Yahoo Tuning Groups Ultimate Backup tuning Shruti

Date: Sat, Dec. 23, 2000
From: Haresh BAKSHI <hareshbakshi@hotmail.com>
Subject: Shruti

The spelling: sruti (the first letter "s" as the diacritical Roman
symbol) has been used by Sir Monier-Williams, Walter Kaufmann, Ashok
Da Ranade, L. Subramaniam; shruti, by Alain Danielou, Swami
Prajnananda and several others. It is a question of how we prefer to
represent the Sanskrit sound, as in the English word "shrewd", for
transliteration into the Roman alphabet.

Definitions: Lack of standardisation is the hallmark of Sanskrit
terms. Shruti has been variously translated as: microtone,
microtonic interval, interval, step etc. It is mainly determined
through fine auditory perception or grasp. The number of sounds
falling within a scale is infinite; but the number which can be
differntiated (or perceived, or grasped) as musically useful, is 22.
These 22 shruti-s are further grouped into five classes, bringing in
the aesthetic concept that a shruti is not only an auditory
perception, but also a distinct expression to the listener's mind.
There is a broad agreement that shruti-s are not equal; that the
seven notes are the sounds selected from among these 22 shruti-s;
that each note is established on its first shruti; that shruti-s can
be reproduced on a string instrument, in terms of various vibrating
lengths; that shruti-s can, therefore, be stated in terms of
frequencies, though Indian music, being truly modal, is built on the
relationship of sounds with the tonic (also called the reference
note, key etc.).
References: The earliest mention appears to have been made in Bharat
muni's Natyashastra (about 500 B.C.). Later references include
Narada of Shiksha (1st century A.D.), Kohala (quoted in Brihaddeshi)
Dattila's Brihaddeshi, and several later works.

Remarks: Shruti is a part of tonal concept. Talented singers use
shruti-s, regularly and unknowingly, as part of their musical
repertoire. Quite often, even the raga-s have their notes
established on shruit-s, rather than the usual notes. For example,
the "komal Re" (Db, if C is the tonic) of the raga Todi, is really a
microtone, lower than the usual location of "komal Re" on the Indian
scale. Even the singers of a capella, use the notes of the harmonic
scale, not the notes of the tempered scale; this is because of
absence of accompaniment on piano or any instrument with fixed,
tempered scale. So, singing correct microtones comes naturally to
the voice of a talented singer.

Haresh BAKSHI <hareshbakshi@hotmail.com>

----------------------------
Haresh BAKSHI
www.SoundOfIndia.com
4771 Silverberry LN NW
ACWORTH GA 30102-3559 USA
Tel: (678) 354-4678
-----------------------------

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

--- In tuning@egroups.com, hareshbakshi@h... wrote:

> For example,
> the "komal Re" (Db, if C is the tonic) of the raga Todi, is really
a
> microtone, lower than the usual location of "komal Re" on the
Indian
> scale.

According to the usual (Wilson, Ramanthan, etc.) 5-limit
specification of the srutis, which can be seen as a 22-tone
periodicity block
(http://www.ixpres.com/interval/td/erlich/srutipblock.htm), Todi That
would be

1/1 256/243 32/27 45/32 3/2 128/81 243/128 1/1

while the usual "komal Re" is usually given as 16/15 or a tempered
tone between 16/15 and 135/128, rather than 256/243.

Is this what you are referring to?

I am curious as to how this specification of Todi comes about -- I am
skeptical as to an acoustical derivation, since on a lattice of
consonances it would be highly disconnected, while a stricly
Pythagorean derivation of this and other Thats would fail to account
for the presence of 45/32, unless this is taken as a schismatic
shorthand for 1024/729 (which would mean that an interval of 10
srutis was being used to approximate a consonant one of 9
srutis) . . . ?

