Re: The Middle Path

--- In tuning@y..., jacky_ligon@y... wrote:

> Also, don't forget there's a "middle path"
> between ETs and JI: linear temperaments, planar temperaments, etc.

> Just so I understand these terms, will you explain what these are?

In an ET, we have just one generator s for all notes--for instance in
the 12-et, that would be s = 2^(1/12), and any note will be of the
form s^a for some integer exponent a. In JI, we have as many
generators as primes--in fact, the generators may as well be primes,
so that in the 5-limit, we have three generators 2,3, and 5, and any
note will be of the form 2^a 3^b 5^c for integer exponents a, b and c.

A regularly generated temperament is in between these extremes. For
instance, in the 5-limit we can write notes with two generators, not
one or three; if one of the generatorsis the octave, any note will be
of the form 2^a s^b; if s is e.g. a meantone fifth, this will give us
a meantone temperament. Because we don't count 2, this is
called "linear"; however any temperament with two generators
is "linear", any with three is "planar" and so forth. It isn't
logical, but it is standard.

In fact, 2 does not need to be a generator; for example with two ets
we may have a corresponding "linear" temperament, so that we have the
7-12 temperament where we write each note as r^a s^b, where a is the
number of scale steps in the 7-et, and b is the number in the 12 et,
and r and s are chosen to give good tunings. It is not the choice of
generators which matters so much as the choice of notes and tunings.

--- In tuning@y..., jacky_ligon@y... wrote:
> Paul Erlich in MMM says:
>
> Also, don't forget there's a "middle path"
> between ETs and JI: linear temperaments, planar temperaments, etc.
>
>
> Paul,
>
> Just so I understand these terms, will you explain what these are?
>
> Best,
>
> Jacky

Hopefully, Gene explained this well. I assume you understand how
meantone is an example of linear temperament, generated by a fifth
that is narrowed by some fraction of a syntonic comma. You can think
of it as a "middle path" in several different ways, each of which
implies a different meantone temperament.

1) You can pick one JI consonance to keep in JI, and temper the other
intervals. Examples are 1/4-comma meantone (with just major thirds,
minor sixths) and 1/3-comma meantone (with just major sixths, minor
thirds).

2) You can pick a chromatic unison vector to keep in JI (recall that
in periodicity block theory, the diatonic scale is defined by the
81:80 syntonic comma, along with one chromatic unison vector).
Examples are 2/7-comma meantone (with 25:24 just) and 1/7-comma
meantone (with 135:128 just).

3) You can forget about any specific intervals remaining in JI, but
optimize the approximations of all 5-limit JI consonances. Examples
include 7/26-comma meantone (equal-weighted least squares) and 3/14-
comma meantone (inverse-limit-weighted least squares).

Besides meantone, other examples of linear temperaments, most geared
for approximating some set of JI intervals within the 15-limit, are
listed by Graham Breed here:

http://x31eq.com/catalog.htm

I wrote:

> 3/14-
> comma meantone (inverse-limit-weighted least squares).

Sorry, 3/14-comma meantone is inverse-limit-weighted minimax. The
inverse-limit-weighted least squares solution is 63/250-comma
meantone.

P. S. If you're not aware of the great importance of meantone
temperament (and hence the Middle Path) for Western music, see, for
example,

http://home.earthlink.net/~kgann/histune.html

--- In tuning@y..., jacky_ligon@y... wrote:

> So "Linear" is MOS

If it's based on a strict Fokker periodicity block, yes. If it's
based on a periodicity block of some other shape, then it may contain
alterations from the MOS pattern -- but will have the same number of
notes.

> - or MP,

Even if it's MOS, it doesn't necessarily have Myhill's Property,
since the interval of repetition may repeat itself some integer
number of times (>1) within the interval of equivalence (usually 2:1).

> and three step sizes (or perhaps more)
> is "Planar"?

Well, a planar temperament will have at most a two-dimensional
subspace of JI intervals, and there must be at least one dimension
that correspond to tempered intervals (otherwise you're really
talking about planar JI -- not a "Middle Path").

