Ratios and Intonation (warning: long & complicated!)
Warren Senders
..........in the interest of further work-avoidance, I sat down and
did the calculations for some of the intonational relationships
which we've been discussing (and from which I digressed a
few days ago in order to make my previous post in response
to the Shivaranjani fracas). In particular I investigated the
various intervallic structures involved in the scale known as
Todi thaat, because Praful had been making some statements
about the intonation of Re in Multani and Todi.
Let me state some preliminary assumptions:
1 - The human ear prefers simple whole-number ratios
to complex ones, and the singer or instrumental performer
will gravitate towards intervallic relationships which can be
understood as simple ratios. These relationships are
perceived as "more consonant" than those which have
complex ratios or ones which are irreducible to whole
numbers (such as the logarithmically-derived settings of
ANY equal-tempered system). This is a widely recognized
phenomenon; researchers (at Harvard? I can't remember
at the moment) note that newborns respond more
positively to small-whole-number ratio consonances
than to large-number or irrational intervals.
2 - Sung or played intervals can be described in two ways:
"OVERTONAL" Intervals, which are derived by:
tuning the fundamental (or any 2/1 octave harmonic
thereof) of the performed note to a given harmonic of
a fixed starting frequency, and
"RECIPROCAL" Intervals, which are derived by:
matching a given harmonic of the performed note
with the starting frequency (or any 2/1 octave harmonic thereof).
For example, the Justly Intoned Major Third (Shuddh
Ga) is an "overtonal" interval - its proper tuning depends
on matching the sung note (or its "octave" harmonic)
with the 5th partial of the Sa.
The Justly Intoned Minor Sixth (Komal Dha), on the other hand,
is a "reciprocal" interval - its proper tuning depends on
matching the 5th partial of the sung note with the Sa
(or its "octave" harmonic).
Are you confused yet?
3 - There are multiple positions available for almost
every interval in the pitch continuum. Even the seemingly
invariate Sa (1/1), Pa (3/2) and Ma (4/3) intervals have
multiple positions (but the discrepancy between the very
small-number ratios given above and the much larger
numbers involved in their alternative placements means
that those versions are rarely heard in music which does
not involve modulation to distant key centers). In western
music theory the multiple intonations of particular intervals
are known as "enharmonic equivalents." Each of these
intonations is in some sense a "consonance," and its
effectiveness or lack thereof is determined by the
context within which it is heard.
To cite just one example, let's examine two positions
available in Just Intonation for the natural sixth (Shuddh
Dha). One is the so-called Pythagorean Sixth. This is
obtained by taking a series of 3/2 (perfect fifth) intervals
from Sa: SA (1/1), PA (3/2), RE (9/8), DHA (27/16). This Sixth
is thus, so to speak, the "PA of RE." Keep that one in mind
for a second, and let's look at the other important Sixth, which
is obtained by taking a 5/4 (major third) interval from Ma (which
is a reciprocal 3/2 from Sa). This Sixth is thus a "GA of MA," with
a ratio of 5/3. In a raga without Ma, THIS sixth will be somewhat
weaker, because there's nothing else in the melodic frame to
give it a context. Thus the Dha in Shivaranjani is assumed to
be "tuned to RE." In a raga with an absent Re (but with Shuddh
Ma) the opposite is true. Thus the Dha in Kaushi Dhani is
operationally "tuned to MA." In a raga with both Re and Ma,
the Dha is ambiguated, and can be tuned to one or the other
depending on melodic context. For example, in a MPDMP
phrase in Durga, we assume the 5/3 Dha, but in a SRSRPDP
line, the 27/16 version of the interval is more likely. On the
other hand, in Hem Kalyan, the Ma is confined to the Kamod-
ang GMP/GMRS phrase, while the SRS and PDP phrases
are isomorphs; the 27/16 would be the default...and in
Bageshri, the Re is confined to cadential descending
sequences, resulting in a default 5/3 (Ma-based) intonation.
That 5/3 Dha is also a reciprocal 3/2 from Ga, which means
that in some cases, a strong shuddh Ga will push the Dha
a perfect fifth below it towards that position, even when there
is no Ma present in the raga (see my discussion of Bhoopali,
below).