🔗graham@microtonal.co.uk

🔗hareshbakshi@hotmail.com

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> --- In tuning@egroups.com, hareshbakshi@h... wrote:
>
> > For example,
> > the "komal Re" (Db, if C is the tonic) of the raga Todi, is
really
> a
> > microtone, lower than the usual location of "komal Re" on the
> Indian
> > scale.
>
> According to the usual (Wilson, Ramanthan, etc.) 5-limit
> specification of the srutis, which can be seen as a 22-tone
> periodicity block
> (http://www.ixpres.com/interval/td/erlich/srutipblock.htm), Todi
That
> would be
>
> 1/1 256/243 32/27 45/32 3/2 128/81 243/128 1/1
>
> while the usual "komal Re" is usually given as 16/15 or a tempered
> tone between 16/15 and 135/128, rather than 256/243.
>
> Is this what you are referring to?
>
> I am curious as to how this specification of Todi comes about -- I
am
> skeptical as to an acoustical derivation, since on a lattice of
> consonances it would be highly disconnected, while a stricly
> Pythagorean derivation of this and other Thats would fail to
account
> for the presence of 45/32, unless this is taken as a schismatic
> shorthand for 1024/729 (which would mean that an interval of 10
> srutis was being used to approximate a consonant one of 9
> srutis) . . . ?

Haresh BAKSHI responds:
Hi Paul, At the outset, let me say that the Tuning List is doing
great service to Indian music by posting discussions on the concept
of Indian shruti. I am sure more and more Indian musicians will try
to appreciate, and even adopt -- as I am trying to do -- this
methodology to get more gleanings from this vast subject.

From your response to my posting, it appears to me that I have
created confusion by writing about the "lower Re in Todi" etc. I
should have mentioned that I was referring to the RAGA Todi, not
the "thaat" Todi. A "thaat" is a scale, the 7-note ladder. We have
10 thaat-s. With the note C as the tonic, they are, in natural scale
(not tempered scale): Names: Bilawal, C-D-E-F-G-A-B; Khamaj, C-D-E-F-
G-A-Bb; Bhairav, C-Db-E-F-G-Ab-B; Kafi, C-D-Eb-F-G-A-Bb; Asavari, C-D-
Eb-F-G-Ab-Bb; Bhairavi, C-Db-Eb-F-G-Ab-Bb; Kalyan, C-D-E-F#-G-A-B;
Marwa, C-Db-E-F#-G-A-B; Poorvi, C-Db-E-F#-G-Ab-B; Todi, C-Db-Eb-F#-G-
Ab-B. The thaat-s are the raw material from which raga-s are
formed. The thaat-s are bereft of any aesthetic value by themselves;
they are NEVER performed on the stage. Now, it so happens that we
have raga-s which have the same names as the thaat-s. Thus to a
performing Indian musician like me, "Todi" refers to the raga Todi,
NOT the thaat Todi. This is how I confused one with the other. I
apologize, and stand corrected. So, then, what is a raga? Let me
recapitulate some basic tenets:

The concept associated with the term" raga" is highly developed and
complex. It defies any definition or description. We can, at best,
enumerate some of the characteristics of the raga. Even then its
concept eludes us. The only way to grasp the concept of the raga is
to listen to its development over prolonged periods of time and
repeatedly. It is like learning a new language: However enjoyable,
it requires consistent and continuous effort. Let us take up some of
the characteristics of the raga:
1. The raga must be aesthetically pleasing. It must delight the
heart of the listener. It must be capable of existing in its own
right, as an individual entity, with unique aesthetical value.
2. The development of a raga is. in principle, constant attainment
of aesthetic tensions, followed by resolutions, through various
degrees of sonance (consonance and dissonance). This is achieved by
the use of the permissible notes, singly or in combination, through
the movements in three octaves. This is, of course, all, and only,
improvisation. There are regulations that govern these movements.
3. Basically, a raga is capable of being developed into an infinite
number of melodic patterns, following certain guidelines.
4. A raga should be capable of being classified into one of the thaat-
s.
5. A raga cannot be formed out of less than 5 notes. "Sa" [C] has to
be present as the reference note. Both "Ma" (the fourth) and "Pa"
(the fifth) cannot be simultaneously omitted from the raga. So, it
must have the following notes: SA; one of the Re [D] or Ga [E]; Ma
[F] or Pa [G]; one of the Dha [A] or Ni [B], Further, if only one of
the Ma/Pa is present, the raga must include either both Re and Ga, or
both Dha and Ni, so as to achieve a total of 5 notes.
5. It cannot include two states [out of the three possible states:
shuddha (natural), komala (flat), and teevra (sharp)] of same note
consecutively.
6. The raga must follow the time theory. This means that a raga can
be performed only at its stipulated time of the day (or night, or a
season of the year). And so on and on..........

