Whups--I forgot

I used a program which I had forgotten I had not yet made some
changes to, so that my results for n+m;46 when n and m are even are
wrong. I was wondering why the 12+34 system didn't show better
results for triads, as I expected it to do!

The correct mapping for 34+12 is [1,-2,-8,11], which is of course
completely different, and shows that the Gamut is well-supplied with
complete triads. The 34+12 system, rather than being mildly
interesting, is one of the very important 5-limit systems, comparable
to meantone, since the most significant 5-limit comma here is the
diaschisma, 2048/2025, and this is a diaschismic system. Again, this
makes sense in terms of 34 and 12, and suggests that the diaschisma
has some significant role in Indian musical theory, or at least
should have. I've just checked Graham's web page, and he says it may
have relevance, so someone must have noticed this connection.

The correct mapping for 24+22 is [2,-4,7,-1,-7,2] which also is
better, and makes the cure for quarter-tone music a more vital one.

--- In tuning@y..., genewardsmith@j... wrote:
> I used a program which I had forgotten I had not yet made some
> changes to, so that my results for n+m;46 when n and m are even are
> wrong. I was wondering why the 12+34 system didn't show better
> results for triads, as I expected it to do!
>
> The correct mapping for 34+12 is [1,-2,-8,11],

As I thought it should be. The hypothetical shruti scale is a chain
of 21 fifths, ignoring the diaschisma!

> which is of course
> completely different, and shows that the Gamut is well-supplied
with
> complete triads.

And the hypothetical shruti scale even more so.

> The 34+12 system, rather than being mildly
> interesting, is one of the very important 5-limit systems,
comparable
> to meantone, since the most significant 5-limit comma here is the
> diaschisma, 2048/2025, and this is a diaschismic system. Again,
this
> makes sense in terms of 34 and 12, and suggests that the diaschisma
> has some significant role in Indian musical theory, or at least
> should have. I've just checked Graham's web page, and he says it
may
> have relevance, so someone must have noticed this connection.

It does seem, according to my research, that shruti #2 probably
functioned as both 135/128 and 16/15. Also, the standard JI
interpretation of the shruti scale is a PB with a diaschisma as one
of the unison vectors:

http://www.ixpres.com/interval/td/erlich/srutipblock.htm

> The correct mapping for 24+22 is [2,-4,7,-1,-7,2] which also is
> better, and makes the cure for quarter-tone music a more vital one.

Well _now_ we see why Graham said the 11-limit complexity was 22: (7-
(-4))*2 = 22. I do get 11-limit hexads on my Shrutar though --
because I'm using the omnitetrachordal rather than MOS arrangement,
and/or because the Shrutar scale is actually a fretting over which
I'm tuning open strings to both 1/1 (0) and 3/2 (27).

Actually, rather than exact 46-tET, Dave Keenan proposed a fretting
which incorporates some JI notes (the Indian diatonic scale and a few
others), some notes tempered by a fraction of the diaschisma, and
some notes tempered by a fraction of 896/891. I think I'll use this
proposal when I get the instrument made.

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

> It does seem, according to my research, that shruti #2 probably
> functioned as both 135/128 and 16/15. Also, the standard JI
> interpretation of the shruti scale is a PB with a diaschisma as one
> of the unison vectors:

> http://www.ixpres.com/interval/td/erlich/srutipblock.htm

A more plausible looking 46-tone 5-limit block would combine the
diaschisma with 2^2 3^9 5^(-7), the difference between seven major
sixths and two 6s.

> Actually, rather than exact 46-tET, Dave Keenan proposed a fretting
> which incorporates some JI notes (the Indian diatonic scale and a
few
> others), some notes tempered by a fraction of the diaschisma, and
> some notes tempered by a fraction of 896/891. I think I'll use this
> proposal when I get the instrument made.

For authenticity? I thought you were not much of a believer in
extreme JI.

In-Reply-To: <9rq69d+jvfe@eGroups.com>
Gene wrote:

> The correct mapping for 34+12 is [1,-2,-8,11], which is of course
> completely different, and shows that the Gamut is well-supplied with
> complete triads.

