13-limit Saggital?

A familiar "notation" in my sense for the 5-limit has a basis
<16/15, 25/24, 81/80>; this can readily be related to the diatonic scale. If we look at <16/15, 25/24, 81/80, 64/63, 33/32>^(-1) we get
[h7,h5,h3,-v7,v11] where h3, h5, and h7 are the standard 11-limit versions of the 3,5, and 7 ets, and v7 and v11 are the 7-adic and
11-adic valuations. If we have either some way of symbolizing both major and minor semitones or equivalently a semitone and the minor diesis of 128/125, we have a way of representing the 5-limit without involving higher limits. Perhaps George has already done this and I failed to understand.

In any case, having done this the 7 and 11 limits are now extensions of the 5-limit system so obtained in the simplest way imaginable, namely by means of the 7 and 11-adic valuations which simply check for the exponent of 7 and 11 in the prime factorization. It seems to me that if saggital doesn't already work this way, then perhaps it should; it would give us a nice system which would cover the JI
11-limit, but also reduce neatly to the 7 or 5 limits.

Another aspect of interest is that this can easily be extended to the 13-limit, so that there is a nice 13-limit saggital availble if anyone wants it. We simply take [h7,h5,h3,v7,v11,v13]^(-1) and obtain <16/15,25/24,81/80,63/64,33/32,39/40>, from which we see that adding a symbol for 40/39 extends this scheme of notation.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> A familiar "notation" in my sense for the 5-limit has a basis
> <16/15, 25/24, 81/80>; this can readily be related to the diatonic
>scale. If we look at <16/15, 25/24, 81/80, 64/63, 33/32>^(-1) we get
> [h7,h5,h3,-v7,v11] where h3, h5, and h7 are the standard 11-limit
>versions of the 3,5, and 7 ets, and v7 and v11 are the 7-adic and
> 11-adic valuations. If we have either some way of symbolizing both
>major and minor semitones or equivalently a semitone and the minor
>diesis of 128/125, we have a way of representing the 5-limit without
>involving higher limits. Perhaps George has already done this and I
>failed to understand.

I think it would be extremely valuable if you could examine George's
notation proposal and try to understand it. Of course, no one's
paying you for it :(

> In any case, having done this the 7 and 11 limits are now
>extensions of the 5-limit system so obtained in the simplest way
>imaginable, namely by means of the 7 and 11-adic valuations which
>simply check for the exponent of 7 and 11 in the prime
>factorization. It seems to me that if saggital doesn't already work
>this way, then perhaps it should; it would give us a nice system
>which would cover the JI
>11-limit, but also reduce neatly to the 7 or 5 limits.

Gene, I'm still thinking that a "notation" in the sense George is
considering comes with an essential basis of a 7-note Pythagorean
chain, which form the categories into which all the other pitches are
notated. Is yours a "notation" in this sense? In some other, similar
sense? I'm not trying to be picky about terminology -- just trying to
get a handle on it all.

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

> I think it would be extremely valuable if you could examine George's
> notation proposal and try to understand it. Of course, no one's
> paying you for it :(

I'm gleaning what I've gleaned from his list of commas and his diagram of saggital as related to 72-et. From the latter, it appears that a sort of 12-equal standard is the starting point, so that we have a way of representing numbers of semitones as given, so far as I can see, by h5+h7=g12 in the 11-limit. It looks to me as if it must be a 225/224~1 system, but I'm waiting for George to respond to all this this stuff I've put out to see for certain.

> Gene, I'm still thinking that a "notation" in the sense George is
> considering comes with an essential basis of a 7-note Pythagorean
> chain, which form the categories into which all the other pitches are
> notated. Is yours a "notation" in this sense?

I think actually it *doesn't* come from a 7-note Pythagorean chain; the "13 limit univeral" proposal I made would do that. His 72-et diagram seems to me to show that enharmonic equivalents are the same, and simply provide a way of counting semitones.

One can start from a 7-tone basis or a 12-tone basis, but one needs to decide which it will be. Which is easier on the brain of people used to the present system, I wonder? When I suggested meantone should be easier to learn than 72-et, I was told I was wrong.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > I think it would be extremely valuable if you could examine
George's
> > notation proposal and try to understand it. Of course, no one's
> > paying you for it :(
>
> I'm gleaning what I've gleaned from his list of commas and his
>diagram of saggital as related to 72-et.

You should look closely at his examples of how chords are actually
notated.

