta said that the focus on shruti value is much more prominent
in the alap portion - when the speed has picked up, he said this degree of exact
intonation can not be achieved on fast passages, fast taans, toras, etc.
Keith Erskine
I don't speak for HP.
Warren Senders
>for well tempered major 3rd which is sharper than a just/pythagorean major
>3rd.
>These differences, while real, are only discernible to a well trained ear.
Keith, be careful. The just major third is 5/4; the Pythagorean M3 is 81/64;
your post suggests that they are the same. Helmholtz gives the values in cents
of just, equally-tempered and Pythagorean major thirds as 386, 400 and 408
respectively. Note that the Pythagorean third is 8 cents sharper than
the tempered; conflating just and Pythagorean tunings is a grave
intonational solecism.
The difference between the just and tempered M3 is readily audible: play
a medium low note on the piano, adjust your voice to a pure just M3 against
that note, and then hit the corresponding piano key. The 14 or so cents by
which the equally-tempered third exceeds the just third is powerfully
audible, and may in fact be painful.
I have demonstrated this difference literally hundreds of times, and I have
never yet had a listener who was unable to hear it. It does not
require a "well-trained" ear; 6th graders can readily hear the discrepancy.
The Pythagorean third is much closer to the tempered, and because the
frequency ratio is rather complex, it's hard to hear unless there is a
supporting tone a 3/2 below (a 27/16 shuddh Dha built as a 3/2 off the
9/8 Re). It is correspondingly more difficult to differentiate the Pythagorean
and Tempered major thirds -- both are sharp to the just position, and
both are complex relationships relative to the simplicity of the 5/4 ratio
(although of course E-T is exponentially more complicated than Pythagorean!).
Warren
Keith Erskine
: >The differences in pitch are minute, 0.8% is the maximum deviation (8/1000),
: >for well tempered major 3rd which is sharper than a just/pythagorean major
: >3rd.
: >These differences, while real, are only discernible to a well trained ear.
: Keith, be careful. The just major third is 5/4; the Pythagorean M3 is 81/64;
: your post suggests that they are the same.
You're entirely correct, I was typing too quickly. I meant well tempered is
sharper than just tempered by 0.8%.
: The difference between the just and tempered M3 is readily audible: play
: a medium low note on the piano, adjust your voice to a pure just M3 against
: that note, and then hit the corresponding piano key. The 14 or so cents by
: which the equally-tempered third exceeds the just third is powerfully
: audible, and may in fact be painful.
: I have demonstrated this difference literally hundreds of times, and I have
: never yet had a listener who was unable to hear it. It does not
: require a "well-trained" ear; 6th graders can readily hear the discrepancy.
How do you KNOW they heard it? Did you do a blindfold test asking the subject
to identify the ET 3rd vs. just 3rd? It is much easier to SAY one can discern
the difference than to repeatedly, accurately identify which is which.
Your earlier experiment asks one to sing a just tempered major 3rd. I'll
remind you that most people (non-musicians) can't even sing a major 3rd on
demand, let alone within 1% accuracy. I think we have a very different idea of
what a "well-trained" ear (or voice) is.
I do agree that some can hear the sharpness of the well tempered
major 3rd, but remember, this is by far the most egregious of equal tempered
intervals. This is for 2 reasons, because the % error is the highest, and
because of the prominence of the 3rd in the overtone series (octave, fifth,
octave, 3rd).
Do you think students could identify a equal tempered shuddha Re vs.
just, where the error is only 0.22 % (2 parts in 1000)? Even identifying
the most prominent overtone, the perfect fifth, just vs. equal tempered fifth,
where the error is only 0.11 % (1 part in 1000) is a signficant challenge.
It can usually only be discerned via slowly oscillating beat frequencies
rather than based on absolute pitch. (This too would be a good experiment -
testing people on intonational accuracy/detection for serially sounded pitches
rather than simultaneously sounded pitches)
Thus, when I say it takes a well trained ear to discern just vs. equal,
I mean for ALL of the swaras, not JUST (hyuk hyuk) the major 3rd, the poor
whipping boy the shruti cogniscenti prefer to beat (hyuk hyuk) upon.
In general, I'll be a cynic and say in general much of the talk of shrutis,
the "serious" shortcomings of the Western tuning, and the superiority of
Indian intonation is much ado about nothing. I would wager that the
standard deviation, 1 sigma, of intonation of a given swara by a given performer
of the highest repute within a given raga on a given performnace is on par, if
not greater, than the less than 1% discrepancies in frequency of equal-tempered
vs. just. It was very refreshing to hear Pt. DasGupta, a mechanical engineer
as well as renowned sarodiya, well versed in the sciences, candidly state
delicacies of shruti intonation go out the window quickly after the alap.
What is the intonational accuracy of the human voice, anyway? Is the
periodicity of the waveform within 1% accuracy, or does it drift from
cycle to cycle?
I've heard so much bravado and "my ICM is better than WCM" by those who know
virtually nothing of either, based merely on mystical allusions to shruti.