4 - Because of the vagaries of instrumental construction,
performance technique, vocal control (and psychoacoustic
factors, natch), musicians operating in real time do not
render these intervallic relationships exactly. Rather, their
intonation APPROXIMATES the various consonances, or,
to use a more apt term, IMPLIES them. When we discuss
HCM intonation in the abstract, therefore, we are considering
the intervallic structures which are IMPLIED by a given artist,
as opposed to those which s/he has actually sung or played
(which may fall short of or overshoot the intended location).
Studies of actual HCM performance have shown that
the intervals sung or played by musicians vary widely
from one rendering to another. Even notes which are
reinforced by the drone are not always sung accurately
(as any of us who practice regularly can attest!). But the
listening mechanism is more charitable; it will accept a
slightly mistuned interval and mentally adjust it to be
closer to the presumed target position. Mathieu did a
wonderful demonstration of this many years ago --
starting with C as the tonal center, he played a repeated
high A on the piano, while moving from a D chord (in
which the A was a "Pa of Re") to an F chord (in which
the A was a "Ga of Ma"). ALL the listeners in the room
agreed that they had distinctly HEARD the A shifting pitch
slightly at the time of the chord change -- an impossibility
on the piano, but not in the psychoacoustic realm! Thus
we can assume a flexibility on the part of the listener which
will skew the perceived interval in the direction of a simple
whole-number relationship with SOME element in the overall
tonal/harmonic structure of the raga. This means that all the
intonations noted in the previous paragraph are PERCEIVED
relationships, but not necessarily PERFORMED ones; the
actual sung or played notes may move within a fairly wide
range (as much as 25 cents either way according to Levy),
and yet continue to SUGGEST the appropriate small-number
relationships to the listener.
With these governing assumptions and their elaborations
in mind, we are ready to proceed in our examination of one
of the more intricate intervallic structures of Hindustani music,
the scale form of Todi thaat.
The simplest whole-number ratios available for Todi thaat
(excluding intervals derived from the application of prime-
numbered overtones above the 5th partial) are as follows:
SA: 1/1
RE-b: 16/15 (a reciprocal 3/2 from the 8/5 Dha-b)
GA-b: 6/5 (a reciprocal 5/4 from 3/2 Pa)
MA-#: 45/32 (an overtonal 3/2 from 15/8 Ni)
PA: 3/2
DHA-b: (a reciprocal 5/4 from 1/1 Sa)
NI: 15/8 (an overtonal 5/4 from 3/2 Pa)
Keep this list of interval ratios handy for a minute. I'm
going to do something that will seem like a digression.
Trust me; it's not -- but hang on tight, okay?
----------------------------------------------------
The simplest whole number ratios available for the "Major
Pentatonic Scale" (that's Bhoopali, folks) are as follows:
SA: 1/1
RE: 9/8
GA: 5/4
PA: 3/2
DHA: 5/3 or 27/16 depending on context (see above)
SA: 2/1
This scale is one of the world's commonest, and may be
presumed to be a "default" setting for the listening ear.
Even when sung with imprecise pitch, it is recognizable,
and melodies set to this scale form are readily graspable
and memorizable IN CORRECT INTONATION! That is, if
I listen to somebody singing a folk song in the Bhoopali
scale, I'll remember and reproduce it with (more or less)
accurate small-number ratio intervals...even if the original
performer was singing her/his notes with a wide intonational
standard deviation. (this is actually a fascinating phenomenon
that would reward extensive study. I wish I had some grad
students around; I'd sic 'em on the problem in a Hot New York
Minute!)
----------------------------------------------------
Back to Todi. Let's consider one of the commonest scales
found under Todi thaat's aegis: the fifth-less form known as
Gujari Todi.
