My top 5--for Paul

#1

2^-90 3^-15 5^49

This is not only the the one with lowest badness on the list, it is the smallest comma, which suggests we are not tapering off, and is evidence for flatness.

Map:

[ 0 1]
[49 -6]
[15 0]

Generators: a = 275.99975/1783 = 113.00046/730; b = 1

I suggest the "Woolhouse" as a name for this temperament, because of the 730. Other ets consistent with this are 84, 323, 407, 1053 and 1460.

badness: 34
rms: .000763
g: 35.5
errors: [-.000234, -.001029, -.000796]

#2 32805/32768 Schismic badness=55

#3 25/24 Neutral thirds badness=82

#4 15625/15552 Kleismic badness=97

#5 81/80 Meantone badness=108

It looks pretty flat so far as this method can show, I think.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> #1
>
> 2^-90 3^-15 5^49
>
> This is not only the the one with lowest badness on the list, it is
the smallest comma, which suggests we are not tapering off, and is
evidence for flatness.
>
> Map:
>
> [ 0 1]
> [49 -6]
> [15 0]
>
> Generators: a = 275.99975/1783 = 113.00046/730; b = 1
>
> I suggest the "Woolhouse" as a name for this temperament,

Tricky -- "Woolhouse temperament" clearly means 7/26-comma meantone
to me. So this one falls inside the cutoff but the 612-tET-related
one doesn't? I'd favor reeling in the cutoff . . . schismic is pretty
complex already . . . as long as Ennealimmal makes it into the 7-
limit list, we're searching out far enough, as far as I'm concerned.
Also, I'm thinking a badness cutoff around 300 might be good, but
I'll hold off until I see more results. Finally, I'd like to
reinstate my strong belief that the "g" measure should be _weighted_.

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

> > I suggest the "Woolhouse" as a name for this temperament,
>
> Tricky -- "Woolhouse temperament" clearly means 7/26-comma meantone
> to me. So this one falls inside the cutoff but the 612-tET-related
> one doesn't?

That's what happens with cutoffs--this one has an even lower badness, not that it makes a difference at this point.

I don't know if it deserves a name; I agree Woolhouse won't do, and it's really more of a 1783 system anyway. I tried to give it a name because of its very low badness, but it's kind of absurd.

I'd favor reeling in the cutoff . . . schismic is pretty
> complex already . . .

I could continue analyzing them until we are clearly running off the rails into idiocy, which is why I suggested the other cutoff.

Finally, I'd like to
> reinstate my strong belief that the "g" measure should be _weighted_.

You think 3 should counts for more than 5, etc?

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

> Finally, I'd like to
> > reinstate my strong belief that the "g" measure should be
_weighted_.
>
> You think 3 should counts for more than 5, etc?

That's right. Harmonic progression by 3 is more comprehensible than
progression by 5 or 5/3. I suggest weights of 1/log(3), 1/log(5), and
1/log(5), to conform with the geometry of the lattice and with what
seems to me to be a properly octave-reduced Tenney metric.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> I don't know if it deserves a name;

You're right. It doesn't.

> I tried to give it a name
> because of its very low badness, but it's kind of absurd.

Yes. It is a fine example of the musical irrelevance of a flat badness
measure. I think musicians would rate it somewhere between 5 and
infinity times as bad as the other four you listed. 50 notes for one
triad? The problem, as usual is that an error of 0.5 c is
imperceptible and so an error of 0.0002 c is no better, and does not
compensate for a huge number of generators. Sorry if I'm sounding like
a stuck record.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > I don't know if it deserves a name;
>
> You're right. It doesn't.
>
> > I tried to give it a name
> > because of its very low badness, but it's kind of absurd.
>
> Yes. It is a fine example of the musical irrelevance of a flat
badness
> measure. I think musicians would rate it somewhere between 5 and
> infinity times as bad as the other four you listed. 50 notes for
one
> triad? The problem, as usual is that an error of 0.5 c is
> imperceptible and so an error of 0.0002 c is no better, and does
not
> compensate for a huge number of generators. Sorry if I'm sounding
like
> a stuck record.

Let's not make decisions for musicians. Many theorists have delved
into systems such as 118, 171, and 612. We would be doing no harm to
have something to say about this range, even if we don't personally
feel that it would be musically useful.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Yes. It is a fine example of the musical irrelevance of a flat
> badness
> > measure. I think musicians would rate it somewhere between 5 and
> > infinity times as bad as the other four you listed. 50 notes for
> one
> > triad? The problem, as usual is that an error of 0.5 c is
> > imperceptible and so an error of 0.0002 c is no better, and does
> not
> > compensate for a huge number of generators. Sorry if I'm sounding
> like
> > a stuck record.
>
> Let's not make decisions for musicians. Many theorists have delved
> into systems such as 118, 171, and 612. We would be doing no harm to
> have something to say about this range, even if we don't personally
> feel that it would be musically useful.

But Paul! You _are_ making decisions for musicians! You can't help but
do so. Unless you plan to publish an infinite list of temperaments,
the fact that you rate cases like this highly means that you will
include fewer cases having more moderate numbers of gens per
consonance. Shouldn't the question be rather whether you are making a
_good_ decision for musicians?

There's nothing terribly personal about the fact that an error of 0.5
c is imperceptible by humans.

Theorists have delved into systems such as 118, 171, 612-tET, but has
anything musical ever come of it? And if it has or does, surely we
would be looking at subsets, not the entire 118 notes per octave etc.
i.e. we'd be looking at temperaments within these ETs where consonant
intervals are produced by considerably fewer than 50 notes in a chain
(or chains) of generators.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> There's nothing terribly personal about the fact that an error of
0.5
> c is imperceptible by humans.

It sure is if you're playing with a loud or distorted sound system --
my favorite!

> Theorists have delved into systems such as 118, 171, 612-tET, but
has
> anything musical ever come of it? And if it has or does, surely we
> would be looking at subsets, not the entire 118 notes per octave
etc.
> i.e. we'd be looking at temperaments within these ETs where
consonant
> intervals are produced by considerably fewer than 50 notes in a
chain
> (or chains) of generators.

Perhaps not yet . . . but what harm comes from _informing_ musicians
of these systems? I'd love it if a genius musician did make use of
not considerably fewer than 50 notes per octave -- oh, wait a minute,
my lips are a little partched today . . . and when I make lattices
for these systems, you can be sure I'm going to start with the
simplest and work my way up until the impenetrable thickets of notes
make me decide a single line of data from Gene would be more
appropriate.

Hey Dave, why not look at Gene's list of 5-limit temperaments and see
if he's missed anything?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Theorists have delved into systems such as 118, 171, 612-tET, but has
> anything musical ever come of it?

The 118 et is a good one for schismic temperament, which we hear on the tuning list seems to have been hit on at one time, so I'd hardly dismiss it out of hand. Things such as ennealimmal haven't been tried, but until quite recently, neither had miracle been tried.

And if it has or does, surely we
> would be looking at subsets, not the entire 118 notes per octave etc.
> i.e. we'd be looking at temperaments within these ETs where consonant
> intervals are produced by considerably fewer than 50 notes in a chain
> (or chains) of generators.

This is the 21st century--there is no particular obstacle to using 612 notes, other than that is lot of notes to get around to.

I just took a look at the 118 et, and find it has the ragisma
(4375/4374) and the shisma as commas, and also what Manuel calls the "gamelan residue" of 1029/1024, and 3136/3125. Does anyone know where the name "gamelan resiude" comes from? In any case each of these separately, or more than one in combination, produce some interesting temperaments.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > There's nothing terribly personal about the fact that an error of
> 0.5
> > c is imperceptible by humans.
>
> It sure is if you're playing with a loud or distorted sound system
--
> my favorite!

OK. So choose a lower number, it will still be higher than the 0.0002
c or whatever it was of that supposed number-one temperament that even
Gene described as "absurd".

> > Theorists have delved into systems such as 118, 171, 612-tET, but
> has
> > anything musical ever come of it? And if it has or does, surely we
> > would be looking at subsets, not the entire 118 notes per octave
> etc.
> > i.e. we'd be looking at temperaments within these ETs where
> consonant
> > intervals are produced by considerably fewer than 50 notes in a
> chain
> > (or chains) of generators.
>
> Perhaps not yet . . . but what harm comes from _informing_ musicians
> of these systems? I'd love it if a genius musician did make use of
> not considerably fewer than 50 notes per octave -- oh, wait a
minute,
> my lips are a little partched today . . .

Whether you take Partch's 41 to be schismic-41 plus 2 or miracle-45
minus 2, no consonant interval is produced by a generator-chain of
more than 23 notes. I consider 23 to be considerably less than 50.

> and when I make lattices
> for these systems, you can be sure I'm going to start with the
> simplest and work my way up until the impenetrable thickets of notes
> make me decide a single line of data from Gene would be more
> appropriate.

OK. But don't some of the simplest ones have such large errors as to
be absurd too?

> Hey Dave, why not look at Gene's list of 5-limit temperaments and
see
> if he's missed anything?

This would be a lot easier for me if he would deign to give the
optimum generator (whether rms or max-absolute), in cents.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> This is the 21st century--there is no particular obstacle to using
612 notes,

For that matter there's no technical obstacle to using an
essentially continuous spectrum.

> other than that is lot of notes to get around to.

That's the one! Namely the limitation is in human cognition. The
composer can have computer assistance, but the listener can't.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> This would be a lot easier for me if he would deign to give the
> optimum generator (whether rms or max-absolute), in cents.

Ummm...ever heard of mulitplication? Graham likes it in octaves, and lately I've been packaging information more compactly by picking and appropriate et and giving it relative to that. The hard work is finding a rational number the generator corresponds to, and I've been doing that. I don't know how seriously you meant this, but of course I could do cents also, but so could you.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > This is the 21st century--there is no particular obstacle to using
> 612 notes,
>
> For that matter there's no technical obstacle to using an
> essentially continuous spectrum.

None whatever. I often don't use a scale. However, using a temperament differs from not using one; I can imagine a composer wanting to use ennealimmal approximations in what is in effect JI music, and 171 or 612 could be appropriate for that.

> > other than that is lot of notes to get around to.
>
> That's the one! Namely the limitation is in human cognition. The
> composer can have computer assistance, but the listener can't.

The listener doesn't need to sort out 612 notes, this is a red herring.

> None whatever.

That's been true since the 15th century.

>I often don't use a scale. However, using a temperament differs
>from not using one;

You bet. Also, using a scale differs from not using one.