Coming back to the point. It would appear to me, that this forum
chiefly concerns itself with tuning and temperament (as its name,
obviously, suggests). This would involve treatment of microtones and
thaat-s, but definitely, not raga-s. Is this correct? I want to know
the the line of demarkation.

Even if it is so, it does not in any way make it less important for
Indian music. Since time immemorial, hair-splitting has been
Indians' national pastime -- especially the never-ending and futile
type. It is time for more and more Indian musicians to get thorough
exposure to new ideas and new methodologies of research. By the way,
do I have here the dubious distinction of being the only Indian
musician on the Tuning list? -- I hope not.

In the next posting, I hope to give a summary of "standard Indian
text-book type" treatment of the subject of shruti.
With regards,

Haresh BAKSHI
www.SoundOfIndia.com
Acworth GA 30102-3559

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗hareshbakshi@hotmail.com

--- In tuning@egroups.com, graham@m... wrote:
> Haresh BAKSHI wrote:
>
> > The spelling: sruti (the first letter "s" as the diacritical
Roman
> > symbol) has been used by Sir Monier-Williams, Walter Kaufmann,
Ashok
> > Da Ranade, L. Subramaniam; shruti, by Alain Danielou, Swami
> > Prajnananda and several others. It is a question of how we
prefer to
> > represent the Sanskrit sound, as in the English word "shrewd",
for
> > transliteration into the Roman alphabet.
>
> The Oxford Hindi-English dictionary gives the "u" and "i" as short
vowels.
> So that's right, is it? I originally thought it was pronounced
> "shrootee" but it appears not.
>
> Also, while you're around, what are the correct Hindi or Sanskrit
plurals
> for "shruti" and "raag"?
>
> Just curious,
>
> Graham

Hi Graham, I am sorry for this delayed response.
Yes, "u" and "i" are short vowels. So, "shrooti" would be incorrect.
The plural forms for "shruti" are:-
Sanskrit: shrutayaH (sounds like shrutayah); Hindi:shrutiyaaN (nasal)
The plural forms for "raag" are:-
Sanskrit: raagaaH (sounds like raagaahaa); Hindi: raaga (same as the
singular form).
However, their respective dual forms in Sanskrit are:-
shrutee; raagau. There are no Hindi dual forms.
I hope this helps. Thanks.
Haresh.

🔗graham@microtonal.co.uk

hareshbakshi@hotmail.com () wrote:

> Hi Graham, I am sorry for this delayed response.
> Yes, "u" and "i" are short vowels. So, "shrooti" would be incorrect.
> The plural forms for "shruti" are:-
> Sanskrit: shrutayaH (sounds like shrutayah); Hindi:shrutiyaaN (nasal)
> The plural forms for "raag" are:-
> Sanskrit: raagaaH (sounds like raagaahaa); Hindi: raaga (same as the
> singular form).
> However, their respective dual forms in Sanskrit are:-
> shrutee; raagau. There are no Hindi dual forms.
> I hope this helps. Thanks.

Well, it means I can casually drop them into my posts to pretend I know
what I'm talking about. Presumably the "dual form" is used for exactly
two items? It's interesting that Sanskrit words used in Hindi should have
their own, non-standard plurals. Or is there some pattern I don't know
about?

So, do you have any more thoughts on the shruti scale? The way I
understand it, the 22 notes made sense when Bharata used them, and were
roughly equally tempered. But the more recent tradition uses a 12 note
scale and pretends there's a continuity with the older one. So the idea
of 22 shrutayaH is still around, and people have tried to find a relevance
for it. Hence the 5-limit scale that is reported as an ideal for tuning,
and the general use of "shruti" to mean any small pitch difference.

This is the impression I get, but I haven't made any great study, and it
isn't my culture. It would be interesting to get another perspective.

You also said in a previous post 'For example, the "komal Re" (Db, if C is
the tonic) of the raga Todi, is really a microtone, lower than the usual
location of "komal Re" on the Indian scale.' How common is it for a raga
to include a sruti in its definition like this? In this case it would
appear to be a leading tone to the tonic, but I wonder if 7-limit
intervals are ever implied.