Listening to Gujari Todi, it becomes very clear that much of the
time what is being performed is in some sense a "Bhoopali from
Ni" -- because of the plethora of NRGMDNDMGRN sequences,
in which Sa is omitted, the Ni is the starting point of that old
familiar interval sequence! It's only at the point of resolving to
Sa that the Bhoopali-ness of the melody gives way. If you ask
someone to sing this raga without a tamboura drone (and to
eschew sargamization, which helps establish the existence
of an implicit tonic identity), you'll find that they appear to be
singing a straight vanilla pentatonic scale which suddenly
shifts identity at rare cadential moments.
If we take the whole-number ratios of Bhoopali, and start
them from the 15/8 Ni, we get a pretty complicated set of numbers:
NI: 15/8
RE-b: 135/128
GA-b: 75/64
MA-#: 45/32
DHA-b: 25/16 or 405/256 depending on context
NI: 30/8
But in performance of Gujari Todi, these rather large
relationships may well be the most consonant ones available
in the prevailing Ni-based tonality!
When we examine Mian ki Todi, or pure Todi (recognizing and
for the purposes of discussion eliding the ongoing nomenclatural
dispute about the differences between these forms), the addition
of Pa in greater or lesser degree increases the likelihood that
the performer's intonation implies the simpler intervals (as
stated several paragraphs ago). The more Sa-based intonation,
the more likely that Sa-based whole-number ratios will be
perceived by the listener. Finally, there is the additional fact
of melodic direction's influence on intonation, typically in the
sense that ascending melodies tend to push pitches higher,
descending melodies to push them lower.
This suggests, then, that perceived intonation in a
performance of Todi is enormously dependent on
melodic context (goodness, gracious! What a surprise!).
In a NRGMD phrase the intervallic implications will tend
toward the Ni-based pentatonic intervals...but in a
SRGRGMGRS phrase we'll be more likely to hear the
notes in their Sa-based just forms. Remember, I'm talkin'
perception -- actual measurement of performed intervals
will probably place them all over the intonational map.
There are multiple implied positions available for many of
the intervals found in this melodic system. Let us take
komal Re as an example; there are three possible
"consonant" placements of this interval within a 5-limit matrix.
From lowest to highest they are:
25/24
(a reciprocal 3/2 from the 25/16 Dha-b, itself a reciprocal
3/2 from the 75/64 Ga-b, which is an overtonal 5/4 from the 15/8 Ni);
135/128
(an overtonal 9/8 from the 15/8 Ni);
16/15
(a reciprocal 3/2 from the 8/5 Dha-b)
This suggests that, for example, an ascending D-R-S or D-G-R-S
sequence will imply the16/15 ("Just") placement (because it's
in the context of the 8/5 Dha, itself heard in the context of Sa).
Likewise, a phrase like S--RG---RGR-----S would skew perception
toward this version of the note, because of the implied reciprocal 9/8
with Ga. A N-R-GR-N--R-- phrase, on the other hand, will edge
perception toward the 135/128 interval, because of the implied 9/8
relationship to Ni. And in a strongly Ni-based context, a
N-RG--RNDNR--- line *could* push listener perception even
lower, towards the 25/24 placement. This last is harmonically
most distant from Sa, and is least likely as a perceived intervallic
location...but who knows? Circumstances alter cases.
Let us look at Multani, quite a different avatara of the Todi structure.
Multani tends to emphasize Sa and Pa much more
heavily than Todi, Mian-ki-Todi, or Gujari Todi. Thus
the Ni-based intonation is less likely to be perceived than
the Sa-based -- and it is in this context that Praful's contention
that Multani's Re-komal is higher than Todi's suddenly makes
sense. Re in Multani is much more often heard in the
context of Ga than as part of N-R phrases, and thus is most
likely to be heard in its highest 16/15 position.
However, some treatments of Multani may ambiguate this
interval considerably. In Abdul Waheed Khan's recording,
for example, he spends a fair amount of time doing merukhandish
swara permutations not only on S/G/M, but on N/G/M as well,
before eventually resolving to Sa (by way, naturally, of Re). The
first group implies Sa-based intonation, which means a
straightforward 6/5 komal Ga, and a 16/15 (higher position) Re...
but N-G-M, on the other hand, will inevitably move listener
perception toward a 4:5:6 structure from a 15/8 interval,
which yields 15/8 ; 75/64 ; 45/32 -- a simple major triad
based on the seventh! In this context the perceived
intonation of the resolving Re should be lower, moving
toward the 135/128 placement.