>> That's the one! Namely the limitation is in human cognition. The
>> composer can have computer assistance, but the listener can't.
>
>The listener doesn't need to sort out 612 notes, this is a red
>herring.

The listener will sort out notes one way or the other, and
far fewer of them 171 or 612. Dave's point, I think, is
that he or she wouldn't be able to tell the difference.

-Carl

> This would be a lot easier for me if he would deign to give the
> optimum generator (whether rms or max-absolute), in cents.

Dave, I'll do the arithmetic for you and give you the RMS optima for
the ones with RMS less than 20 cents and g<8. Is there anything
_missing_ that is as good as one of these?

Generator 522.86¢, Period 1 oct.
Generator 505.87¢, Period 1/4 oct.
Generator 163.00¢, Period 1 oct.
Generator 491.20¢, Period 1/3 oct.
Generator 379.97¢, Period 1 oct.
Generator 503.83¢, Period 1 oct.
Generator 494.55¢, Period 1/2 oct.
Generator 442.98¢, Period 1 oct.
Generator 387.82¢, Period 1 oct.
Generator 271.59¢, Period 1 oct.
Generator 317.08¢, Period 1 oct.
Generator 498.27¢, Period 1 oct.

Did I miss any, Gene?
Thanks for checking, Dave.

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

> Did I miss any, Gene?

Apparently not; however I notice that Orwell came close to not making the cut. As probably the only person with experience using it, I can tell you it's a lot more practical than that would suggest.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Dave, I'll do the arithmetic for you and give you the RMS optima for
> the ones with RMS less than 20 cents and g<8. Is there anything
> _missing_ that is as good as one of these?
>
> Generator 522.86¢, Period 1 oct.
> Generator 505.87¢, Period 1/4 oct.
> Generator 163.00¢, Period 1 oct.
> Generator 491.20¢, Period 1/3 oct.
> Generator 379.97¢, Period 1 oct.
> Generator 503.83¢, Period 1 oct.
> Generator 494.55¢, Period 1/2 oct.
> Generator 442.98¢, Period 1 oct.
> Generator 387.82¢, Period 1 oct.
> Generator 271.59¢, Period 1 oct.
> Generator 317.08¢, Period 1 oct.
> Generator 498.27¢, Period 1 oct.

Thanks Paul. First note that I am only looking at those with a period
of 1 octave at this stage. As a 5-limit approximation the 522.86c
generator is junk. It has an error of 25 c in the 2:3. So there are
plenty of other temperaments as good as this. If you omitted this and
163.00 c I would agree with the list, and I would probably only find a
few more as good as 163.00 c. I'll let you know what these are when I
have more time.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Did I miss any, Gene?
>
> Apparently not; however I notice that Orwell came close to not
>making the cut. As probably the only person with experience using
>it, I can tell you it's a lot more practical than that would suggest.

In the 5-limit??

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > Dave, I'll do the arithmetic for you and give you the RMS optima
for
> > the ones with RMS less than 20 cents and g<8. Is there anything
> > _missing_ that is as good as one of these?
> >
> > Generator 522.86¢, Period 1 oct.
> > Generator 505.87¢, Period 1/4 oct.
> > Generator 163.00¢, Period 1 oct.
> > Generator 491.20¢, Period 1/3 oct.
> > Generator 379.97¢, Period 1 oct.
> > Generator 503.83¢, Period 1 oct.
> > Generator 494.55¢, Period 1/2 oct.
> > Generator 442.98¢, Period 1 oct.
> > Generator 387.82¢, Period 1 oct.
> > Generator 271.59¢, Period 1 oct.
> > Generator 317.08¢, Period 1 oct.
> > Generator 498.27¢, Period 1 oct.
>
> Thanks Paul. First note that I am only looking at those with a
period
> of 1 octave at this stage.

What do you mean? You did a whole spreadsheet with graphs for the 1/2-
octave period case.

> As a 5-limit approximation the 522.86c
> generator is junk. It has an error of 25 c in the 2:3.

On behalf of Herman Miller, Margo Schulter, Bill Sethares, and the
entire island of Java, let me just say #(@*$& ?@#>$, and then let me
just say, go play with this scale for a while. 'Junk' my &$$.

> So there are
> plenty of other temperaments as good as this.

By _as good as_, I mean having an equal or lower RMS error ANS and
equal or lower 'gens' measure.

> If you omitted this and
> 163.00 c I would agree with the list, and I would probably only
find a
> few more as good as 163.00 c. I'll let you know what these are when
I
> have more time.

I'd appreciate it. You'd be making a positive contribution.

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

> > Apparently not; however I notice that Orwell came close to not
> >making the cut. As probably the only person with experience using
> >it, I can tell you it's a lot more practical than that would suggest.
>
> In the 5-limit??

Weeel...I stayed mostly in the 7-limit, with excursions into the
11-limit. With a generator of 7/6, it's pretty hard to treat Orwell simply as a 5-limit system. However, the point I was making is that setting g<8 may be too low even if you have pretty strict ideas about what is practical.

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

> Weeel...I stayed mostly in the 7-limit, with excursions into the
> 11-limit.

Well, then.

>With a generator of 7/6, it's pretty hard to treat Orwell >simply as
>a 5-limit system.

Gee whiz, well you can't really think of it as 7/6 if you're talking
about 5-limit systems, can you? It's a one-third-of-a-minor-sixth
generator, really, in this context.

>However, the point I was making is that >setting g<8 may be too low
>even if you have pretty strict ideas >about what is practical.

I just did that because Dave Keenan so far has tended to look at
chains of up to 14 generators within a period of one octave, and
chains of up to 8 generators within a period of 1/2 pctave. So, since
I was asking him if anything was missing, it would have been useless
to include the more complex systems, which I agree are valuable for
certain musical styles, just as the simple systems with large errors
are.

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

> >However, the point I was making is that >setting g<8 may be too low
> >even if you have pretty strict ideas >about what is practical.
>
> I just did that because Dave Keenan so far has tended to look at
> chains of up to 14 generators within a period of one octave, and
> chains of up to 8 generators within a period of 1/2 pctave. So,
since
> I was asking him if anything was missing, it would have been useless
> to include the more complex systems, which I agree are valuable for
> certain musical styles, just as the simple systems with large errors
> are.

I'm now looking at up to 36 generators per prime, with an octave
period.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Thanks Paul. First note that I am only looking at those with a
> period
> > of 1 octave at this stage.
>
> What do you mean? You did a whole spreadsheet with graphs for the
1/2-
> octave period case.

I just haven't got to that yet.

> > As a 5-limit approximation the 522.86c
> > generator is junk. It has an error of 25 c in the 2:3.
>
> On behalf of Herman Miller, Margo Schulter, Bill Sethares, and the
> entire island of Java, let me just say #(@*$& ?@#>$, and then let me
> just say, go play with this scale for a while. 'Junk' my &$$.

Paul, I think you're severely distorting what I wrote. I didn't say
pelog is junk. I said "as a 5-limit approximation ..."

Is there really any evidence that pelog is a 5-limit temperament?
Specifically that it exists (even partly) because it is a 7 note chain
of generators such that a single generator approximates a 2:3, -3
generators approximates a 4:5 and 4 generators approximates a 5:6?
e.g. Do they play a lot of the approximate 1:3:5 triads in this
temperament? Or is it perhaps an approximate 7-tET, for mostly melodic
reasons, with inharmonic timbres to make the "fifths" sound ok, and
little or no importance placed on any approximate ratios of 5? I'm
only guessing. I know very little about pelog. I'm just very wary of
JI "explanations" for things like pelog and slendro.

> > So there are
> > plenty of other temperaments as good as this.
>
> By _as good as_, I mean having an equal or lower RMS error ANS and
> equal or lower 'gens' measure.

Why can't I use my own criteria for "as good as"?

It would be evidence that pelog actually is this 5-limit temperament
if, when a pelog scale departs from 7-tET it does so by making all
it's fifths but one, even narrower than the 7-tET fifth, i.e. even
further from a 2:3. A 7-tET fifth is 685.7 c. The rms optimum "fifth"
in this temperament is 677.1 c. A chain of 6 of these would leave a
super-wide wolf of 737.2 c.

Do pelog scales really tend to do this; have 6 fifths that are up to
25 c narrow and one that is up to 35 c wide (of 2:3)?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Do pelog scales really tend to do this; have 6 fifths that are up to
> 25 c narrow and one that is up to 35 c wide (of 2:3)?

OK. I checked it out myself in the Scala archives and the answer is
yes! They really do.

I said I didn't know much about pelog, but I'm blowed if I know where
I got that approx 7-tET idea.

So, although pelog is well represented as a chain of very uneven (+-25
c) generators averaging 523 c +-15 c, I'm still waiting to learn
whether -1, 3 and -4 generators traditionally represent the
consonances of the system?

Meanwhile, I'll asume this apparent 5-limit approximation is real and
will weight the gens by 1/log(max-odd-factor) and give you the list of
those with whole octave period that I consider equal or better than
this.

I wrote...

>The listener will sort out notes one way or the other, and
>far fewer of them 171 or 612. Dave's point, I think, is
>that he or she wouldn't be able to tell the difference.

For ets, it probably craps out somewhere around 282-tET,
where the 19-limit is consistently represented to within
a cent rms, and no interval has more than 2 cents absolute
error.

-Carl

I've modified my badness measure in ways that I hope take into account
the fact (assuming it is one) that pelog is some kind of 5-limit
temperament. I give the following possible ranking of 5-limit
temperaments having a whole octave period.

Gen Gens in RMS err Name
(cents) 3 5 (cents)
------------------------------------
503.8 [-1 -4] 4.2 meantone
498.3 [-1 8] 0.3 schismic
317.1 [ 6 5] 1.0 kleismic
380.0 [ 5 1] 4.6
163.0 [-3 -5] 8.0
387.8 [ 8 1] 1.1
271.6 [ 7 -3] 0.8 orwell
443.0 [ 7 9] 1.2
176.3 [ 4 9] 2.5
339.5 [-5 -13] 0.4
348.1 [ 2 8] 4.2
251.9 [-2 -8] 4.2
351.0 [ 2 1] 28.9
126.2 [-4 3] 6.0
522.9 [-1 3] 18.1 pelog?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> I've modified my badness measure in ways that I hope take into account
> the fact (assuming it is one) that pelog is some kind of 5-limit
> temperament. I give the following possible ranking of 5-limit
> temperaments having a whole octave period.