I don't know what this is intended to mean, as you seem to be deriving a
linear temperament from two inconsistent equal temperaments. You need
more information in there! Taking nearest prime approximations, I get [1,
-2, -8, -6]. 46+58 is [1, -2, -8, -12].

> The 34+12 system, rather than being mildly
> interesting, is one of the very important 5-limit systems, comparable
> to meantone, since the most significant 5-limit comma here is the
> diaschisma, 2048/2025, and this is a diaschismic system. Again, this
> makes sense in terms of 34 and 12, and suggests that the diaschisma
> has some significant role in Indian musical theory, or at least
> should have. I've just checked Graham's web page, and he says it may
> have relevance, so someone must have noticed this connection.

As 12 and 22 note scales are used with 5-limit justification, it's
obviously tempting to look for a diaschismic interpretation. But I don't
know of any contemporaneous documents suggesting this.

> The correct mapping for 24+22 is [2,-4,7,-1,-7,2] which also is
> better, and makes the cure for quarter-tone music a more vital one.

I get this as the 22+24 or 46+22 mapping taking nearest prime
approximations. 46 and 22 are at least consistent in the 11-limit.

Graham

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <9rq69d+jvfe@e...>

> > The correct mapping for 34+12 is [1,-2,-8,11], which is of course
> > completely different, and shows that the Gamut is well-supplied
with
> > complete triads.

> I don't know what this is intended to mean, as you seem to be
deriving a
> linear temperament from two inconsistent equal temperaments.

There you are, shaking that consistency fetish at my nose again. If
you want, you can consider this to be h46=h34+g12, where g12(11)=41.

> You need
> more information in there!

Actually, I don't, and I don't need to work out what vals seem to be
involved, as I did above. From 34 and 12 I get 1/6 < 4/23 < 3/17 on
the 23rd row Farey sequence, and that's all I need, in fact more than
I need. However it doesn't hurt to know that the 50th row looks like
1/6<8/47<7/41<6/35<5/29<4/23<7/40<3/17 if I want to learn about the
neighborhood, so to speak, and locate 80 and 58 in relation to this
system.

> Taking nearest prime approximations, I get
> [1, -2, -8, -6].

This is what I get for 34, not 46.

> 46+58 is [1, -2, -8, -12].

This is 58 (and 46 with the generators regarded as defined mod 23,
which is how it should be viewed) but not 46+58=104, so it seems we
are not on the same wavelength so far as notation goes.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > It does seem, according to my research, that shruti #2 probably
> > functioned as both 135/128 and 16/15. Also, the standard JI
> > interpretation of the shruti scale is a PB with a diaschisma as
one
> > of the unison vectors:
>
> > http://www.ixpres.com/interval/td/erlich/srutipblock.htm
>
> A more plausible looking 46-tone 5-limit block would combine the
> diaschisma with 2^2 3^9 5^(-7), the difference between seven major
> sixths and two 6s.

More plausible than what? The block there has 22 tones, not 46 tones.
Can you clarify?
>
> > Actually, rather than exact 46-tET, Dave Keenan proposed a
fretting
> > which incorporates some JI notes (the Indian diatonic scale and a
> few
> > others), some notes tempered by a fraction of the diaschisma, and
> > some notes tempered by a fraction of 896/891. I think I'll use
this
> > proposal when I get the instrument made.
>
> For authenticity?

Partially.

> I thought you were not much of a believer in
> extreme JI.

I'd like the Indian diatonic plus a few other notes in JI rather than
46-tET because

(a) I'd like to have the central chords as just as possible
(b) I'd like the primary whole steps to be closer in size than they
are in 46-tET.

Think of it as a sort of "well-temperament" with the tempering
distributed unequally so that the "drone key" is essentially just.