>From the latter, it appears that a sort of 12-equal standard is the
>starting point,

Then it wouldn't apply to 31-tET and 41-tET, as stated.
>
> > Gene, I'm still thinking that a "notation" in the sense George is
> > considering comes with an essential basis of a 7-note Pythagorean
> > chain, which form the categories into which all the other pitches
are
> > notated. Is yours a "notation" in this sense?
>
> I think actually it *doesn't* come from a 7-note Pythagorean chain;

Clearly the very *position* -- the most salient quality of the
notation -- of any pitch is determined by relating it to the 7-note
Pythaogorean chain -- can you deny that this is central to the
notation?

>One can start from a 7-tone basis or a 12-tone basis, but one needs
>to decide which it will be. Which is easier on the brain of people
>used to the present system, I wonder? When I suggested meantone
>should be easier to learn than 72-et, I was told I was wrong.

Yes, as far as modern musicians are concerned, I must side with
Joseph and say that, unfortunately, 12-tET enharmonic equivalence is
a firmly entrenched habit.

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

> >From the latter, it appears that a sort of 12-equal standard is the
> >starting point,

> Then it wouldn't apply to 31-tET and 41-tET, as stated.

A 7-et standard is the basis of ordinary musical notation, which does not stop it from being applicable to 12, 19, 31 etc. The basic semitone count is the 12-et part, and to it you add 81/80, 64/63
33/32 (and 40/39, if you want to keep going.) This lets you notate
31 or 41 et readily enough--you could start with a periodicy block.

> > I think actually it *doesn't* come from a 7-note Pythagorean chain;
>
> Clearly the very *position* -- the most salient quality of the
> notation -- of any pitch is determined by relating it to the 7-note
> Pythaogorean chain -- can you deny that this is central to the
> notation?

I think it is *not* central. D-arrow-up and E-arrow-down are exactly the same.

> >One can start from a 7-tone basis or a 12-tone basis, but one needs
> >to decide which it will be. Which is easier on the brain of people
> >used to the present system, I wonder? When I suggested meantone
> >should be easier to learn than 72-et, I was told I was wrong.
>
> Yes, as far as modern musicians are concerned, I must side with
> Joseph and say that, unfortunately, 12-tET enharmonic equivalence is
> a firmly entrenched habit.

Then George may have the right plan.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > >From the latter, it appears that a sort of 12-equal standard is
the
> > >starting point,
>
> > Then it wouldn't apply to 31-tET and 41-tET, as stated.
>
> A 7-et standard is the basis of ordinary musical notation, which
>does not stop it from being applicable to 12, 19, 31 etc.

Exactly! But _Sagittal_ is meant to apply to 31-tET and 41-tET.

> > > I think actually it *doesn't* come from a 7-note Pythagorean
chain;
> >
> > Clearly the very *position* -- the most salient quality of the
> > notation -- of any pitch is determined by relating it to the 7-
note
> > Pythaogorean chain -- can you deny that this is central to the
> > notation?
>
> I think it is *not* central. D-arrow-up and E-arrow-down are
>exactly the same.

Not in 31-tET and 41-tET, two tunings that Sagittal is intended for!

> > >One can start from a 7-tone basis or a 12-tone basis, but one
needs
> > >to decide which it will be. Which is easier on the brain of
people
> > >used to the present system, I wonder? When I suggested meantone
> > >should be easier to learn than 72-et, I was told I was wrong.
> >
> > Yes, as far as modern musicians are concerned, I must side with
> > Joseph and say that, unfortunately, 12-tET enharmonic equivalence
is
> > a firmly entrenched habit.
>
> Then George may have the right plan.

Have you even read George's posts?

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

> Have you even read George's posts?

It sounds as if you think I've missed something obvious, so you may as well say what. If you mean his remarks about 41-et, in fact no more symbols are required. He has symbols for 81/80, 64/63 and 33/32, and together with a way of notating g12, that is all you need for a
225/224~1 system such as 41-et.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Have you even read George's posts?
>
> It sounds as if you think I've missed something obvious, so you may
> as well say what.

I haven't been trying to keep anything from you! Maybe I'm just a
complete dunce.

>If you mean his remarks about 41-et, in fact no >more symbols are
>required.

But how can enharmonic equivalence of the 12-tET be said to be at the
basis of the whole system, as you claimed?

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

> But how can enharmonic equivalence of the 12-tET be said to be at the
> basis of the whole system, as you claimed?

The 12 notes, with enharmonic equivalents, play just the same role as A-G do. I think "D-up" or "E-down" should in effect be regarded as a single symbol.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > But how can enharmonic equivalence of the 12-tET be said to be at
the
> > basis of the whole system, as you claimed?
>
> The 12 notes, with enharmonic equivalents, play just the same role
>as A-G do. I think "D-up" or "E-down" should in effect be regarded
>as a single symbol.