How are you ranking them, and how are you finding them?

Here is your table, with my annotations:

> Gen Gens in RMS err Name
> (cents) 3 5 (cents)
> ------------------------------------
> 503.8 [-1 -4] 4.2 meantone
> 498.3 [-1 8] 0.3 schismic
> 317.1 [ 6 5] 1.0 kleismic
> 380.0 [ 5 1] 4.6 magic
> 163.0 [-3 -5] 8.0 maxidiesic
> 387.8 [ 8 1] 1.1 wuerschmidt
> 271.6 [ 7 -3] 0.8 orwell
> 443.0 [ 7 9] 1.2 minidiesic
> 176.3 [ 4 9] 2.5 not on my list; badness=649
> 339.5 [-5 -13] 0.4 AMT
> 348.1 [ 2 8] 4.2 meantone
> 251.9 [-2 -8] 4.2 meantone
> 351.0 [ 2 1] 28.9 neutral thirds
> 126.2 [-4 3] 6.0 not on my list; badness=728
> 522.9 [-1 3] 18.1 pelogic

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > > As a 5-limit approximation the 522.86c
> > > generator is junk. It has an error of 25 c in the 2:3.
> >
> > On behalf of Herman Miller, Margo Schulter, Bill Sethares, and
the
> > entire island of Java, let me just say #(@*$& ?@#>$, and then let
me
> > just say, go play with this scale for a while. 'Junk' my &$$.
>
> Paul, I think you're severely distorting what I wrote. I didn't say
> pelog is junk. I said "as a 5-limit approximation ..."

But it's really the ratio of *3* that had the error you objected to.

> Is there really any evidence that pelog is a 5-limit temperament?

I think there's strong evidence it at least relates to the 3-limit,
and that's the error you objected to. As far as 5-limit, it's
definitely a matter of opinion, but I'm referring to Herman Miller's
use of the scale, not necessarily the traditional one. I'm also
referring to Margo and Bill's use of consonant sonorities where the
departures from 5-limit JI are even larger than this.

> > > So there are
> > > plenty of other temperaments as good as this.
> >
> > By _as good as_, I mean having an equal or lower RMS error ANS
and
> > equal or lower 'gens' measure.
>
> Why can't I use my own criteria for "as good as"?

Well, we're trying to find out if Gene is missing anything with his
methods. But if you'd like to suggest a different measure of cents
error and/or a different complexity measure, I'd hope Gene could be
accomodating . . .

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Do pelog scales really tend to do this; have 6 fifths that are up
to
> > 25 c narrow and one that is up to 35 c wide (of 2:3)?
>
> OK. I checked it out myself in the Scala archives and the answer is
> yes! They really do.

Well, this should be mentioned in our paper, I think.
>
> So, although pelog is well represented as a chain of very uneven (+-
25
> c) generators averaging 523 c +-15 c, I'm still waiting to learn
> whether -1, 3 and -4 generators traditionally represent the
> consonances of the system?

Gamelan music doesn't operate with Western notions of "consonance"
and "dissonance". There is lots of simultaneity though, so the issue
wouldn't seem to be completely irrelevant . . .

> Meanwhile, I'll asume this apparent 5-limit approximation is real
and
> will weight the gens by 1/log(max-odd-factor) and give you the list
of
> those with whole octave period that I consider equal or better than
> this.

Thanks! Hopefully, Gene can either locate all the ones you give in
his own terms, or figure out why he missed them.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> I've modified my badness measure in ways that I hope take into
account
> the fact (assuming it is one) that pelog is some kind of 5-limit
> temperament. I give the following possible ranking of 5-limit
> temperaments having a whole octave period.
>
> Gen Gens in RMS err Name
> (cents) 3 5 (cents)
> ------------------------------------
> 503.8 [-1 -4] 4.2 meantone
> 498.3 [-1 8] 0.3 schismic
> 317.1 [ 6 5] 1.0 kleismic
> 380.0 [ 5 1] 4.6
> 163.0 [-3 -5] 8.0
> 387.8 [ 8 1] 1.1
> 271.6 [ 7 -3] 0.8 orwell
> 443.0 [ 7 9] 1.2
> 176.3 [ 4 9] 2.5
> 339.5 [-5 -13] 0.4
> 348.1 [ 2 8] 4.2
> 251.9 [-2 -8] 4.2
> 351.0 [ 2 1] 28.9
> 126.2 [-4 3] 6.0
> 522.9 [-1 3] 18.1 pelog?

Dave, I was hoping that, instead of doing this, you would think in
terms of two separate badness factors, an 'error' factor and
a 'complexity' factor -- and let us know if you could find anything
that was better on _both_ factors than _any_ of the temperaments I
listed, but was not in the list anywhere . . . see?

> I've modified my badness measure in ways that I hope take into
> account the fact (assuming it is one) that pelog is some kind
> of 5-limit temperament.

Was there evidence for this, or is this just an assumption for
further exploration? It strikes me as extremely unlikely that
any Indonesian tuning is a 5-limit temperament.

-Carl

>>I've modified my badness measure in ways that I hope take into
>>account the fact (assuming it is one) that pelog is some kind
>>of 5-limit temperament.
>
>Was there evidence for this, or is this just an assumption for
>further exploration? It strikes me as extremely unlikely that
>any Indonesian tuning is a 5-limit temperament.

Posted this before I saw the bit on six narrow and one wide
fifths. But:

() The Scala scale archive is not a good source of actual pelogs,
or any other ethnic tunings for that matter.

() There may be many explanations for this pattern of fifths,
including something like Sethares' treatment... have you seen
his derivation of gamelan tunings in his book? While far from
conclusive, it's the best treatment I've seen, and the approach
strikes me as making sense... The long decay of gamelan
instruments, the style of Indonesian music, the timbre of
metalophones, and the ubiquity of the pythagorean scale suggest
some chain of fifths over which the total sensory dissonance
has been minimized. The 'experimental' way in which actual
instances of these ensembles are tuned (as opposed to fixed
tunings which are written down) fits this theory.

() In any case, because Indonesian music doesn't use 5-limit
consonances -- let alone modulate them -- I'd call it an abuse
of terminology to say they use a 5-limit temperament, even if
the data do match up. Since there is such wide variation in
Indonesian tunings, it isn't very difficult to get the data to
match, either...

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> () There may be many explanations for this pattern of fifths,
> including something like Sethares' treatment... have you seen
> his derivation of gamelan tunings in his book?

Looks totally contrived, and what about harmonic entropy?

> () In any case, because Indonesian music doesn't use 5-limit
> consonances -- let alone modulate them

It modulates plenty, as we've recently discussed on the tuning list.
And, listen to some Pelog-scale Indonesian music. It doesn't evoke 5-
limit harmony to your ears?

> I'd call it an abuse
> of terminology to say they use a 5-limit temperament, even if
> the data do match up. Since there is such wide variation in
> Indonesian tunings, it isn't very difficult to get the data to
> match, either...

At least, we can call it a "creative interpretation" of Pelog, which
Herman Miller has used effectively in his music, and the tuning of
which is by no means precluded as a "statistical center" for actual
Pelog tunings.

>>() There may be many explanations for this pattern of fifths,
>>including something like Sethares' treatment... have you seen
>>his derivation of gamelan tunings in his book?
>
>Looks totally contrived,

As opposed to what we've seen here recently?

>and what about harmonic entropy?

You'd have to plug in all the partials. The timbres are too
out there to just plug in the fundamentals as we do normally.
IOW, I'm not sure harmonic entropy is so significant for this
music.

>>() In any case, because Indonesian music doesn't use 5-limit
>>consonances -- let alone modulate them
>
>It modulates plenty, as we've recently discussed on the tuning
>list. And, listen to some Pelog-scale Indonesian music. It
>doesn't evoke 5-limit harmony to your ears?

It has in the past, but I attributed that to my cultural
conditioning.

>At least, we can call it a "creative interpretation" of Pelog,

Def. Just don't want that important qualifier to be left out.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>() There may be many explanations for this pattern of fifths,
> >>including something like Sethares' treatment... have you seen
> >>his derivation of gamelan tunings in his book?
> >
> >Looks totally contrived,
>
> As opposed to what we've seen here recently?

Huh? You have something to say? Please, this was a strange pairing of
instruments Sethares used, and all the evidence is that Indonesian
music _cultivates_ beating, rather than trying to minimize it.

> >and what about harmonic entropy?
>
> You'd have to plug in all the partials. The timbres are too
> out there to just plug in the fundamentals as we do normally.
> IOW, I'm not sure harmonic entropy is so significant for this
> music.

Try an experiment. Get three bells or gongs or whatever, as long as
they each have a clear pitch (I guess you can use a synth for this).
Tune them to a Pelog major triad. You don't hear any sense of
integrity? I sure do.

>
> >At least, we can call it a "creative interpretation" of Pelog,
>
> Def. Just don't want that important qualifier to be left out.
>
Fine. If you look at what Wilson did with Pelog, I don't think we're
crossing any lines. The creative potential of this is not to be
trifled. Listen to Blackwood's 23-tET etude, where he emulates
Indonesian music. You don't hear 5-limit harmony there? Isn't it
beautiful?

>>Looks totally contrived,
>>
>>As opposed to what we've seen here recently?
>
>Huh? You have something to say?

No way. Just asking what you thought the differences were.
Scanning the scala archive, and reporting, "they do"?

>Please, this was a strange pairing of instruments Sethares
>used,

Really? My memory is that he based the stuff on DATs he
made while visiting. I'd look it up, but I've left his
book in Montana.

>and all the evidence is that Indonesian music _cultivates_
>beating, rather than trying to minimize it.

I did not know that. They certainly get plenty of it.
But can't they minimize roughness and still have plenty of
beating? Roughness is pretty unpleasant. Beating can
be pleasant, though.

>>>and what about harmonic entropy?
>>
>>You'd have to plug in all the partials. The timbres are too
>>out there to just plug in the fundamentals as we do normally.
>>IOW, I'm not sure harmonic entropy is so significant for this
>>music.
>
>Try an experiment. Get three bells or gongs or whatever, as long
>as they each have a clear pitch (I guess you can use a synth for
>this).

I don't have a synth that does inharmonic additive sythesis.
Besides, many gamelan instruments don't evoke a clear sense
of pitch to me at all. At least, I've usually done analytical
listening (I forget Sethares' term -- where you try to listen
to the partials) when I've enjoyed Balinese/Javanese music.

>Tune them to a Pelog major triad. You don't hear any sense of
>integrity? I sure do.