Graham

🔗hareshbakshi@hotmail.com

--- In tuning@egroups.com, graham@m... wrote:
> hareshbakshi@h... () wrote:
>
> > Hi Graham, I am sorry for this delayed response.
> > Yes, "u" and "i" are short vowels. So, "shrooti" would be
incorrect.
> > The plural forms for "shruti" are:-
> > Sanskrit: shrutayaH (sounds like shrutayah); Hindi:shrutiyaaN
(nasal)
> > The plural forms for "raag" are:-
> > Sanskrit: raagaaH (sounds like raagaahaa); Hindi: raaga (same as
the
> > singular form).
> > However, their respective dual forms in Sanskrit are:-
> > shrutee; raagau. There are no Hindi dual forms.
> > I hope this helps. Thanks.
>
> Well, it means I can casually drop them into my posts to pretend I
know
> what I'm talking about. Presumably the "dual form" is used for
exactly
> two items? It's interesting that Sanskrit words used in Hindi
should have
> their own, non-standard plurals. Or is there some pattern I don't
know
> about?

Haresh BAKSHI's response:
Yes, the dual form in Sanskrit is for "two"; Hindi has derived much
of its vocabulary from Sanskrit, but has evolved in a different
direction. Hindi does not have provision for the dual form -- we have
to use the plural form instead. India has 20 major languages and 500
dialects. Most of those have their roots in Sanskrit. I can travel
less than 300 miles and reach another state whose popular language I
do not know. Hindi and English are the common "bridges" for people
in 25 different states, each with its own language. The people in the
villages use the regional dialect mostly. Villages constitute more
than 80% of India.

>
> So, do you have any more thoughts on the shruti scale? The way I
> understand it, the 22 notes made sense when Bharata used them, and
were
> roughly equally tempered. But the more recent tradition uses a 12
note
> scale and pretends there's a continuity with the older one. So the
idea
> of 22 shrutayaH is still around, and people have tried to find a
relevance
> for it. Hence the 5-limit scale that is reported as an ideal for
tuning,
> and the general use of "shruti" to mean any small pitch difference.
>
> This is the impression I get, but I haven't made any great study,
and it
> isn't my culture. It would be interesting to get another
perspective.
>
> You also said in a previous post 'For example, the "komal Re" (Db,
if C is
> the tonic) of the raga Todi, is really a microtone, lower than the
usual
> location of "komal Re" on the Indian scale.' How common is it for
a raga
> to include a sruti in its definition like this? In this case it
would
> appear to be a leading tone to the tonic, but I wonder if 7-limit
> intervals are ever implied.

I do have many things to say about shruti, which will make up one
more perspective. I am going to try to re-group and re-state many
of my ideas in the light of what is going on in the Tuning postings.
This is a difficult task for me, primarily because I am, as yet,
hardly even a beginner here. I am going through various sites and
all the old postings and have started learning new concepts,
approaches, and vocabulary. My chief concern is that, I have started
feeling that, "thaat" and "raga" are perhaps being used
synonymously. Needless to say, they are ENTIRELY different. I am
sure my apprehension will go away as study more postings.
Thanks. Haresh.

>
>
> Graham

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

--- In tuning@egroups.com, hareshbakshi@h... wrote:

> I do have many things to say about shruti, which will make up one
> more perspective. I am going to try to re-group and re-state many
> of my ideas in the light of what is going on in the Tuning postings.
> This is a difficult task for me, primarily because I am, as yet,
> hardly even a beginner here. I am going through various sites and
> all the old postings and have started learning new concepts,
> approaches, and vocabulary. My chief concern is that, I have started
> feeling that, "thaat" and "raga" are perhaps being used
> synonymously. Needless to say, they are ENTIRELY different. I am
> sure my apprehension will go away as study more postings.
> Thanks. Haresh.

Haresh, I have no idea why you have this concern. Everyone who has posted on Indian music
here has had a very clear understanding of the difference between Thaats and Ragas. In fact, as
Thaats (10 scale types) were simply a proposal of one isolated theorist with no importance to
musicians, while the Ragas are essentially the "compositions" off of which all their improvisations
are created, couldn't we forget about Thaats altogether and concentrate on Ragas? Or are there
shruti theories connected to the Thaat theory?