If the scale is based on a strict Fokker periodicity block, then
there should be no more than 4 step sizes, 4 sizes of "third" (that
is, the interval of two steps), 4 sizes of "fourth" (that is, the
interval of three steps), etc. Again, if it's based on a PB of some
other shape, there may be alterations to this hyper-MOS pattern --
but the scale will still have the same number of notes.
>
> Use'm all the time

Really?! Can you give some examples?

--- In tuning@y..., jacky_ligon@y... wrote:

> Paul (and Gene),
>
> I'm not sure how well these will accord with the definitions but
here
> are a couple of scales I find to be musically interesting:

[...]

Hi Jacky.

The "middle path" I was referring to basically works like this: ETs
take a bunch
of JI intervals and set them equal to a bunch of others. The "middle
path" does this,
but with fewer of the JI intervals, so one does not get an ET, and
one generally
stays closer to JI.

I don't know if this is what's going on in your Phi scales, but I
don't see it. Do you have a JI basis in mind that you're trying to
approximate with these scales?

Regards,
Paul

--- In tuning@y..., jacky_ligon@y... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., jacky_ligon@y... wrote:
> > Hi Jacky.
> >
> > The "middle path" I was referring to basically works like this:
ETs
> > take a bunch of JI intervals and set them equal to a bunch of
> others. The "middle path" does this, but with fewer of the JI
> intervals, so one does not get an ET, and ne generally stays closer
> to JI.
>
> Could you kindly provide me with at least one example in cents?

I take it you understand meantone temperament, and cents values for
it can be found in many places, some of which I'm sure you're
familiar with . . . other than that, there are myriad other
examples . . . are there any particular properties you're looking for?
>
> > I don't know if this is what's going on in your Phi scales, but I
> > don't see it. Do you have a JI basis in mind that you're trying
to
> > approximate with these scales?
>
> Well - actually these were just a few of many such Phi scales, and
> that the one had some pretty good 5 limit approximations was
*there*
> to be discovered, but not necessarily my first intention.

Well I would say that that differentiates your approach from
the "middle path". By "middle path", I'm thinking of those who use
strict JI periodicity-block scales at one extreme (this would include
Terry Riley, Harry Partch, Erv Wilson, Ben Johnston, etc.), and those
who use ETs specifically for their many JI approximations as the
other extreme (this would include Joel Mandelbaum, Easley Blackwood,
sometimes Ivor Darreg, sometimes Neil Haverstick, sometimes Paul
Rapoport) . . . there's a "middle path" between the two extremes.

> Always fun
> to me to notice these kinds of convergences in various systems.

Absolutely.

--- In tuning@y..., jacky_ligon@y... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > I take it you understand meantone temperament, and cents values
for
> > it can be found in many places, some of which I'm sure you're
> > familiar with . . . other than that, there are myriad other
> > examples . . . are there any particular properties you're looking
> for?
>
> How about a tuning which optimizes:
>
> 11/10
> 11/9
> 11/8
> 11/7
> 11/6
>
All of these intervals are present in a 6:7:8:9:10:11 chord, which is
Partch's hexad. So I'll take it you're interested in optimizing what
Partch calls the 11-limit, which is what we call the 11-odd-limit.
It's the set of ratios you list above intepreted as intervals, plus
the set of intervals you find between the ratios you list above
intepreted as pitches, plus the octave and all octave inversions and
extensions of said intervals.

Dave Keenan created a planar microtoemperament with 31 notes that is
excellent for the 11-limit.

For linear temperaments, the MIRACLE temperament is probably best
(generator 116.7 cents). This is Graham Breed's list of the top 10
octave-repeating linear temperaments for the 11-limit. The number of
notes per octave is enough to give at least 1 complete hexad, give an
MOS, and not exceed 60:

generator intv. of repetition # of notes per octave max. error

116.7¢ octave 31 or 41 3.3¢
183.2¢ 1/2 octave 46 2.4¢
271.1¢ octave 22 or 31 or 53 9.3¢
216.7¢ 1/2 octave 50 3.1¢
16.9¢ 1/29 octave 58 3.7¢
83.2¢ 1/2 octave 44 or 58 2.9¢
310.2¢ octave 27 or 31 or 58 5.4¢
193.2¢ octave 56 2.8¢
585.1¢ octave 31 or 39 or 41 4.1¢
165.2¢ 1/2 octave 36 or 58 5.1¢

I'd be happy to elaborate on any of these systems should one interest
you in particular.