How can that be? They refer to different pitches in 41-tET and in 31-
tET.

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

> How can that be? They refer to different pitches in 41-tET and in 31-
> tET.

Possibly as George really does it, though that hasn't really been explained, but not the way I would use the system, since no extra symbols are needed. We know how it looks in 72-et, and I can tell you how to get it to work in 41 or 31 et in a way consistent with what he has said. Just use 12 symbols as the g12 semitone counter, in the same way as the 7 symbols A-G notate the h7 tone counter.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > How can that be? They refer to different pitches in 41-tET and in
31-
> > tET.
>
> Possibly as George really does it, though that hasn't really been
>explained, but not the way I would use the system, since no extra
>symbols are needed. We know how it looks in 72-et, and I can tell
>you how to get it to work in 41 or 31 et in a way consistent with
>what he has said. Just use 12 symbols as the g12 semitone counter,
>in the same way as the 7 symbols A-G notate the h7 tone counter.

It would be lovely if you elaborated that into a full system. But
Gene, George's system, or anything like it, lives on a standard
staff, where position indicates one of seven values -- this is at the
very core of the system. And until you say what the precise meaning
of "natural" is for those seven values (or twelve, in your case), you
don't have a notation yet.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_33330.html#33339

> One can start from a 7-tone basis or a 12-tone basis, but one needs
to decide which it will be. Which is easier on the brain of people
used to the present system, I wonder? When I suggested meantone
should be easier to learn than 72-et, I was told I was wrong.

***I would say unequivocally 12...

??

Opps, I equivocated...

JP

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

/tuning/topicId_33330.html#33342

> Yes, as far as modern musicians are concerned, I must side with
> Joseph and say that, unfortunately, 12-tET enharmonic equivalence
is a firmly entrenched habit.

****Well, of course, we needn't say "unfortunately" :) since 12-tET
is pretty "ummm, ummmm gooood" at least when viewed by the famous
Paul Erlich chart of "good" ETs!

And a *few* people over the years have seemed to do something with
it... :)

But, now, as a basis for something even *greater,* an exploration
into JUST INTONATION (yes, in all caps, just like a cultist) we have
it as a *springboard* to 72-tET, a decent approximation!

Seems like a *natural* evolution to *me*!

(and, of course, *everybody else* showed me this... I wish I had
invented it myself... :) )

JP

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

> It would be lovely if you elaborated that into a full system.

I looked at George's system more closely, and I'm afraid instead of working for 225/224~1 it seems to me to work only for systems where the Pythagorean comma is a unison, which isn't very interesting for a universal system as it stands. However, by adding another symbol for the schisma, we could take the 12-tone Pythagorean scale as our base, and inflect it with saggital symbols plus the schisma, and get a system for JI in the 11-limit. This, of course, could then spell any equal temperament up to the 11-limit.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > It would be lovely if you elaborated that into a full system.
>
> I looked at George's system more closely, and I'm afraid instead of
>working for 225/224~1 it seems to me to work only for systems where
>the Pythagorean comma is a unison

This contradicts George's own claim that it works for 31 and 41.

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

> > I looked at George's system more closely, and I'm afraid instead of
> >working for 225/224~1 it seems to me to work only for systems where
> >the Pythagorean comma is a unison

> This contradicts George's own claim that it works for 31 and 41.

It also contradicts what he just posted by way of an explanation, but I think we are getting there. If the system is based on seven tones, it presumably will be very close to Manuel's notation, and to the Pythagorean system I was talking about. This would make his and Manuel's suggestion of 27/26 for the 13-limit make sense, and would suggest how to extend it.

Two systems which seem to make sense are

<256/243, 2187/2048, 81/80, 64/63, 33/32, 27/26, 2187/2176, 513/512>

and

<245/243, 2187/2048, 81/80, 64/63, 33/32, 27/26, 4131/4096, 513/512>

both for the 19-limit. The inverses of the above as matricies are

[h7-v17, h5-v17, -v5, -v7, v11, -v13, -v17, v19]

and

[h7, h5, -v5, -v7, v11, -v13, -v17, v19]

The idea would not be to have symbols for the two Pythagorean semitones, but to translate the 3-limit part into A-G plus sharps and flats. The maps h7 and h5, or h7-v17 and h5-v17, would tell you what the Pythagorean part should be, and it could then be inflected with symbols to give the prime limits up to 19. However, we would have a different staff position or accidental for a 17-limit interval depending on which of the above systems we used.