🔗hareshbakshi@hotmail.com

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> --- In tuning@egroups.com, hareshbakshi@h... wrote:
>
> > I do have many things to say about shruti, which will make up one
> > more perspective. I am going to try to re-group and re-state
many
> > of my ideas in the light of what is going on in the Tuning
postings.
> > This is a difficult task for me, primarily because I am, as yet,
> > hardly even a beginner here. I am going through various sites
and
> > all the old postings and have started learning new concepts,
> > approaches, and vocabulary. My chief concern is that, I have
started
> > feeling that, "thaat" and "raga" are perhaps being used
> > synonymously. Needless to say, they are ENTIRELY different. I
am
> > sure my apprehension will go away as study more postings.
> > Thanks. Haresh.
>
> Haresh, I have no idea why you have this concern. Everyone who has
posted on Indian music
> here has had a very clear understanding of the difference between
Thaats and Ragas. In fact, as
> Thaats (10 scale types) were simply a proposal of one isolated
theorist with no importance to
> musicians, while the Ragas are essentially the "compositions" off
of which all their improvisations
> are created, couldn't we forget about Thaats altogether and
concentrate on Ragas? Or are there
> shruti theories connected to the Thaat theory?

Haresh BAKSHI responds:
Thanks a lot, Paul. You have made it amply clear, once for all: We
are all dealing with the raga-s here. It will be much easier for me
to start participating in the proceedings now, though following the
various responses to my postings will remain, for quite some time,
difficult for me. May I crave indulgence of all ?
Haresh BAKSHI
www.SoundOfIndia.com

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗hareshbakshi@hotmail.com

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗hareshbakshi@hotmail.com

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Haresh,

I suspect we are just muddling about here, not really knowing how much of
this is really related to Indian tuning. We need people like you to tell us
this.

Because the Shrutar tuning is a temperament (but with many just intervals),
no single lattice, without duplicates, can indicate all the relationships.

Here's one 11-limit lattice for it, with no duplicates.

5/4----15/8----45'32
/ \ / \ / \
7,6 /7/4\ / \ / \
/11/8 \ /33/32\ /14,9 \
4/3-----1/1-----3/2-----9/8----27/16
/ \16,11/ \12/11/ \18/11/
/ \ / \ / \9/7/ 27'14
/ \ / \ / \ /
16;15----8,5-----6/5-----9/5

I'm using an ad hoc notation where I replace the slash / with
" to mean raised about 10 cents
' to mean raised about 5 cents
, to mean lowered about 5 cents
; to mean lowered about 10 cents

Notice that 16,11 = 81'56 and 14,9 = 99'64

Here's a 7-limit lattice with many duplicates, showing some of the ways the
lattice wraps around.
9/7/---27'14---16,11
/ \ / \ / \
/ \ / \ / \5/4
/ \ / \ / \ \
11/8----33/32---14,9-----7,6-----7/4\
/ \ / / \
/ \ / 4/3-----1/1-----3/2
/ \ / / \ / \ /
27'14---16,11---12/11---18/11 / \ / \ /
/ \ / \ / / \ / \ /
/ \ / \5/4/---15/8----45'32---16;15----8,5-----6/5
/ \ / \ / / \ / \ /
14,9-----7,6-----7/4\ / \ / \ / 11/8----33/32
/ \ / \ / \ / / \ /
4/3-----1/1-----3/2-----9/8----27/16 / \ /
/ \ / \ / \ / / \ /
/ \ / \ / \9/7/---27'14---16,11---12/11---18/11
/ \ / \ / \ / / \ / \ /
16;15----8,5-----6/5-----9/5\ / \ / \ /
/ \ / \ / \ /
11/8----33/32---14,9-----7,6-----7/4

One of Paul's specifications was that the tuning had to preserve the
following scales with no more than 35 cents difference between major and
minor whole steps. Choose one note from each line.

3/2
8,5 27/16
9/5 15/8
1/1
16;15 9/8
6/5 5/4
4/3 45'32
3/2

In strict 5-limit JI they would only be 22 cents different. The current
proposal has about 31 cents difference [between major and minor whole
tones]. There are some additional diatonics available. One is a minor mode
from 15/8. Some barely usable diatonics with about 37c difference between
major and minor whole-tones are based on 14,9 and 27'14.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> Haresh,
>
> I suspect we are just muddling about here, not really knowing how
much of
> this is really related to Indian tuning. We need people like you to
tell us
> this.