Yours,
Paul

In-Reply-To: <9q2gm8+dpf9@eGroups.com>
Jacky:

> > How about a tuning which optimizes:
> >
> > 11/10
> > 11/9
> > 11/8
> > 11/7
> > 11/6

Paul:

> All of these intervals are present in a 6:7:8:9:10:11 chord, which is
> Partch's hexad. So I'll take it you're interested in optimizing what
> Partch calls the 11-limit, which is what we call the 11-odd-limit.

Or perhaps Jacky means what he says, and is only interested in the
11-identity. The program then spits out

generator intv. of repetition # of notes per octave max. error

16.2 cents 1/8 octave 32 1.2 cents
78.2 cents octave 12 8.5 cents
495.6 cents octave 12 8.8 cents
116.6 cents octave 31 2.4 cents
216.7 cents half octave 50 1.5 cents

Graham

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <9q2gm8+dpf9@e...>
> Jacky:
>
> > > How about a tuning which optimizes:
> > >
> > > 11/10
> > > 11/9
> > > 11/8
> > > 11/7
> > > 11/6
>
> Paul:
>
> > All of these intervals are present in a 6:7:8:9:10:11 chord,
which is
> > Partch's hexad. So I'll take it you're interested in optimizing
what
> > Partch calls the 11-limit, which is what we call the 11-odd-
limit.
>
> Or perhaps Jacky means what he says, and is only interested in the
> 11-identity.

I'm unclear on what you're thinking here. The 11-identity is just a
single note having a certain function in a chord.

> The program then spits out
>
> generator intv. of repetition # of notes per octave max. error
>
> 16.2 cents 1/8 octave 32 1.2 cents
> 78.2 cents octave 12 8.5 cents
> 495.6 cents octave 12 8.8 cents
> 116.6 cents octave 31 2.4 cents
> 216.7 cents half octave 50 1.5 cents

So what exactly do these optimize?

Paul wrote.

> I'm unclear on what you're thinking here. The 11-identity is just a
> single note having a certain function in a chord.

Oh. So what are all the 11-limit intervals with an 11 in them called?

Graham

--- In tuning@y..., graham@m... wrote:
> Paul wrote.
>
> > I'm unclear on what you're thinking here. The 11-identity is just
a
> > single note having a certain function in a chord.
>
> Oh. So what are all the 11-limit intervals with an 11 in them
called?

Ratios of 11.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., graham@m... wrote:
> > Paul wrote.
> >
> > > I'm unclear on what you're thinking here. The 11-identity is
just
> a
> > > single note having a certain function in a chord.
> >
> > Oh. So what are all the 11-limit intervals with an 11 in them
> called?
>
> Ratios of 11.

Graham, for your investigation into this latter interpretation of
Jacky's specification, what did you use as your complexity measure?

In-Reply-To: <9q504j+vgcc@eGroups.com>
Paul wrote:

> Graham, for your investigation into this latter interpretation of
> Jacky's specification, what did you use as your complexity measure?

The usual one for ranking. Largest number of generators for an interval
in the set, multiplied by the number of periods to an octave. Although I
think I reported the smallest MOS, because it seemed to match your
figures. Here's the code:

import temper

consonances = [
(0,0,0,1), # 11/8
(-1,0,0,1), # 11/6
(0,-1,0,1), # 11/10
(0,0,-1,1), # 11/7
(-2,0,0,1), # 11/9
]

consist = temper.getLimitedETs(consonances)
temperaments = temper.getLinearTemperaments(consist, worstError=0.01)
print temperaments[:10]

At least, that should have been the code. I forgot 11:8 before. So we
now have

MOS ratio ppo generator smallest MOS minimax error
------------------------------------------------------
2/19 8 16.2 32 1.3
2/31 1 78.2 12 8.5
31/75 1 495.6 12 8.8
7/72 1 116.6 31 2.4
17/47 2 216.7 50 1.5

Where ppo is periods per octave, and the generator and error are in cents.

Graham