I encourage interested readers to try these sequences
out, remembering that what is involved is a sort of
psychoacoustical game...just sing or play the raga to
the best of your ability, while monitoring your perception
of the various intervals involved and their consonance with
one another.
Or, for that matter, apply this process to any of your
favorite recordings. Please let me know the results!
Cheers,
Warren Senders
Lorraine B Cahn
Please let me know when your book is coming out.
Your posts are witty and brilliant and gentlemanly.(is that a word?)
Thanks.
Vibhooti
buh...@ecn.ab.ca
: Dear Warren
: Please let me know when your book is coming out.
: Your posts are witty and brilliant and gentlemanly.(is that a word?)
: Thanks.
: Vibhooti
agreed!
: Warren Senders wrote:
<snip very engaging post>
: > Cheers,
: >
: > Warren Senders
i take it, then, that i'm not the only one who hopes warren finds more
home renovation work that he doesn't want to do ;-)
--
------------------------------------------------------------------------
Andrew Buhr buh...@freenet.edmonton.ab.ca
Savour the Irony! bu...@infinity.gmcc.ab.ca
http://freenet.edmonton.ab.ca/~buhrger
Iyengar
I am also looking for published materials on the same lines. Please e-mail
or post if you know of publishings that co-relate the aesthetics and the
mathematical structures in the ragas and their use in presentations.
Thanks
Iyengar
"Warren Senders" <war...@aol.comqwerty> wrote in message
news:20011013231827...@mb-mk.aol.com...
Praful Kelkar
> ..........in the interest of further work-avoidance,
Dear Warren-ji, First of all I will never hire you to fix up my house
:-) Now I am getting hooked on work avoidance as well.......
Firstly your writing is good, thoughtful as always. However, you may
be oversimplifying. You seem to be implying that all there is
essentially no difference in the shrutis but is only a "perceived"
difference based on what other relationships are invoked and that
these invokations happen based certain umproven assumptions. The
number game is curious but, is there more or we have explained all
there to it?
When notes are sung/played correctly there no need to invoke any
ratios at all. The first kR when sung correnctly right after Sa
tells you it is Shree, doesn't it? Does that come from "what's
happening around the note (more on that later)" or does it also have
to do with "shruti"?
I suggest following experiement. Start with Sa - then sing Multani
kR and then wait a few min lock into Todi mood and then do todi Sa-Re
- what is the difference?
Or let's experiment with single notes -- let the taanpura he beard and
then sing one note - kR - try that for Bhairav, Todi, Shree, Multani,
Lalit -- what is going on? Again, for sarod players Arnab, Jodi - can
you track your nail touching the string. What is going on?
.............
In terms of shruti mapping previous studies have shown that the pitch
changes between as well with during a performance. But that doesn't
immediately mean there is no stability within shruti system. E.g. If
one were to plot the shruti of kD in Todi and compare it with kD of
Lalit - I would predict that the Lalit kD scatter lies higher than
Todi.
We can think of looking at mean, standard deviation, etc. but just
because there is variance therefore to assume there is no pattern is
incorrect. The previous studies you mentioned have not done anything
as sofisticated as that. They were simple measurements and their data
are not analyzed to look for hidden patterns. Any statisticians out
there??
Does the scatterogram of Todi kR look different from Multani kR? What
kind of scatterogram sounds soorila and within the raag, and what
would be besura or betray the raag?
...................
Another set of experiment that should be done (after when I get fired
for work avoiding) is to map out shrutis of Dhrupadiyas. I think
they may be more "pure" in terms of intonation (let's not start a war
over holi purity here, we have gone thru that already). ZM Dagar in
particular comes to mind.
Of the living Gundechas would do well. I spent a lovely morning with
them last year just on this - and they are good in their control of
intonation. What I observed is this - In one note they were able to
separate different raags - and that difference also was due to
differneces in pitch/shruti. There were no other notes to invoke
ratios.