Well, I did take the criteria partially from my studies of Indian
music. With all due respect to Haresh, I believe we've already gone
over all the relevant assumptions with him.

> Because the Shrutar tuning is a temperament (but with many just
intervals),
> no single lattice, without duplicates, can indicate all the
relationships.
>
> Here's one 11-limit lattice for it, with no duplicates.
>
> 5/4----15/8----45'32
> / \ / \ / \
> 7,6 /7/4\ / \ / \
> /11/8 \ /33/32\ /14,9 \
> 4/3-----1/1-----3/2-----9/8----27/16
> / \16,11/ \12/11/ \18/11/
> / \ / \ / \9/7/ 27'14
> / \ / \ / \ /
> 16;15----8,5-----6/5-----9/5
>
> I'm using an ad hoc notation where I replace the slash / with
> " to mean raised about 10 cents
> ' to mean raised about 5 cents
> , to mean lowered about 5 cents
> ; to mean lowered about 10 cents
>
> Notice that 16,11 = 81'56 and 14,9 = 99'64

and, most importantly, 16;15 = 135"128.
>
> Here's a 7-limit lattice with many duplicates, showing some of the
ways the
> lattice wraps around.
> 9/7/---27'14---16,11
> / \ / \ / \
> / \ / \ / \5/4
> / \ / \ / \ \
> 11/8----33/32---14,9-----7,6-----7/4\
> / \ / / \
> / \ / 4/3-----1/1-----3/2
> / \ / / \ / \ /
> 27'14---16,11---12/11---18/11 / \ / \ /
> / \ / \ / / \ / \ /
> / \ / \5/4/---15/8----45'32---16;15----8,5-----6/5
> / \ / \ / / \ / \ /
> 14,9-----7,6-----7/4\ / \ / \ / 11/8----33/32
> / \ / \ / \ / / \ /
> 4/3-----1/1-----3/2-----9/8----27/16 / \ /
> / \ / \ / \ / / \ /
> / \ / \ / \9/7/---27'14---16,11---12/11---18/11
> / \ / \ / \ / / \ / \ /
> 16;15----8,5-----6/5-----9/5\ / \ / \ /
> / \ / \ / \ /
> 11/8----33/32---14,9-----7,6-----7/4
>
> One of Paul's specifications was that the tuning had to preserve the
> following scales with no more than 35 cents difference between
major and
> minor whole steps. Choose one note from each line.
>
> 3/2
> 8,5 27/16
> 9/5 15/8
> 1/1
> 16;15 9/8
> 6/5 5/4
> 4/3 45'32
> 3/2
>
> In strict 5-limit JI they would only be 22 cents different. The
current
> proposal has about 31 cents difference [between major and minor
whole
> tones].

I'm almost ready to settle for this proposal and have the instrument
built, but I wonder: what if I were to say that 31 cents is too much?
What if I were to insist on the septimal comma (64:63) as the maximum
difference?

P.S. For those just following along, the tuning is actually produced
by a given fret pattern on a 1/1 string and a 3/2 string, so there
are often some slightly different pitches available on different
strings.

🔗Haresh BAKSHI <hareshbakshi@hotmail.com>

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Paul Erlich wrote:
>I'm almost ready to settle for this proposal and have the instrument
>built, but I wonder: what if I were to say that 31 cents is too much?
>What if I were to insist on the septimal comma (64:63) as the maximum
>difference [between major and minor whole tones]?

We can ignore ratios of 7 and 11 and only need to consider this lattice.

5/4----15/8----45'32---16;15
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
4/3-----1/1-----3/2-----9/8----27/16
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
16;15----8,5-----6/5-----9/5

Sorry I'm not familiar with the Indian names. I'll refer to the lattice below.