I wrote,

> --- In tuning@y..., genewardsmith@j... wrote:
> > I used a program which I had forgotten I had not yet made some
> > changes to, so that my results for n+m;46 when n and m are even
are
> > wrong. I was wondering why the 12+34 system didn't show better
> > results for triads, as I expected it to do!
> >
> > The correct mapping for 34+12 is [1,-2,-8,11],
>
> As I thought it should be. The hypothetical shruti scale is a chain
> of 21 fifths, ignoring the diaschisma!

Oops, that's not correct. The hypothetical shruti scale is a chain
of 21 fifths, ignoring the schisma. But in this case, it would have
been more relevant to say that the hypothetical shruti scale two
chains of fifths, one of 11 fifths and one of 9 fifths, ignoring the
diaschisma. See my srutiblock page again (Gene, you didn't comment
when I asked you what you meant about there being a "better" 46-tone
block, since the block there is only a 22-tone block).

So Gene, take the PB on my srutiblock page, which is the accepted JI
specification for the shrutis. Treat the diaschisma as the commatic
unison vector. Is the result equivalent to (12+10);12+34? In message
29815, you said this would be a good paultone variant . . . but I
think not, as the ratios of 7 are almost completely absent here! Or
maybe that's what you meant by "variant".

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

> More plausible than what? The block there has 22 tones, not 46
tones.
> Can you clarify?

Sorry, I thought the idea was to get a 46-tone 5-limit block, where
my proposal would be a good one.

> I'd like the Indian diatonic plus a few other notes in JI rather
than
> 46-tET because
>
> (a) I'd like to have the central chords as just as possible
> (b) I'd like the primary whole steps to be closer in size than they
> are in 46-tET.

There's always the possibility of MOS-tempering 46 notes; for instance
46;53+46 tunes 46 to the excellent 7-limit of the 99-et; the
generator map is [-5,-13,17,-9,6] so we have a 5-limit complexity of
13; 7, 9, 11 and 13 limit of 30; and 15,17,19 and 21-limit of 35.

--- In tuning@y..., genewardsmith@j... wrote:

> > (a) I'd like to have the central chords as just as possible
> > (b) I'd like the primary whole steps to be closer in size than
they
> > are in 46-tET.
>
> There's always the possibility of MOS-tempering 46 notes; for
instance
> 46;53+46 tunes 46 to the excellent 7-limit of the 99-et; the
> generator map is [-5,-13,17,-9,6] so we have a 5-limit complexity of
> 13; 7, 9, 11 and 13 limit of 30; and 15,17,19 and 21-limit of 35.

But the diaschisma no longer vanishes (and what about the 891:896),
so this doesn't work for the Shrutar scale.

Me:
> > I don't know what this is intended to mean, as you seem to be
> deriving a
> > linear temperament from two inconsistent equal temperaments.

Gene:
> There you are, shaking that consistency fetish at my nose again. If
> you want, you can consider this to be h46=h34+g12, where g12(11)=41.

You can call them g or h or z or w or whatever you like, but for the
benefit of the rest of us you'll have to say what they mean!

Me:
> > You need
> > more information in there!

Gene:
> Actually, I don't, and I don't need to work out what vals seem to be
> involved, as I did above. From 34 and 12 I get 1/6 < 4/23 < 3/17 on
> the 23rd row Farey sequence, and that's all I need, in fact more than
> I need. However it doesn't hurt to know that the 50th row looks like
> 1/6<8/47<7/41<6/35<5/29<4/23<7/40<3/17 if I want to learn about the
> neighborhood, so to speak, and locate 80 and 58 in relation to this
> system.

I think I understand all the words there, but not how they fit together.

> > Taking nearest prime approximations, I get
> > [1, -2, -8, -6].
>
> This is what I get for 34, not 46.

Well, you can get anything you like from 34 because, again, it isn't
consistent.

>
> > 46+58 is [1, -2, -8, -12].
>
> This is 58 (and 46 with the generators regarded as defined mod 23,
> which is how it should be viewed) but not 46+58=104, so it seems we
> are not on the same wavelength so far as notation goes.

46+58 is the linear temperament consistent with both 46- and 58-equal.