Even notes like Ma in Kedar and Bhimpalasi seemed to have different
pitches! Furthermore, they were able to clearly and consitently show
use of different shrutis of Ma in Bhimpalasi - very beatiful play on
shrutis. Are you implying Warren that you are missing stuff like
this, which I would find difficult to believe.
...........
Finally, where is this all coming from? A simple question is why
should the shrutis not be different for different raags? After all a
well trained human ear is able to hear many microtones in an octave
(the last time I was playing with this I came up 40-50, but this was
not a controlled experiment). Why then should the raag system not use
a play on these shrutis? Just beacuse someone else found varience in
their simple experiments shouldn't make us disregard the entire
system.
......................
Is there any money for doing this Warren?
...
> Are you confused yet?
Elementary my dear Warren :-)
Praful
Praful Kelkar
Sounds like until there is a good way to map out and analyze the data
about actual frequencies we will be left with "perceived" perception.
The word Shrutis translates into "that what is heard" - implying
perceived intonation rather than a notion of fixed pitch. So it may
be fine to talk about raag-specific shrutis
(raag-specific-what-intonation-is-heard), whether or not one knows the
explanation in terms of ratios or knows it just by the gut feeling. I
like this definition of shrutis.
There may be ways to think of ratios bringing in the "shrutis" of kR
in Todi, Bhairav, Shree, Multani - there seem to be at least 4
distinct ones that one hears so I presuming they exist as actual
entities! Shree kR is harder to put in a ratio it too comes with a
heavy P and S but sits at a different spot. - may be it sits at a
non-ratio place, thus the eeriness ???
So it seems fair afterall to talk about higher/lower - hence it is
fine to say that kR used in Multani most commonly sits below kR in
Todi.? - acknowledging that diff shrutis will occur depending on the
context, phrasing, and what is actually being played may be something
else. But this will hold true by and large.
Praful
AKTALWAR
>Of the living Gundechas would do well. I spent a lovely morning with
>them last year just on this - and they are good in their control of
>intonation. What I observed is this - In one note they were able to
>separate different raags - and that difference also was due to
>differneces in pitch/shruti. There were no other notes to invoke
>ratios.
>
>Even notes like Ma in Kedar and Bhimpalasi seemed to have different
>pitches! Furthermore, they were able to clearly and consitently show
>use of different shrutis of Ma in Bhimpalasi - very beatiful play on
>shrutis. Are you implying Warren that you are missing stuff like
>this, which I would find difficult to believe.
>...........
>
Sruti magazine carried an article by Gundecha brothers titled “The Adverse
Effects Of Notating Raga-s” in the April 2001 issue. (It was translated from
the original Hindi by Nancy Lesh). They state that if you start with Raag
Bhoopali and adopt the note ga as the tonic then the resulting scale will have
the superficial sound of Malkauns. To get true Malakauns we would have to
adjust the intonation of the pitches slightly. Elsewhere in the article, they
state “.......For example, nowadays we have established the phrase ni sa dha
ni re as the catch phrase of Raga Jaijaiwanti.......We have been persuaded to
believe that Jaijaiwanti should be identified by this phrase, whereas we have
forgotten that the true nature of Jaijaiwanti comes from the slow glide (meend)
from the gandhara to the rishabha and finally resting on that note. Indeed we
are no longer taught to pay attention to this subtle glide as the
distinguishing feature of Jaijaiwanti.
In this way the subtleties of the entire raga system are
disappearing...........and reduced them to a very simple and low level
reproducible even on the harmonium.”
Ashok
>
>I have been looking for materials - as in the following post. Thanks.
>
>I am also looking for published materials on the same lines. Please e-mail
>or post if you know of publishings that co-relate the aesthetics and the
>mathematical structures in the ragas and their use in presentations.
>
>Thanks
>Iyengar
A physicist based in Bangalore, Dr. Varadarangan, has written
a book, titled Shruti Lakshana Prakashini, on the mathematical
aspects of the shruti system and claims to have a software that
outputs Shruti Bheda raagas for any input raaga. Currently, the
book is in Kannada, but the author says that he is working on an
English translation.