A ----- E ----- B ----- F#
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
Bb----- F ----- C ----- G ----- D
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
Gb----- Db----- Ab----- Eb

Major whole steps are: (difference from 8:9 in current proposal)
Gb:Ab (+10c)
Db:Eb (+5c)
Bb:C
F:G
C:D
A:B (+5c)
E:F# (+10c) (optional)

Minor whole steps are: (all are 9:10 in current proposal)
Ab:Bb
Eb:F
G:A
D:E

If we must
(a) distribute the diaschisma (~20 cents) so that F# and Gb are the same
note, and
(b) keep Ab:C and C:E as just 4:5s,
then we see that the diaschisma can only be distributed between the Gb:Ab
and E:F# whole tones.

So one way to acheive your proposal above, is to sacrifice inversional
symmetry and the E:F# whole tone. We can give it a 14 cent error while only
giving Gb:Ab 6 cents. If we are also to have no more than 6 cents error in
A:B, then the B:F# fifth must have an error of 8 cents. Can you accept that?

Or are you willing to drop requirement (b) above? If we widen those major
thirds by 4c each, then we can acheive your proposal with only 3c errors in
any fifth and have no major whole step more than 6c wide of just (including
E:F#).

Alternatively we could leave the major whole steps as they are and widen
all the minor whole steps by 4c. This could be done by flattening Bb and A
by 4c and sharpening Eb and D by 4c. This means that the central minor
thirds will no longer be just.

I think those 3 suggestions pretty much cover the possibilities.
1. "sacrifice" the B:F# fifth and E:F# whole tone
2. temper the central major thirds
3. temper the central minor thirds
or some combination of the above.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:

> If we must
> (a) distribute the diaschisma (~20 cents) so that F# and Gb are the
same
> note, and
> (b) keep Ab:C and C:E as just 4:5s,
> then we see that the diaschisma can only be distributed between the
Gb:Ab
> and E:F# whole tones.
>
> So one way to acheive your proposal above, is to sacrifice
inversional
> symmetry and the E:F# whole tone. We can give it a 14 cent error

error relative to what?

> while only
> giving Gb:Ab 6 cents. If we are also to have no more than 6 cents
error in
> A:B, then the B:F# fifth must have an error of 8 cents. Can you
accept that?

Maybe, but moving any fret will affect not only a note on the 1/1
string but also on the 3/2 string. So what else is affected here?

> Or are you willing to drop requirement (b) above?

Quite possibly.

> If we widen those major
> thirds by 4c each, then we can acheive your proposal with only 3c
errors in
> any fifth and have no major whole step more than 6c wide of just
(including
> E:F#).

Sounds pretty good.
>
> Alternatively we could leave the major whole steps as they are and
widen
> all the minor whole steps by 4c. This could be done by flattening
Bb and A
> by 4c and sharpening Eb and D by 4c. This means that the central
minor
> thirds will no longer be just.

Possibly . . .

> or some combination of the above.

What's an "optimal" combination? Please reply to tuning-
math@yahoogroups.com -- this is getting too "specialized".

🔗Haresh BAKSHI <hareshbakshi@hotmail.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > So one way to acheive your proposal above, is to sacrifice
> inversional
> > symmetry and the E:F# whole tone. We can give it a 14 cent error
>
> error relative to what?

Relative to JI, as usual. It's a major whole tone. So relative to 8:9.

I guess I forgot to point out that if minor tones remain JI then max
diff between major and minor tones of a septimal comma (approx 27c)
means max widening of major tones is about 6c.

> >If we are also to have no more than 6 cents error in
> > A:B, then the B:F# fifth must have an error of 8 cents. Can you
> accept that?
>
> Maybe, but moving any fret will affect not only a note on the 1/1
> string but also on the 3/2 string. So what else is affected here?

I don't want to spend the time to answer this, particularly if you are
willing to temper central thirds (and thereby have no strict-JI
chords).

> What's an "optimal" combination? Please reply to tuning-
> math@y... -- this is getting too "specialized".