What's your setup?

>Fine. If you look at what Wilson did with Pelog, I don't think
>we're crossing any lines.

Wilson strongly disclaims making any conclusions about what is
actually going on, says he's doing a creative interp, though his
papers do leave out this important disclaimer...

>The creative potential of this is not to be trifled.

No argument here!

>Listen to Blackwood's 23-tET etude, where he emulates Indonesian
>music. You don't hear 5-limit harmony there? Isn't it beautiful?

I seem to remember thinking it was triadic, and quite beautiful.
I'll listen again tonight.

-Carl

I wrote,

> crossing any lines. The creative potential of this is not to be
> trifled. Listen to Blackwood's 23-tET etude, where he emulates
> Indonesian music. You don't hear 5-limit harmony there? Isn't it
> beautiful?

To say nothing of the potential of such systems were a combination of
adaptive tempering and adaptive timbring to be used.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > I've modified my badness measure in ways that I hope take into
account
> > the fact (assuming it is one) that pelog is some kind of 5-limit
> > temperament. I give the following possible ranking of 5-limit
> > temperaments having a whole octave period.
>
> How are you ranking them,

gens * exp((err/7.4c)^0.5)

where
gens = sqrt((gens(1:3)/ln(3))^2 +
(gens(1:5)/ln(5))^2 +
(gens(3:5)/ln(5))^2)
/(1/ln(3) + 2/ln(5))

and
err = sqrt(err(1:3)^2 +
err(1:5)^2 +
err(3:5)^2)
/3

and how are you finding them?

Brute force search of all generators from 0 to 600 c in increments of
0.1 c. Three passes, limiting the max absolute number of gens for any
prime to 4, then 10, then 36 gens.

>
> Here is your table, with my annotations:
>
> > Gen Gens in RMS err Name
> > (cents) 3 5 (cents)
> > ------------------------------------
> > 503.8 [-1 -4] 4.2 meantone
> > 498.3 [-1 8] 0.3 schismic
> > 317.1 [ 6 5] 1.0 kleismic
> > 380.0 [ 5 1] 4.6 magic
> > 163.0 [-3 -5] 8.0 maxidiesic
> > 387.8 [ 8 1] 1.1 wuerschmidt
> > 271.6 [ 7 -3] 0.8 orwell
> > 443.0 [ 7 9] 1.2 minidiesic
> > 176.3 [ 4 9] 2.5 not on my list; badness=649
> > 339.5 [-5 -13] 0.4 AMT

What does that stand for?

> > 348.1 [ 2 8] 4.2 meantone
> > 251.9 [-2 -8] 4.2 meantone

Not really.

> > 351.0 [ 2 1] 28.9 neutral thirds

Simple neutral thirds? as opposed to the complex ones above?

> > 126.2 [-4 3] 6.0 not on my list; badness=728
> > 522.9 [-1 3] 18.1 pelogic

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > > 348.1 [ 2 8] 4.2 meantone
> > > 251.9 [-2 -8] 4.2 meantone
>
> Not really.

You have two meantone systems, and you can't pass from one to the other using a consonant interval. I don't want to count these, since I think they are pointless, but other people do. I'd like to hear what the point is.

> > > 351.0 [ 2 1] 28.9 neutral thirds
>
> Simple neutral thirds? as opposed to the complex ones above?

If 25/24 is a unison, then 6/5~5/4, and that is the basis of this temperament.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Paul, I think you're severely distorting what I wrote. I didn't
say
> > pelog is junk. I said "as a 5-limit approximation ..."
>
> But it's really the ratio of *3* that had the error you objected to.

Sure but ratios of 3 are part of the 5 limit and the ratios of 5 are
supposedly explaining why the ratio of 3 is so bad. It's supposedly to
get good enough ratios of 5 with only 3 and 4 gens (simpler than
meantone).

> > Is there really any evidence that pelog is a 5-limit temperament?
>
> I think there's strong evidence it at least relates to the 3-limit,
> and that's the error you objected to.

Yes but by claiming that the 523 c temperament isn't junk (as a
5-limit temperament) because it corresponds closely enough to pelog,
you are claiming something more than that.

> As far as 5-limit, it's
> definitely a matter of opinion, but I'm referring to Herman Miller's
> use of the scale, not necessarily the traditional one. I'm also
> referring to Margo and Bill's use of consonant sonorities where the
> departures from 5-limit JI are even larger than this.
>
> > > > So there are
> > > > plenty of other temperaments as good as this.
> > >
> > > By _as good as_, I mean having an equal or lower RMS error ANS
> and
> > > equal or lower 'gens' measure.
> >
> > Why can't I use my own criteria for "as good as"?
>
> Well, we're trying to find out if Gene is missing anything with his
> methods. But if you'd like to suggest a different measure of cents
> error and/or a different complexity measure, I'd hope Gene could be
> accomodating . . .

He might be missing something because of his badness measure as well
as because of his method. I'll just give my list and let others decide
whether the temperaments in it are worth examining.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Dave, I was hoping that, instead of doing this, you would think in
> terms of two separate badness factors, an 'error' factor and
> a 'complexity' factor -- and let us know if you could find anything
> that was better on _both_ factors than _any_ of the temperaments I
> listed, but was not in the list anywhere . . . see?

Ok. When I get time.

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>I've modified my badness measure in ways that I hope take into
> >>account the fact (assuming it is one) that pelog is some kind
> >>of 5-limit temperament.
> >
> >Was there evidence for this, or is this just an assumption for
> >further exploration? It strikes me as extremely unlikely that
> >any Indonesian tuning is a 5-limit temperament.
>
> Posted this before I saw the bit on six narrow and one wide
> fifths. But:
>
> () The Scala scale archive is not a good source of actual pelogs,
> or any other ethnic tunings for that matter.

Why not? What are the pelog tunings Dave used? And Dave, what are the
means and standard deviations of the two sizes of thirds?

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> I did not know that. They certainly get plenty of it.
> But can't they minimize roughness and still have plenty of
> beating? Roughness is pretty unpleasant. Beating can
> be pleasant, though.

OK, that may be part of it.

> >>You'd have to plug in all the partials. The timbres are too
> >>out there to just plug in the fundamentals as we do normally.
> >>IOW, I'm not sure harmonic entropy is so significant for this
> >>music.
> >
> >Try an experiment. Get three bells or gongs or whatever, as long
> >as they each have a clear pitch (I guess you can use a synth for
> >this).
>
> I don't have a synth that does inharmonic additive sythesis.
> Besides, many gamelan instruments don't evoke a clear sense
> of pitch to me at all.

It's a matter of _how_ clear. Typically, according to Jacky, the 2nd
and 3rd partials are about 50 cents from their harmonic series
positions. That spells increased entropy (yes, timbres have entropy),
but still within the "valley" of a particular pitch.

> >Tune them to a Pelog major triad. You don't hear any sense of
> >integrity? I sure do.
>
> What's your setup?

A cheesy Ensoniq, or listen to real Gamelan music, or the Blackwood
piece. Look, I'm not saying the tuning is _designed_ to approximate
the major triad and its intervals, but statistically (pending further
analysis) it sure seems to be playing a shaping role.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> Sure but ratios of 3 are part of the 5 limit and the ratios of 5
are
> supposedly explaining why the ratio of 3 is so bad. It's supposedly
to
> get good enough ratios of 5 with only 3 and 4 gens (simpler than
> meantone).

I wouldn't put it past the Indonesians.

>
> > > Is there really any evidence that pelog is a 5-limit
temperament?
> >
> > I think there's strong evidence it at least relates to the 3-
limit,
> > and that's the error you objected to.
>
> Yes but by claiming that the 523 c temperament isn't junk (as a
> 5-limit temperament) because it corresponds closely enough to
pelog,
> you are claiming something more than that.

That's not the only reason I was claiming that. Note that I
referenced Margo Schulter for example. These 'errors' are well within
the acceptable norm under quite a few interesting circumstances --
the only time they really get in the way is with sustained harmonic
timbres in the absense of adaptive tuning. Try it!

>>>Try an experiment. Get three bells or gongs or whatever, as long
>>>as they each have a clear pitch (I guess you can use a synth for
>>>this).
>>
>>I don't have a synth that does inharmonic additive sythesis.
>>Besides, many gamelan instruments don't evoke a clear sense
>>of pitch to me at all.
>
>It's a matter of _how_ clear. Typically, according to Jacky, the
>2nd and 3rd partials are about 50 cents from their harmonic
>series positions. That spells increased entropy (yes, timbres
>have entropy),

Which is why I suggested that the plug-in-the-fundamentals-only
shortcut shouldn't be applied.

The variety of instruments in the gamelan is huge. I find that
the main melodic insts. have a fairly clear sense of timbre at
the attack (which outlines the melody and rhythm of the music),
but often three or four distinct pitches over the rest of the
envelope (which constitutes the harmony of the music, and thus
my view of Sethares' treatment).

>but still within the "valley" of a particular pitch.

Have you ever tried a bell timbre? Wonder what the curve
looks like...

>A cheesy Ensoniq, or listen to real Gamelan music, or the Blackwood
>piece. Look, I'm not saying the tuning is _designed_ to approximate
>the major triad and its intervals, but statistically (pending
>further analysis) it sure seems to be playing a shaping role.

I've got a share of gamelan music, thanks to Kraig Grady's
suggestions. I'm listening to the Blackwood now. The Blackwood
sounds more triadic than the gamelan music.

-Carl

>> () The Scala scale archive is not a good source of actual pelogs,
>> or any other ethnic tunings for that matter.
>
>Why not?

Have you ever looked at it? It's basically everything that ever
entered anybody's fancy. There are scales in there named after
me I don't even remember making up.

I admit I've never looked at the pelogs. But I'll bet eggs
benedict there are some of Wilson's in there! And I have looked
very closely indeed at the bagpipe and mbira tunings. Just
whatever anybody was dreaming, in ratios -- exactly the kind of
thing you've pined against so often.

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > > > 348.1 [ 2 8] 4.2 meantone
> > > > 251.9 [-2 -8] 4.2 meantone
> >
> > Not really.
>
> You have two meantone systems, and you can't pass from one to the
> other using a consonant interval.

I don't understand what you mean here, or the point you are making.

> I don't want to count these, since I
think they are pointless, but other people do. I'd like to hear what
the point is.
>

They are different melodically from meantone chains and they are
better than some other temperaments that you are including.

They form different MOS scales. Here are denominators of convergents
and (semiconvergents).

