Shruti (was: what a dope (user))

Thank you so much, Herman Miller, for coming to my rescue!

--- In tuning@y..., "Haresh BAKSHI" <hareshbakshi@h...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > > . . . perhaps in earlier times the shrutis
> > > were closer to equal (they are in the Shrutar I've been
> > > contemplating), but today they are extremely unequal.
> >
> > Yes the latest "11-limit well-tempered" Shrutar tuning being
> > contemplated, has shruti ranging only from 44c to 71c, as opposed
> to 22c to 90c. It is Rothenberg-proper where the modern 5-limit JI
> Indian shruti scale is not. The proposed Shrutar tuning also has
more
> pitches that are justly intoned against 1/1 and 3/2 drones.
> > -- Dave Keenan >>>>>>>
>
> Hi Dave, Can you enlighten me on (1) why 11-limit?, (2) why reduce
> shruti range to 44-71c limit?, (3) 5-limit JI Indian shruti scale is
> not Rothenberg-proper -- can you elucidate for someone new like me?
> (4) How do we know that shruti-s in earlier times were closer to
> equal?

Hi Haresh,

I don't know the reasons for supposing that shruti-s in earlier times
were closer to equal?. I'll let Paul answer that since it was his
"perhaps". There are a number in the Scala archive with "indian" and
"old" in their name that do have this property, and Paul's and my
proposed Shrutar tuning does resemble them more than it resembles the
one simply called "indian" (the modern 5-limit JI).

Rothenberg-propriety (usually just called "propriety") simply means
that there are no intervals that are smaller than others while
containing more shrutis. Or to put it the other way around, intervals
composed of a certain number of shrutis are all smaller than (or equal
in size to) all intervals containing more shrutis. If we can drop the
"or equal" then it can be called "strictly proper".

Paul can answer this better than I, but I don't think that propriety,
or a smaller range of shruti sizes, were major design goals for the
Shrutar tuning. It mostly just came out that way. Propriety is more
likely to be important for recognition of the individual raga than it
is for the entire 22-tone gamut (Is this correct usage of "raga" and
"gamut"?).

Nor was it initially intended to include pitches which correspond to
the 11-th harmonic of the drones. 7-th harmonics yes. But when the
other goals had been acheived it turned out that very little extra
tempering was required, in order to include 11s. So we did.

Here's Paul's seminal post on the topic.
/tuning/topicId_16723.html#16723

So you see the important requirements initially were that it contain
the 5-limit 12-note lattice below, and that it add as many ratios of 7
to this as possible within 22 notes, without being too uneven.

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

Here's the current proposal

! shrutar.scl
!
Paul Erlich's Shrutar tuning (from 9th fret) tempered with Dave Keenan

22
!
33/32
101.95500 ! 16/15 / (2048/2025)^(1/2)
12/11
9/8
262.02693 ! 7/6 / (896/891)^(1/2)
6/5
5/4
9/7
4/3
11/8
595.11186 ! 45/32 * (2048/2025)^(1/4)
643.83808 ! 16/11 / (896/891)^(1/2)
3/2
760.07192 ! 14/9 / (896/891)^(1/2)
808.79814 ! 8/5 / (2048/2025)^(1/4)
18/11
27/16
7/4
9/5
15/8
1141.88308 ! 27/14 * (896/891)^(1/2)
2/1

2048/2025 (19.55 c) is the diaschisma. Perhaps 891:896 (9.69 c) should
be called the undecimal semicomma.

You can see that the above 5-limit lattice is preserved, apart from a
5 cent sharpening of the 45/32, a 5 cent flattening of the 8/5 and a
10 cent flattening of the 16/15, all on the periphery of the lattice.
This distributes the diaschisma and allows the approximate 16/15 to
also serve as an approximate 135/128 (a fifth above 45/32), in this
fashion:

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

Let me know if you want the 7-limit lattice with the other 10 notes.

-- Dave Keenan

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

Hi Dave, Thank you for your post #26294. This is extremely
interesting and informative. Thanks.

Yes, please, I do want the 7-limit lattice with the other 10 notes.
This will expand the lattice diagram to make it look "complete", in
addition to providing the remaining data.

In the mean time, I will study the information contained in your
message -- it is, indeed, a strong message; it tells me, in
threateningly clear terms that already too much has gone on in the
area of SHRUTI, before my arrival in the U.S., and I had better
accelerate my studying all that. I shall definitely attempt to.

Regards,
Haresh.

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:> Hi Haresh,
>
> I don't know the reasons for supposing that shruti-s in earlier
times
> were closer to equal?. I'll let Paul answer that since it was his
> "perhaps".

Well, it's just that Bharata measured 22 shrutis and not 12 or
53 . . . so one tends to guess at something closer to a Rothenberg-
proper 22.

> There are a number in the Scala archive with "indian" and
> "old" in their name that do have this property,

Wish I knew more about those.