Graham

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

> So Gene, take the PB on my srutiblock page, which is the accepted
JI
> specification for the shrutis. Treat the diaschisma as the commatic
> unison vector. Is the result equivalent to (12+10);12+34?

It can't be, because a classic Fokker block will always temper
symmerically. However, you do can (11+11);12+34 in this way. The
(12+10);12+34 is what I got by applying h46 to your shruti steps,
which means either I did it wrong or that it might be a Paul block
rather than a Fokker block.

A basis for the 5-kernel of the 22-et is the diaschisma d=2048/2025
and the maximal diesis m=250/243; other kernel elements are the small
diesis, m/d=3125/3072 and your m*d=20480/19683. If we take the
diashisma, the small diesis and the comma of 81/80 as a basis, we get
a useful notation from it and the 12,19, and 22 ets. We can start
with blocks, either yours or another using a diashisma and something
else, and temper out the diashisma. This means tossing 19 out of the
notation, leaving us with 12 and 22. Any of the above blocks now
become the 22 notes [i, floor(6i/11)] as i runs from 0 to 21, where i
is the number of 22-step intervals and floor(6i/11) is the number of
12-step intervals. We can rearrange this into two circles of fifths
of 11 notes each, one starting at [0, 0] and proceeding by [13, 7]
steps (reduced modulo [22, 12]) up to [20 10], and the second
starting from [11 6]. If we send [i, j] to i+j, which means we
consider the 12 and 22 intervals to be of the same size, we get the
34-et version of this, if we take 2i+j we get the 46 et and 3i+j
gives 58.

If instead we had tossed 22, we would get meantone, whereas tossing
12 (did I hear cheers?) leads to magic.

In message
> 29815, you said this would be a good paultone variant . . . but I
> think not, as the ratios of 7 are almost completely absent here! Or
> maybe that's what you meant by "variant".

I meant a 5-limit variant, yes.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > So Gene, take the PB on my srutiblock page, which is the accepted
> JI
> > specification for the shrutis. Treat the diaschisma as the
commatic
> > unison vector. Is the result equivalent to (12+10);12+34?
>
> It can't be, because a classic Fokker block will always temper
> symmerically.

How far down the page did you read? The accepted JI specification for
the shrutis takes one note the Fokker block and transposes it by a
chromatic unison vector. Go back and look at it again.

> I meant a 5-limit variant, yes.

OK, cool.

--- In tuning@y..., graham@m... wrote:

> > There you are, shaking that consistency fetish at my nose again.
If
> > you want, you can consider this to be h46=h34+g12, where g12(11)
=41.

> You can call them g or h or z or w or whatever you like, but for
the
> benefit of the rest of us you'll have to say what they mean!

I denote by h46 the map sending primes to the nearest 48-et value,
with everything else following from that, and similarly for h34. In
that case, in the 11-limit we don't have h46=h34+h12, but if we
instead send 11 to 41 steps of the 12-et, we have a map g12(2)=12,
g12(3)=19, g12(5)=28, g12(7)=34 g12(11)=41 which tells us how many
scale steps we use for anything in the 11-limit, and then we do have
h46=h34+g12, as well as consistency.

If I want to know
> > > Taking nearest prime approximations, I get
> > > [1, -2, -8, -6].
> >
> > This is what I get for 34, not 46.
>
> Well, you can get anything you like from 34 because, again, it
isn't
> consistent.

I can get anything I like from anything, consistent or not, but why
should I want to? The above is derived from the nearest prime
approximations for 34, it seems to me.

> > > 46+58 is [1, -2, -8, -12].
> >
> > This is 58 (and 46 with the generators regarded as defined mod
23,
> > which is how it should be viewed) but not 46+58=104, so it seems
we
> > are not on the same wavelength so far as notation goes.

> 46+58 is the linear temperament consistent with both 46- and 58-
equal.

This is definately something we want a notation for, but so is the et
and generator defined by 46 and 58, which is what I've been using it
for. What would you say to 46&58 or 46^58 to mean the temperament
obtained by intersecting the kernels of 46 and 58?