503.8 c 5 7 12 19 31 50 81
348.1 c 7 (10 17 24) 31 (38 69 100)
251.9 c 5 (9 14) 19 (24 43 62) 81 (100

> > > > 351.0 [ 2 1] 28.9 neutral thirds
> >
> > Simple neutral thirds? as opposed to the complex ones above?
>
> If 25/24 is a unison, then 6/5~5/4, and that is the basis of this
temperament.

Sure. But if we call this temperament "neutral thirds temperament"
without qualification, this conflicts with the usage in Graham's
catalog and the other neutral thirds temperament above.

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >> () The Scala scale archive is not a good source of actual pelogs,
> >> or any other ethnic tunings for that matter.
> >
> >Why not?
>
> Have you ever looked at it? It's basically everything that ever
> entered anybody's fancy. There are scales in there named after
> me I don't even remember making up.
>
> I admit I've never looked at the pelogs. But I'll bet eggs
> benedict there are some of Wilson's in there!

Did you use any of those, Dave? I would assume those would be
specified by ratios. I assumed Dave used some "observed" examples.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "clumma" <carl@l...> wrote:
> > () The Scala scale archive is not a good source of actual pelogs,
> > or any other ethnic tunings for that matter.
>
> Why not? What are the pelog tunings Dave used?

All the 7-note ones with names pelog*.scl. There are 10 or so. Only a
few didn't fit the pattern and they looked like "theoretical" ones.

> And Dave, what are the
> means and standard deviations of the two sizes of thirds?

No time to investigate this now, sorry. But it certainly should be
looked at.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > >
> > > Simple neutral thirds? as opposed to the complex ones above?
> >
> > If 25/24 is a unison, then 6/5~5/4, and that is the basis of this
> temperament.
>
> Sure. But if we call this temperament "neutral thirds temperament"
> without qualification, this conflicts with the usage in Graham's
> catalog

Really? Why so? He also uses a generator of a neutral third in
his "neutral thirds temperament".

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>>Try an experiment. Get three bells or gongs or whatever, as long
> >>>as they each have a clear pitch (I guess you can use a synth for
> >>>this).
> >>
> >>I don't have a synth that does inharmonic additive sythesis.
> >>Besides, many gamelan instruments don't evoke a clear sense
> >>of pitch to me at all.
> >
> >It's a matter of _how_ clear. Typically, according to Jacky, the
> >2nd and 3rd partials are about 50 cents from their harmonic
> >series positions. That spells increased entropy (yes, timbres
> >have entropy),
>
> Which is why I suggested that the plug-in-the-fundamentals-only
> shortcut shouldn't be applied.
>
Why not? Afraid of a little assymetry?
>
> >but still within the "valley" of a particular pitch.
>
> Have you ever tried a bell timbre? Wonder what the curve
> looks like...

That's an instrument where the hear fundamental is not even in the
spectrum . . . the Gamelan instruments are a bit different.

By the way Carl, have you tried any actual _listening experiments_
yet?

> >A cheesy Ensoniq, or listen to real Gamelan music, or the
Blackwood
> >piece. Look, I'm not saying the tuning is _designed_ to
approximate
> >the major triad and its intervals, but statistically (pending
> >further analysis) it sure seems to be playing a shaping role.
>
> I've got a share of gamelan music, thanks to Kraig Grady's
> suggestions. I'm listening to the Blackwood now. The Blackwood
> sounds more triadic than the gamelan music.
>
> -Carl

The gamelan scales sound like they contain a rough major triad and a
rough minor triad, forming a very rough major seventh chord together,
plus one extra note -- don't they?

>>>It's a matter of _how_ clear. Typically, according to Jacky, the
>>>2nd and 3rd partials are about 50 cents from their harmonic
>>>series positions. That spells increased entropy (yes, timbres
>>>have entropy),
>>
>> Which is why I suggested that the plug-in-the-fundamentals-only
>> shortcut shouldn't be applied.
>
>Why not? Afraid of a little assymetry?

Only when it's spelled like that. ;)

Not sure what you mean. The reason I suggested the shortcut not
be applied for inharmonic timbres is because... it is a shortcut.
Which assumes you have clearly resolved fundamentals. No?

>By the way Carl, have you tried any actual _listening
>experiments_ yet?

You mean with a synthesizer? As I explained, I don't have the
right gear -- I've got an additive synth that's stuck in JI.

What do you have in mind? I'm not clear how one would go about
testing anything that's been said here.

>The gamelan scales sound like they contain a rough major
>triad and a rough minor triad, forming a very rough major
>seventh chord together, plus one extra note -- don't they?

Yes, to me, pelog sounds like a I and a III with a 4th in the
middle. But the music seems to use a fixed tonic, with not
much in the way of triadic structure. Okay, let's take a
journey...

"Instrumental music of Northeast Thailand"

Characteristic stop rhythm. Harmonium and marimba-sounding
things play major pentatonic on C# (A=440) or relative minor
on A#. Scale is treated like a chord, not melodically -- tone
cluster on harmonium for drone, melody is essentially a scale
'arpeggio' figure centered on notes of the scale (usually three
notes are used as centers of this pattern, sometimes they form
a 1st-inversion minor triad).

"JAVA Tembang Sunda" (Inedit)

This is unlike the gamelan music I've heard (it's a plucked
string ensemble with vocalists and flute). Jeez, I forgot I
had this CD! There _is_ I -> III, and even I -> IV motion
here.

"Gamelan Semar Pagulingan from Besang-Ababi/Karangasem
Music from Bali"

I suppose there is some argument for triadic structure here
too, but if I hadn't heard the last disc beforehand, I'd
say they were just doing the 'start the figure on different
scale members' thing, as in the first disc. I don't know
Paul, this is not life as we know it (or hear it). I still
say there's nothing here that would turn up an optimized
5-limit temperament! This music bores the hell out of me
when I'm not going for the glassy partial soup that I love
so much.

"The Gamelan of Cirebon"

There's less triadic inuendo here. Their low 'phones have
a more resolved timbre than the Balinese, above, ensemble
had. Also, I haven't heard pelog on this disc. Seems to be
sorog (one chinese, one pelog tetrachord).

I guess it all depends if you consider these tonic changes
or just points of symmetry in a melisma (sp?). If there's
anything that would produce an optimum 5-limit temperament
in there either way, I'll eat my shoe.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> Have you ever looked at it? It's basically everything that ever
> entered anybody's fancy. There are scales in there named after
> me I don't even remember making up.

Not really--I don't think any of my scales are in there.

Gene:
> > > If 25/24 is a unison, then 6/5~5/4, and that is the basis of this
> > temperament.

Dave:
> > Sure. But if we call this temperament "neutral thirds temperament"
> > without qualification, this conflicts with the usage in Graham's
> > catalog

Paul:
> Really? Why so? He also uses a generator of a neutral third in
> his "neutral thirds temperament".

Yes, it looks okay to me. To pick through my description

"Two neutral thirds make up a perfect fifth."

That's true for 6:5=~5:4

"The neutral thirds can be called 11:9."

But don't have to be.

"The `wolf third' to make up a 7-note scale
can be identified with 6:5 or 7:6 or neither."

And in this case neither.

I'd call the temperament with the wolf third as 6:5 the "typical neutral
third temperament" or the "meantone-like neutral third temperament" but
that isn't the only way of doing it. Taking meantone and dividing the
fifths in two, but not calling the result an 11:9, would also be a neutral
third temperament. I'm not sure if this is what some of these examples
are intending.

I should really update that description to specify that the scale be an
MOS generated by the neutral third, or some permutation (like rast).

Graham

--- In tuning-math@y..., graham@m... wrote:
> Paul:
> > Really? Why so? He also uses a generator of a neutral third in
> > his "neutral thirds temperament".
>
> Yes, it looks okay to me. To pick through my description
...

I was referring to your catalog page which I thought did not allow for
5 to map to 1 gen. In any case, when we're talking 5-limit
temperaments I think we should have different names or qualifiers for
the different mappings of the prime 5 when the prime 3 maps to 2 gens.
I suggested "simple neutral thirds temperament" for the [2 1] mapping
and "complex neutral thirds temperament" for the [2 8] mapping. What
are the useful mappings when we go to 7-limit? It would be useful to
have all these mappings listed on your catalog page, Graham.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Dave, I'll do the arithmetic for you and give you the RMS optima for
> the ones with RMS less than 20 cents and g<8. Is there anything
> _missing_ that is as good as one of these?
>
> Generator 522.86¢, Period 1 oct.
> Generator 505.87¢, Period 1/4 oct.
> Generator 163.00¢, Period 1 oct.
> Generator 491.20¢, Period 1/3 oct.
> Generator 379.97¢, Period 1 oct.
> Generator 503.83¢, Period 1 oct.
> Generator 494.55¢, Period 1/2 oct.
> Generator 442.98¢, Period 1 oct.
> Generator 387.82¢, Period 1 oct.
> Generator 271.59¢, Period 1 oct.
> Generator 317.08¢, Period 1 oct.
> Generator 498.27¢, Period 1 oct.
>
> Did I miss any, Gene?
> Thanks for checking, Dave.

Ok Paul, here are all those I have found with a whole octave period
where the rms error is no worse than the worst of these [which is
pelogic (18.2 c)] and the log-odd-limit-weighted rms gens is no worse
than the worst of these [which is orwell (6.3 gens)]. They are listed
in order of generator size.

Gen Gens per
(cents) 3 5
-----------------
78.0 [ 9 5]
81.5 [-6 10]
98.3 [-5 4]
102.0 [-5 -8]
126.2 [-4 3]
137.7 [ 5 -6]
144.5 [ 5 11]
163.0 [-3 -5]
176.3 [ 4 9]
226.3 [ 3 7]
251.9 [-2 -8]
271.6 [ 7 -3]
317.1 [ 6 5]
336.9 [-5 -6]
348.1 [ 2 8]
356.3 [ 2 -9]
380.0 [ 5 1]
387.8 [ 8 1]
414.5 [-7 -2]
443.0 [ 7 9]
471.2 [ 4 11]
490.0 [-1 -9]
498.3 [-1 8]
503.8 [-1 -4]
518.5 [ 6 10]
522.9 [-1 3]
561.0 [-3 -10]
568.6 [-3 7]

Note that this includes those I gave in the earlier list based on my
own badness measure, except for 339.5c [-5 -13] and 351.0c [2 1]. I
still think that earlier list is more relevant.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

There were the usual repetions (meantone, 1/2 fifth meantone, 1/2 fourth meantone, etc) as well as a lot of systems which I ranked pretty low on this list. The worst badness measure belonged to this one:

> 144.5 [ 5 11]

Comma: 200000/177147

Map:

[ 0 1]
[ 5 1]
[11 1]

Generators: a = 3.0066/25 = 144.315 cents; b = 1

badness: 7358
rms: 15.57
g: 7.89
errors: [19.62, 1.15, -18.48]

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> There were the usual repetions (meantone, 1/2 fifth meantone, 1/2
fourth meantone, etc)

To a mathematician focussing on approximation of ratios for harmony
these may be repetitions, but to a musician they are quite distinct
and it is quite wrong to call them "meantones". But it is important to
point out their relationship to meantone.