You tell me your error weighting for every dyad on that 5-limit
lattice and I'll tell you the optimum temperament. :-)

-- Dave Keenan

🔗Paul Erlich <paul@stretch-music.com>

🔗Haresh BAKSHI <hareshbakshi@hotmail.com>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Haresh BAKSHI" <hareshbakshi@h...> wrote:

>>>> Hi Paul, I understand,though I do not recall the source of this
information, that both Sa-Pa and Sa-ma tunings combined, produce 31
intervals. >>>>

[Paul's response]
>>>> I've never heard that accounting, but before we try to reproduce
it, are you really talking about 31 intervals, or 31 _pitches_? If
it's the latter, that doesn't seem right, since the JI lattice for sa-
grama is only shifted by one perfect fifth relative to the lattice
for ma-grama, so there should only be 3 additional pitches added to
the basic 22 when both gramas are taken into account, for a total of
25 pitches. >>>>

Hello Paul, I definitely mean intervals, not pitches. With my books,
and notes destroyed, along with my apartment, in the recent
earthquake in India, I cannot hope to find the reference again.
However, from what I recall, "in the physical spectra, there are only
five musical intervals; but in the subjective inner ear spectra, we
have 19 intervals i Pa-tuning, and 21 intervals in Ma-tuning, both
the tunings together giving 31 different intervals".

I hope your feedback on this will throw some light on my lost link of
facts and figures.

Regards,
Haresh.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "Haresh BAKSHI" <hareshbakshi@h...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Haresh BAKSHI" <hareshbakshi@h...> wrote:
>
> Hello Paul, I definitely mean intervals, not pitches. With my
books,
> and notes destroyed, along with my apartment, in the recent
> earthquake in India, I cannot hope to find the reference again.

My deepest condolences to you for your loss.

> However, from what I recall, "in the physical spectra, there are
only
> five musical intervals;

What does this refer to?

> but in the subjective inner ear spectra, we
> have 19 intervals i Pa-tuning, and 21 intervals in Ma-tuning, both
> the tunings together giving 31 different intervals".
>
> I hope your feedback on this will throw some light on my lost link
of
> facts and figures.
>
Hmm . . . with the usual specification of the 22 srutis, which is
what I thought you were referring to earlier, we have the following
intervals (in cents):

1 sruti: 22, 70, 90
2 srutis: 92, 112, 180
3 srutis: 114, 134, 182, 202
4 srutis: 204, 224, 272
5 srutis: 226, 274, 294
6 srutis: 296, 316, 364, 384
7 srutis: 318, 338, 386, 406
8 srutis: 408, 428, 476
9 srutis: 430, 478, 498
10 srutis: 500, 520, 568, 588
11 srutis: 522, 590, 610, 678

So with 36 intervals less that a half-octave in width, this is
clearly too many to explain the reference above.

However, if we consider the "Pa-tuning" and "Ma-tuning" above to
simply refer to the 7-tone _gramas_ themselves, we get:

sa-grama:
1 degree: 112, 182, 204
2 degrees: 294, 316, 386, 408
3 degrees: 498, 520, 590
4 degrees: 610, 680, 702
5 degrees: 792, 814, 884, 906
6 degrees: 996, 1018, 1088
7 degrees: 1200
That's a total of 21 different intervals within an octave.

ma-grama:
1 degree: 112, 182, 204
2 degrees: 294, 316, 386
3 degrees: 498, 520, 590
4 degrees: 610, 680, 702
5 degrees: 814, 884, 906
6 degrees: 996, 1018, 1088
7 degrees: 1200
That's a total of 19 different intervals within an octave.

So it's seems I've been able to explain your recollection of 19
intervals and 21 intervals.

Now there are two typical ways of combining the two gramas (depending
on how they are transposed relative to one another). The
more "authentic" is to treat sa-grama as being the set of absolute
pitches 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8; and ma-grama as the
same set with 45/32 replacing 4/3. So the combination of the two
would have eight different pitches, and the following intervals:

92, 112, 182, 204, 294, 316, 386, 408, 498, 520, 590, 610, 680, 702,
792, 814, 884, 906, 996, 1018, 1088, 1108, 1200
That's a total of only 23 different intervals within an octave.

The other way of combining the two gramas is to treat ma-grama as the
same as sa-grama but with 5/3 replacing 27/16. So the combination of
the two would again have eight different pitches, and the following
intervals:

22, 112, 182, 204, 294, 316, 386, 408, 498, 520, 590, 610, 680, 702,
792, 814, 884, 906, 996, 1018, 1088, 1178, 1200
Again, that's a total of only 23 different intervals within an octave.

So I'm not sure how the figure of 31 intervals was obtained!