The internet music magazine at http://www.TheMusicMagazine.com
has carried a summary article on the book and later an interview
with the author. Hopefully, the following urls will work. Else,
check their archives for issues earlier this year.
http://www.themusicmagazine.com/bookapr00.html
http://www.themusicmagazine.com/kvranganint2.html
Ashok
Praful Kelkar
> Effects Of Notating Raga-s” in the April 2001 issue. (It was translated from
> the original Hindi by Nancy Lesh). They state that if you start with Raag
> Bhoopali and adopt the note ga as the tonic then the resulting scale will have
> the superficial sound of Malkauns. To get true Malakauns we would have to
> adjust the intonation of the pitches slightly. Elsewhere in the article, they
> state “.......For example, nowadays we have established the phrase ni sa dha
> ni re as the catch phrase of Raga Jaijaiwanti.......We have been persuaded to
> believe that Jaijaiwanti should be identified by this phrase, whereas we have
> forgotten that the true nature of Jaijaiwanti comes from the slow glide (meend)
> from the gandhara to the rishabha and finally resting on that note. Indeed we
> are no longer taught to pay attention to this subtle glide as the
> distinguishing feature of Jaijaiwanti.
> In this way the subtleties of the entire raga system are
> disappearing...........and reduced them to a very simple and low level
> reproducible even on the harmonium.”
Harmonium is a peculiar phenomenon. The vocalists I have spoken to
and asked speicifically in terms of shrutis and harmonium they all
(n=4) said they ignore it and focus on taanpura. As Warren
suggested auditary illusion probably plays a significant role in music
apprecation and in good hands harmonium does indeed sing raags
(although essentially with incorrect intonation) - and up to a point
that is fine. Perhaps too much harmoniumizing (or guitarizing) the
ears may be detrimental to hearing subtlities in shrutis. That's what
some dhrupadiyas seem to think.
But obviously that is not to say that that's all there is to HCM. As
you focus on the subtlties of the swar system in the raags more
complexities arise. I don't think one can find absolute agreement on
placement of shrutis in raags, however, often shrutis sit at
"non-ratio" places.
Specific examples that come to mind of Shrutis that cannot be
exaplined by frequency ratios arising either from Sa or from another
dominant note in the raag are -
1. Yaman - N is lower than Bihag, Shankara. This N and tM are
related as perfect fifth (3/2) - but both are on the lower side.
Other notes in Yaman are beautifully harmonically related to each
other - G is perfect 3rd, P is fifth, and Dha gives Ga as the 5th and
also Sa as the 3rd, and P gives R as the 5th. It is very soothing.
The N and tM don't seem to be related by ratio to other notes - and in
sitting slightly lower they tend to create a certain depth and pathos
that is Yaman. This N is indeed a bit lower than N arising from the
P string (its third).
Bihag implies the same S, G, P - but N is higher so is tM (which is
such a weak note). However, this N receives a punch in that P is at
the start of it and the N keeps wanting to pull toward Sa which
creates a beautiful tension. This N is probably same as 3rd from P.
Shankara N is even higher!!
2. Bhairav - kR brings M as 3rd, kD brings S as 3rd, P is perfect
5th. But G in an odd note. It is slightly higher than the natural
3rd coming from Sa - infact it creates a peculiar beat - and may be
that's what give it the "taste" of bhairav. This G is weaker than M
and often there is sliding between G and M and as you do that the beat
changes creating a most peculiar feeling.
3. Madhamad sarang and Megh both use SRMPnS - there is difference in
weight of M and P. In terms of shrutis the n in Megh is quite a bit
lower. The higher n in M. Sarang pulls it at once into Sarang group.
Similarly R in Megh is lower. Of course there are differneces in
uccharan as beautifully shown by Jhasaheb on Rajan's clip.
I am sure there are many more examples.
Praful
Praful Kelkar
I think I made an error in typing about shrutis in Yaman... the D is
such that it brings in G as the fifth G is 5/4 and D 5/4 x 2/3 = 5/3
(same as 5/6 expressed in the same octave) ....... no such relation
to Sa.
P