>as well as a lot of systems which I ranked
pretty low on this list.

Me too.

>The worst badness measure belonged to this
one:
>
> > 144.5 [ 5 11]
>
> Comma: 200000/177147
>
> Map:
>
> [ 0 1]
> [ 5 1]
> [11 1]
>
> Generators: a = 3.0066/25 = 144.315 cents; b = 1
>
> badness: 7358
> rms: 15.57
> g: 7.89
> errors: [19.62, 1.15, -18.48]

Clearly junk.

But what about those that were on my earlier list (as better than
pelogic), but not on yours. Have you figured out why that is?

I'm preparing to go away with my family in a few days for two weeks on
a coral island, so it doesn't look like I'm going to get to check
those 1/2 octave and 1/3 octave temperaments. Sorry Paul.

By the way, I had some misplaced parentheses in my formulae for rms
error and log-odd-limit-weighted rms gens. The square root operation
should of course be performed last, i.e. _after_ diving by the sum of
the weights.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > There were the usual repetions (meantone, 1/2 fifth meantone, 1/2
> fourth meantone, etc)
>
> To a mathematician focussing on approximation of ratios for harmony
> these may be repetitions, but to a musician they are quite distinct
> and it is quite wrong to call them "meantones". But it is important to
> point out their relationship to meantone.

You have an even and odd set of pitches, meaning an even or odd number of generators to the pitch. You can't get from even to odd by way of consonant 7-limit intervals, so basically we have two unrelated meantone systems a half-fifth or half-fourth apart. You can always glue together two unrelated systems and call it a temperament, and this differs only because it does have a single generator.

> But what about those that were on my earlier list (as better than
> pelogic), but not on yours. Have you figured out why that is?

They weren't junk, but they were below my cutoff; if I raised it from badness 500 to badness 1000 they would have been on it. Should they be?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > > There were the usual repetions (meantone, 1/2 fifth meantone,
1/2
> > fourth meantone, etc)
> >
> > To a mathematician focussing on approximation of ratios for
harmony
> > these may be repetitions, but to a musician they are quite
distinct
> > and it is quite wrong to call them "meantones". But it is
important to
> > point out their relationship to meantone.
>
> You have an even and odd set of pitches, meaning an even or odd
number of generators to the pitch.

You mean even or odd number of generators to the intervals between the
pitches. "Generators making up a pitch" doesn't make sense to me.
Which reminds me: I think it would be a help to readers of your posts
if you adopted the long-standing convention on this list of giving
intervals as m:n (or n:m) and pitches as n/m, and when referring to a
rational part of an octave, writing "n/m oct".

> You can't get from even to odd by
way of consonant 7-limit intervals, so basically we have two unrelated
meantone systems a half-fifth or half-fourth apart. You can always
glue together two unrelated systems and call it a temperament, and
this differs only because it does have a single generator.
>

I see your point now, and it's a very good one. However, they _are_
linear temperaments by all the definitions I am aware of, and they
_are_ very different from meantone melodically, and despite the
doubling of the gens measure relative to meantone they _are_ better
than some others on your list (at least according to me).

> > But what about those that were on my earlier list (as better than
> > pelogic), but not on yours. Have you figured out why that is?
>
> They weren't junk, but they were below my cutoff; if I raised it
from badness 500 to badness 1000 they would have been on it. Should
they be?
>

Ask a musician, e.g Paul. I don't think I've ever seen them before. I
wouldn't miss them. But I do think they look better than pelogic. If
you raised your badness cutoff to 1000 you'd probably end up including
a lot more that I'd consider junk, either because of too many gens or
too big errors.

genewardsmith@juno.com (genewardsmith) wrote:

> You have an even and odd set of pitches, meaning an even or odd number
> of generators to the pitch. You can't get from even to odd by way of
> consonant 7-limit intervals, so basically we have two unrelated
> meantone systems a half-fifth or half-fourth apart. You can always glue
> together two unrelated systems and call it a temperament, and this
> differs only because it does have a single generator.

These are the [2 8] systems. There is some ambiguity, but if you mean the
half-fifth system, isn't that Vicentino's enharmonic? That's 31&24 or
[(1, 0), (1, 2), (0, 8)]. Two meantone scales, only 5-limit consonances
recognize, but neutral intervals used in melody. It may not be a
temperament, but does have a history of both theory and music, so don't
write it off so lightly.

The half-fifth system is 24&19 or [(1, 0), (2, -2), (4, -8)]. There's
also a half-octave system, [(2, 0), (3, 1), (4, 4)]. That's the one my
program would deduce from the octave-equivalent mapping [2 8]. If I had
such a program. If anybody cares, is it possible to write one? Where
torsion's present, we'll have to assume it means divisions of the octave
for uniqueness. Gene said it isn't possible, but I'm not convinced. How
could [1 4] be anything sensible but meantone?

Perhaps the first step is to find an interval that's only one generator
step, take the just value, period-reduce it and work everything else out
from that. But there may be some cases where the optimal value should
cross a period boundary.

But if we could get the periodicity block in pitch-order, we could
reconstruct an equal-tempered mapping and get all the information the
wedge product gives us. Can we do that? Anybody?

If you think it can't be done, show a counter-example: an
octave-equivalent mapping without torsion that can lead to two different
but equally good temperaments.

Graham

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

There is some ambiguity, but if you mean the
> half-fifth system, isn't that Vicentino's enharmonic?

I thought Vicentino was 31-et.

>Gene said it isn't possible, but I'm not convinced. How
> could [1 4] be anything sensible but meantone?

Actually, I said normally there will be a clear best choice.

Me:
> >There is some ambiguity, but if you mean the
> > half-fifth system, isn't that Vicentino's enharmonic?

Gene:
> I thought Vicentino was 31-et.

He never actually says it's equally tempered. Only that the chromatic
semitone is divided into 2 dieses, which follows from the perfect fifth being
divided into two equal neutral thirds. Although he does say the usual diesis
(the difference between a diatonic and chromatic semitone) can be treated
equivalent to the other one, the tuning seems to be two meantone chains,
corresponding to the two keyboards.

The musical examples can all be understood as two meantone chains. He does
obscure this by writing a Gb as F#, but each chord falls entirely on one
keyboard. And they're all normal meantone chords. So the music is fully
described by the meantone-with-neutral-thirds temperament. Although he
mentions, briefly, that he considers neutral thirds as consonant and they may
even be sung in contemporaneous music, he doesn't use them himself in chords.
And he doesn't quite give the 11-limit interpretation.

Graham

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:

Although he
> mentions, briefly, that he considers neutral thirds as consonant and they may
> even be sung in contemporaneous music, he doesn't use them himself in chords.
> And he doesn't quite give the 11-limit interpretation.

If neutral thirds are consonant we are not talking about the 5-limit and the entire argument is moot.

In-Reply-To: <a0tbta+beck@eGroups.com>
Me:
> Although he
> > mentions, briefly, that he considers neutral thirds as consonant and
> > they may even be sung in contemporaneous music, he doesn't use them
> > himself in chords. And he doesn't quite give the 11-limit
> > interpretation.

Gene:
> If neutral thirds are consonant we are not talking about the 5-limit
> and the entire argument is moot.

Neutral thirds are not consonant in Vicentino's enharmonic genus. If you
wish to disagree, give specific references to the musical examples.

On reviewing this, I think I should draw a distinction between the
enharmonic system and the tuning of the archicembalo. The former has a
minor diesis equal to half a chromatic semitone, and a major diesis equal
to a chromatic semitone less a minor diesis. In the latter, the minor
diesis is equal to the difference between a diatonic and chromatic
semitone, and the major diesis is equal to the chromatic semitone.
Vicentino starts off noting this difference, but doesn't always make it
strict in the notation. The reference to neutral thirds being consonant
is in the book on the archicembalo. The books on the diatonic, chromatic
and enharmonic genera only recognize strict 5-limit vertical harmony.

Also, although he does mention somewhere that the whole tone divides into
five roughly equal parts, in the examples of the enharmonic genus he only
divides it into four. So the system, but not always the notation, is
fully consistent with a quartertone scale. Hence 24&31. In Book I of
Music Practice, he's strict about this in the divisions of the whole tone
and examples of the different dieses, but not when he introduces some of
the derived intervals. It's here he says that the enharmonic dieses are
"identical" to the extended meantone intervals on the archicembalo, and
the one can stand in for the other for the sake of "compositional
convenience". In Book III of Music Practice, he spells one of the
enharmonic tetrachords such that the notation won't work in 24-equal, so
must be ignoring the distinctions he made in Book I.

Disclaimer: I'm writing this without the book to hand, but I did check the
details last night.

Graham

In-Reply-To: <memo.582096@cix.compulink.co.uk>
Proof reading time

> On reviewing this, I think I should draw a distinction between the
> enharmonic system and the tuning of the archicembalo. The former has a
> minor diesis equal to half a chromatic semitone, and a major diesis
> equal to a chromatic semitone less a minor diesis. In the latter, the
. ^^^^^^^^^

That should be a *diatonic* semitone less a minor diesis.

Graham

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> Not sure what you mean. The reason I suggested the shortcut not
> be applied for inharmonic timbres is because... it is a shortcut.
> Which assumes you have clearly resolved fundamentals. No?

No. It just assumes that the overtones are pretty close to harmonic,
because they will then lead to the same ratio-intepretations for the
fundamentals as the fundamentals by themselves. If they're 50 cents
from harmonic, they will lead to a larger s value for the resulting
harmonic entropy curve, but that's about it.

> >By the way Carl, have you tried any actual _listening
> >experiments_ yet?
>
> You mean with a synthesizer? As I explained, I don't have the
> right gear -- I've got an additive synth that's stuck in JI.