I've heard a performer introduce Sindhi Bhairavi stating "in this raga all
*72* microtones are utilitized". I've also had non-musicians ICM fans assert
there are 72 shrutis for EACH of the 12 swaras in ICM, for a staggering
72*12 = 864 shrutis! Another time, after a lecture on the shortcomings of
WCM tuning, I was "treated" to a performance of entirely random intonation
swaras on a sitar with frets in absolutely random positions!
: both are complex relationships relative to the simplicity of the 5/4 ratio
: (although of course E-T is exponentially more complicated than Pythagorean!).
^^^^^^^^^^^^^
Little bit of intonational humor there, Warren? :-)
Warren Senders
>: never yet had a listener who was unable to hear it. It does not
>: require a "well-trained" ear; 6th graders can readily hear the discrepancy.
>
>How do you KNOW they heard it? Did you do a blindfold test asking the subject
>to identify the ET 3rd vs. just 3rd? It is much easier to SAY one can discern
>the difference than to repeatedly, accurately identify which is which.
To rephrase: if I sing a just 3rd against a held tone on the piano, and
then play the corresponding key, sixth graders can say: "that's not the
same as the note you're singing!" Some can go further and say that it's
"a little higher."
>Your earlier experiment asks one to sing a just tempered major 3rd. I'll
>remind you that most people (non-musicians) can't even sing a major 3rd
>on demand, let alone within 1% accuracy. I think we have a very different
>idea of what a "well-trained" ear (or voice) is.
Most MUSICIANS can't sing a just major 3rd either. I can only sing one if
I've concentrated all my attention on the intonational environment; paying
any heed to rhythm or melodic arc immediately diminishes intonational
acuity to some degree (as Pt. DasGupta avers).
>In general, I'll be a cynic and say in general much of the talk of shrutis,
>the "serious" shortcomings of the Western tuning, and the superiority of
>Indian intonation is much ado about nothing.
No argument from me. I have no illusions about the "superior intonation" of
any particular musical tradition; all are engaged in artistic deviation from
Platonic ideals of pitch.
>: both are complex relationships relative to the simplicity of the 5/4 ratio
>: (although of course E-T is exponentially more complicated than
Pythagorean!).
> ^^^^^^^^^^^^^
>Little bit of intonational humor there, Warren? :-)
Well, actually it's logarithmically more complicated.... snurk snurk snurk...
WS
Keith Erskine
: >: I have demonstrated this difference literally hundreds of times, and I have
: >: never yet had a listener who was unable to hear it. It does not
: >: require a "well-trained" ear; 6th graders can readily hear the discrepancy.
: >
: >How do you KNOW they heard it? Did you do a blindfold test asking the subject
: >to identify the ET 3rd vs. just 3rd? It is much easier to SAY one can discern
: >the difference than to repeatedly, accurately identify which is which.
: To rephrase: if I sing a just 3rd against a held tone on the piano, and
: then play the corresponding key, sixth graders can say: "that's not the
: same as the note you're singing!" Some can go further and say that it's
: "a little higher."
I believe the students were not actually discerning a qualitatitive
difference in the harmony itself, but rather comparing the new pitch
(equal tempered 3rd on piano) to the prior pitch (sung just 3rd).
It is quite easy to discern whether pitch of a 2nd note is different
than the pitch of a note played immediately before, even if those
pitches are the upper voices of harmony.
What I meant by "hear a just vs. equal tempered 3rd" would be to play
just one harmony, root (Sa) and major 3rd (Ga). Then ask the students
to identify whether the 3rd is just or ET (equal tempered). It would
invalidate the experiment to play just 3rd & equal 3rd back to back,
as they would just be comparing successive pitches. You would have to
play a series of intervals, such as :
1) Perfect fifth (ET)
2) major 3rd (ET)
3) perfect fourth (just)
4) major 7th (ET)
5) major 3rd (just)
If at steps 2) and 7) the students could correctly identify the
ET maj 3rd versus the JUST maj 3rd, that is what I meant by
"being able to hear the difference between just and ET major 3rd"
If untrained ears could truly discern subtle deviations in pitch of
voices in polyphony, then people would be able to tune guitars based
on open intervals of fourths and major 3rds. My personal experience
as a beginning guitarist and teaching guitar for 15 years is that
untrained ears have a hard enough time identifying unison (identical
pitches), let alone idenifying properly tuned intervals of fourths (ma),
fifths (Pa), and thirds (Ga).
Keith Erskine
: Warren Senders (war...@aol.comqwerty) wrote:
: What I meant by "hear a just vs. equal tempered 3rd" would be to play
: just one harmony, root (Sa) and major 3rd (Ga). Then ask the students
: to identify whether the 3rd is just or ET (equal tempered). It would
: invalidate the experiment to play just 3rd & equal 3rd back to back,
: as they would just be comparing successive pitches. You would have to
: play a series of intervals, such as :
: 1) Perfect fifth (ET)
: 2) major 3rd (ET)
: 3) perfect fourth (just)
: 4) major 7th (ET)
: 5) major 3rd (just)
: If at steps 2) and 7) the students could correctly identify the
^^oops, make that step 5),
: ET maj 3rd versus the JUST maj 3rd, that is what I meant by
: "being able to hear the difference between just and ET major 3rd"
: Keith Erskine