..........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