You can synthesize inharmonic sounds, yes? You can use a high-limit
JI scale that sounds like a pelog scale, yes?

> What do you have in mind? I'm not clear how one would go about
> testing anything that's been said here.

Well, what I'm saying seems most clear and powerful to me as a
musician actually playing this stuff.
>
> >The gamelan scales sound like they contain a rough major
> >triad and a rough minor triad, forming a very rough major
> >seventh chord together, plus one extra note -- don't they?
>
> Yes, to me, pelog sounds like a I and a III with a 4th in the
> middle. But the music seems to use a fixed tonic, with not
> much in the way of triadic structure.

How about 5-limit intervals?

> Okay, let's take a
> journey...
>
> "Instrumental music of Northeast Thailand"
>
> Characteristic stop rhythm. Harmonium and marimba-sounding
> things play major pentatonic on C# (A=440) or relative minor
> on A#.

This is clearly not a pelog tuning!

> "JAVA Tembang Sunda" (Inedit)
>
> This is unlike the gamelan music I've heard (it's a plucked
> string ensemble with vocalists and flute). Jeez, I forgot I
> had this CD! There _is_ I -> III, and even I -> IV motion
> here.
>
> "Gamelan Semar Pagulingan from Besang-Ababi/Karangasem
> Music from Bali"
>
> I suppose there is some argument for triadic structure here
> too, but if I hadn't heard the last disc beforehand, I'd
> say they were just doing the 'start the figure on different
> scale members' thing, as in the first disc. I don't know
> Paul, this is not life as we know it (or hear it).

What on earth does that mean?

> I still
> say there's nothing here that would turn up an optimized
> 5-limit temperament!

Forget the optimization. All you need is the mapping -- that chains
of three fifths make a major third and that chains of four fifths
make a minor third. This seems to be a definite characteristic of
pelog! Just as much as the "opposite" is a characteristic of Western
music, regardless of whether strict JI, optimized meantone, 12-tET,
or whatever is used.
>
> I guess it all depends if you consider these tonic changes
> or just points of symmetry in a melisma (sp?).

Why does that matter?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > There were the usual repetions (meantone, 1/2 fifth meantone, 1/2
> fourth meantone, etc)
>
> To a mathematician focussing on approximation of ratios for harmony
> these may be repetitions, but to a musician they are quite distinct
> and it is quite wrong to call them "meantones". But it is important
to
> point out their relationship to meantone.

Hello folks. These systems can be derived from a combined-ET
viewpoint but cannot be derived from a unison vector viewpoint. This
is very reminiscent of torsional blocks, which can be derived from a
unison vector viewpoint but not from a combined-ET viewpoint. So is
this, mathematically, the "dual of torsion"? Can we deal with
torsion, as well as "contorsion" or whatever we call this, in the
beginning of our paper, and leave out the specific examples, as they
follow a fairly obvious pattern??

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Ask a musician, e.g Paul. I don't think I've ever seen them before.
>I
> wouldn't miss them. But I do think they look better than pelogic.

As a musician, I have to say pelogic is one of my favorites. Perhaps
I'm really advocating a gens^4 (weighted, of course) times mistuning
badness measure here . . . but I'm happy as long as pelogic is in
there. Try it with a marimba or other appropriate, inharmonic sound --
it instantly transports you to Bali -- big time! My keyboardist
friend was improvising on a 12-tone mapping of this generator -- now
that was some awesome music!

--- In tuning-math@y..., graham@m... wrote:
> genewardsmith@j... (genewardsmith) wrote:
>
> > You have an even and odd set of pitches, meaning an even or odd
number
> > of generators to the pitch. You can't get from even to odd by way
of
> > consonant 7-limit intervals, so basically we have two unrelated
> > meantone systems a half-fifth or half-fourth apart. You can
always glue
> > together two unrelated systems and call it a temperament, and
this
> > differs only because it does have a single generator.
>
> These are the [2 8] systems.

Not really. It's similar to torsion, but not quite the same.

> There is some ambiguity, but if you mean the
> half-fifth system, isn't that Vicentino's enharmonic? That's 31&24
or
> [(1, 0), (1, 2), (0, 8)]. Two meantone scales, only 5-limit
consonances
> recognize, but neutral intervals used in melody. It may not be a
> temperament, but does have a history of both theory and music, so
don't
> write it off so lightly.

I doubt this reflects Vicentino's practice well at all. For instance,
he didn't base any consonant harmonies on the second meantone scale,
did he?

> The half-fifth system is 24&19 or [(1, 0), (2, -2), (4, -8)].

You mean half-fourth system?

> There's
> also a half-octave system, [(2, 0), (3, 1), (4, 4)]. That's the
one my
> program would deduce from the octave-equivalent mapping [2 8].

From that unison vector? If so, I think you're confusion torsion
with "contorsion".

> If I had
> such a program. If anybody cares, is it possible to write one?
Where
> torsion's present, we'll have to assume it means divisions of the
octave
> for uniqueness.

Huh? Clearly this doesn't work in the Monz sruti 24 case.

> Gene said it isn't possible, but I'm not convinced. How
> could [1 4] be anything sensible but meantone?

Not sure what the connection is.

> Perhaps the first step is to find an interval that's only one
generator
> step, take the just value, period-reduce it and work everything
else out
> from that.

If the half-fifth is the generator, what's the just value?

> But there may be some cases where the optimal value should
> cross a period boundary.

??

> If you think it can't be done, show a counter-example: an
> octave-equivalent mapping without torsion that can lead to two
different
> but equally good temperaments.

Equally good? Under what criteria? Look, why do we care about the
octave-equivalent mapping? Certainly we can't object to asking the
mapping to be octave-specific, can we?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
>
> > There is some ambiguity, but if you mean the
> > half-fifth system, isn't that Vicentino's enharmonic?
>
> I thought Vicentino was 31-et.

Vicentino did a lot of things both inside and outside 31-tET.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., Graham Breed <graham@m...> wrote:
>
> > Although he
> > mentions, briefly, that he considers neutral thirds as consonant
and they may
> > even be sung in contemporaneous music, he doesn't use them
himself in chords.
> > And he doesn't quite give the 11-limit interpretation.
>
> If neutral thirds are consonant we are not talking about the 5->
>limit and the entire argument is moot.

They're not considered consonant on the level of the 5-limit
consonances. And anyway, the music makes the argument not moot, if
Graham's interpretation is reasonable. How far out in the chain of
generators do you have to go to account for V's simplest example
of "enharmonic genus" music?

>No. It just assumes that the overtones are pretty close to harmonic,
>because they will then lead to the same ratio-intepretations for the
>fundamentals as the fundamentals by themselves. If they're 50 cents
>from harmonic, they will lead to a larger s value for the resulting
>harmonic entropy curve, but that's about it.

s represents the blur of the spectral components coming in. How
could an inharmonic timbre change that?

>You can synthesize inharmonic sounds, yes?

No, that's the problem.

>You can use a high-limit JI scale that sounds like a pelog scale,
>yes?

The scale isn't stuck in JI, the timbre is.

>>>The gamelan scales sound like they contain a rough major
>>>triad and a rough minor triad, forming a very rough major
>>>seventh chord together, plus one extra note -- don't they?
>>
>>Yes, to me, pelog sounds like a I and a III with a 4th in the
>>middle. But the music seems to use a fixed tonic, with not
>>much in the way of triadic structure.
>
> How about 5-limit intervals?

Not sure what you're asking.

>> Okay, let's take a
>> journey...
>>
>> "Instrumental music of Northeast Thailand"
>>
>> Characteristic stop rhythm. Harmonium and marimba-sounding
>> things play major pentatonic on C# (A=440) or relative minor
>> on A#.
>
>This is clearly not a pelog tuning!

Right, it's the chinese pentatonic. I threw it in for
completeness.

>>I suppose there is some argument for triadic structure here
>>too, but if I hadn't heard the last disc beforehand, I'd
>>say they were just doing the 'start the figure on different
>>scale members' thing, as in the first disc. I don't know
>>Paul, this is not life as we know it (or hear it).
>
>What on earth does that mean?

It's easy for me to hear triadic structure. I'll bend over
backwards to do it. It's easy for me to hear pelog as a
subset of the diatonic scale, too. Indonesians might hear
it differently.

>>I still say there's nothing here that would turn up an optimized
>>5-limit temperament!
>
>Forget the optimization. All you need is the mapping -- that
>chains of three fifths make a major third and that chains of
>four fifths make a minor third. This seems to be a definite
>characteristic of pelog! Just as much as the "opposite" is a
>characteristic of Western music, regardless of whether strict
>JI, optimized meantone, 12-tET, or whatever is used.

Western music uses progressions of four fifths and expects to
wind up on a major third. I didn't notice anything like this
for the [1 -3] map (right?) on the cited discs.

>>I guess it all depends if you consider these tonic changes
>>or just points of symmetry in a melisma (sp?).
>
>Why does that matter?

One's a harmonic device, the other melodic. All other things
being equal, it wouldn't matter. But I think a lot of the
other stuff that goes along with harmonic music is missing
from this music. Western music requires meantone. The pelog
5-limit map is far more extreme, but what suffers in this
music as we change the tuning from 5-of- 7, to 23, to 16, all
the way to strict JI? I think the tuning on these discs is
closer to JI than 23-tET, and I don't hear them avoiding a
disjoint interval. Do you?

Incidentally, I think Wilson agrees with your point of view
here. While he does caution against eager interps. of his
ethno music theory, I think he thinks that harmonic mapping
is inevitable, and atomic in music. I'm not sure I agree.
Not sure I disagree.

-Carl

In-Reply-To: <a13gbn+gl7s@eGroups.com>
Me:
> > There is some ambiguity, but if you mean the
> > half-fifth system, isn't that Vicentino's enharmonic? That's 31&24
> or
> > [(1, 0), (1, 2), (0, 8)]. Two meantone scales, only 5-limit
> consonances
> > recognize, but neutral intervals used in melody. It may not be a
> > temperament, but does have a history of both theory and music, so
> don't
> > write it off so lightly.

Paul:
> I doubt this reflects Vicentino's practice well at all. For instance,
> he didn't base any consonant harmonies on the second meantone scale,
> did he?

How do you mean? The two meantones fit snugly on the two different
keyboards, and chords in the enharmonic genus typically alternate between
them. As most chords are consonances, there's no other way of getting the
enharmonic melodies right. For you to ask this question suggests either I
didn't understand you, or you don't have a copy of Vicentino's book. It
is worth reading. I thought you had it because you recommended it to
somebody else.

> > The half-fifth system is 24&19 or [(1, 0), (2, -2), (4, -8)].
>
> You mean half-fourth system?

Looks like it.

Me:
> > There's
> > also a half-octave system, [(2, 0), (3, 1), (4, 4)]. That's the
> one my
> > program would deduce from the octave-equivalent mapping [2 8].

Paul:
> >From that unison vector? If so, I think you're confusion torsion
> with "contorsion".

This has nothing directly to do with unison vectors.

Me:
> > If I had
> > such a program. If anybody cares, is it possible to write one?
> Where
> > torsion's present, we'll have to assume it means divisions of the
> octave
> > for uniqueness.

Paul:
> Huh? Clearly this doesn't work in the Monz sruti 24 case.

No, that can't be expressed in this particular octave equivalent system.
It may be possible to include it later, but let's deal with the simple
cases first.

> > Gene said it isn't possible, but I'm not convinced. How
> > could [1 4] be anything sensible but meantone?
>
> Not sure what the connection is.

[1 4] is a definition of meantone: 4 fifths are equivalent to a major
third. Is that a unique definition, or do we have to add "plus two
octaves"?

> > Perhaps the first step is to find an interval that's only one
> generator
> > step, take the just value, period-reduce it and work everything
> else out
> > from that.
>
> If the half-fifth is the generator, what's the just value?

Well, it could be either 5:4 or 6:5. Or 11:9 or 27:22. Or 49:40 or
60:49. But if you mean the case where all consonances are specified in
terms of fifths, but the generator is a half-fifth, I thought I defined
those out of existence above. If not, you can take the square root.

Me:
> > But there may be some cases where the optimal value should
> > cross a period boundary.

Paul:
> ??

Say you have a system that divides the octave into two equal parts, and
7:5 is a single generator steps. It may happen that 7:5 approximates best
to be larger than a half octave, so taking its just value for calculating
the mapping will get the wrong results. This may be a real problem when
the octave is divided into 41 equal parts, like one of the higher-limit
temperaments I came up with, and the generator is a fairly complex
interval.

Me:
> > If you think it can't be done, show a counter-example: an
> > octave-equivalent mapping without torsion that can lead to two
> different
> > but equally good temperaments.

Paul:
> Equally good? Under what criteria? Look, why do we care about the
> octave-equivalent mapping? Certainly we can't object to asking the
> mapping to be octave-specific, can we?

It should be fairly obvious if you get the mapping right because the
errors will be small. If you can find an example that depends on the
choice of reasonable criteria, that'll do as a counterexample.

You were the one originally pushing for octave-equivalent calculations.
If you aren't bothered any more, I'm not; I was only trying to answer your
questions. But it would be elegant to describe systems in the simplest
possible way, and one consistent with Fokker. It's up to you if you don't
want the paper to cover that.

Graham

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >No. It just assumes that the overtones are pretty close to
harmonic,
> >because they will then lead to the same ratio-intepretations for
the
> >fundamentals as the fundamentals by themselves. If they're 50 cents
> >from harmonic, they will lead to a larger s value for the resulting
> >harmonic entropy curve, but that's about it.
>
> s represents the blur of the spectral components coming in. How
> could an inharmonic timbre change that?

When we're dealing with a dyad consisting of complex tones, and
trying to apply harmonic entropy to that dyad, s is decreased below
the value that sine waves in place of the complex tones would imply.
The more inharmonic the timbre, the less s is decreased below the
sine-wave case.

>
> >You can synthesize inharmonic sounds, yes?
>
> No, that's the problem.

Oops.

> >>Yes, to me, pelog sounds like a I and a III with a 4th in the
> >>middle. But the music seems to use a fixed tonic, with not
> >>much in the way of triadic structure.
> >
> > How about 5-limit intervals?
>
> Not sure what you're asking.

Not much in the way of 5-limit intervals?

> >> Okay, let's take a
> >> journey...
> >>
> >> "Instrumental music of Northeast Thailand"
> >>
> >> Characteristic stop rhythm. Harmonium and marimba-sounding
> >> things play major pentatonic on C# (A=440) or relative minor
> >> on A#.
> >
> >This is clearly not a pelog tuning!
>
> Right, it's the chinese pentatonic. I threw it in for
> completeness.

Completeness of what?

> >>I still say there's nothing here that would turn up an optimized
> >>5-limit temperament!
> >
> >Forget the optimization. All you need is the mapping -- that
> >chains of three fifths make a major third and that chains of
> >four fifths make a minor third. This seems to be a definite
> >characteristic of pelog! Just as much as the "opposite" is a
> >characteristic of Western music, regardless of whether strict
> >JI, optimized meantone, 12-tET, or whatever is used.
>
> Western music uses progressions of four fifths and expects to
> wind up on a major third.

These don't have to be triadic, harmonic progression.

> I didn't notice anything like this
> for the [1 -3] map (right?) on the cited discs.

[3 1]. It's not something you should expect to hear as a triadic
harmonic progression. It's simply the way the 5-limit intervals fit
together in the scale. If they didn't, the scale, and the music that
depends on it, wouldn't work.

> >>I guess it all depends if you consider these tonic changes
> >>or just points of symmetry in a melisma (sp?).
> >
> >Why does that matter?
>
> One's a harmonic device, the other melodic.

There are a lot of simultaneities going on, regardless of whether you
consider them to constitute "tonic changes".

> But I think a lot of the
> other stuff that goes along with harmonic music is missing
> from this music. Western music requires meantone. The pelog
> 5-limit map is far more extreme, but what suffers in this
> music as we change the tuning from 5-of- 7, to 23, to 16, all
> the way to strict JI?

23 and 16 give you the Pelog sound. 7 doesn't. Give me a strict JI
scale to try.

> I think the tuning on these discs is
> closer to JI than 23-tET, and I don't hear them avoiding a
> disjoint interval. Do you?

Avoiding a disjoint interval? You mean you hear it as 5-of-7? It
modulates that much?? What exactly do you mean?

> Incidentally, I think Wilson agrees with your point of view
> here. While he does caution against eager interps. of his
> ethno music theory, I think he thinks that harmonic mapping
> is inevitable, and atomic in music. I'm not sure I agree.
> Not sure I disagree.

Well, the idea of this paper that Gene, Dave, Graham, and I are
working on, at least it seems to me, is to start with the assumption
that notes are connected to one another via simple-ratio intervals,
explain periodicity blocks, show that an MOS results when you temper
out all but one of the unison vectors, show that MOSs are linear, and
present the "best" linear temperaments from this point of view. It's
just a paper, not a manifesto, so there's nothing wrong with starting
with a very simple and strong set of assumptions, and seeing where
they lead.

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

> How do you mean? The two meantones fit snugly on the two different
> keyboards, and chords in the enharmonic genus typically alternate
between
> them. As most chords are consonances, there's no other way of
getting the
> enharmonic melodies right. For you to ask this question suggests
either I
> didn't understand you, or you don't have a copy of Vicentino's book.

I don't.

> It
> is worth reading.

I'll have to look for it.

> I thought you had it because you recommended it to
> somebody else.

I did?

>
> Me:
> > > There's
> > > also a half-octave system, [(2, 0), (3, 1), (4, 4)]. That's
the
> > one my
> > > program would deduce from the octave-equivalent mapping [2 8].
>
> Paul:
> > >From that unison vector? If so, I think you're confusion torsion
> > with "contorsion".
>
> This has nothing directly to do with unison vectors.

Then what do you mean by "the octave-equivalent mapping [2 8]"?
>
> Me:
> > > If I had
> > > such a program. If anybody cares, is it possible to write
one?
> > Where
> > > torsion's present, we'll have to assume it means divisions of
the
> > octave
> > > for uniqueness.
>
> Paul:
> > Huh? Clearly this doesn't work in the Monz sruti 24 case.
>
> No, that can't be expressed in this particular octave equivalent
system.

Can it be expressed in any?
>
> > > Gene said it isn't possible, but I'm not convinced. How
> > > could [1 4] be anything sensible but meantone?
> >
> > Not sure what the connection is.
>
> [1 4] is a definition of meantone: 4 fifths are equivalent to a
major
> third. Is that a unique definition, or do we have to add "plus two
> octaves"?

To be completely clear, yes.
>
> > > Perhaps the first step is to find an interval that's only one
> > generator
> > > step, take the just value, period-reduce it and work everything
> > else out
> > > from that.
> >
> > If the half-fifth is the generator, what's the just value?
>
> Well, it could be either 5:4 or 6:5.

We've already mapped these to other intervals.

> Or 11:9 or 27:22. Or 49:40 or
> 60:49.

You can't just bring in 11 or 7 like that -- then you would have a 7-
limit or 11-limit system, with the associated mappings and all, which
you could work out in the normal way.

> But if you mean the case where all consonances are specified in
> terms of fifths, but the generator is a half-fifth, I thought I
defined
> those out of existence above.

Defined those out of existence? I thought you were saying this was
the Vicentino enharmonic case.

> If not, you can take the square root.

That's not a just interval.

> Me:
> > > But there may be some cases where the optimal value should
> > > cross a period boundary.
>
> Paul:
> > ??
>
> Say you have a system that divides the octave into two equal parts,
and
> 7:5 is a single generator steps. It may happen that 7:5
approximates best
> to be larger than a half octave, so taking its just value for
calculating
> the mapping will get the wrong results. This may be a real problem
when
> the octave is divided into 41 equal parts, like one of the higher-
limit
> temperaments I came up with, and the generator is a fairly complex
> interval.

Can you give a specific example?
>
> Me:
> > > If you think it can't be done, show a counter-example: an
> > > octave-equivalent mapping without torsion that can lead to two
> > different
> > > but equally good temperaments.
>
> Paul:
> > Equally good? Under what criteria? Look, why do we care about the
> > octave-equivalent mapping? Certainly we can't object to asking
the
> > mapping to be octave-specific, can we?
>
> It should be fairly obvious if you get the mapping right because
the
> errors will be small.

Granted, but how can we object to asking the mapping to be octave-
specific? Wouldn't it be better to do that from the outset than to
count on the errors being "small"?
>
> You were the one originally pushing for octave-equivalent >
calculations.

I think Gene has convinced be that they won't work. The only way you
can possibly distinguish cases of torsion correctly is with the
octave-specific mapping.

> If you aren't bothered any more, I'm not; I was only trying to
answer your
> questions. But it would be elegant to describe systems in the
simplest
> possible way, and one consistent with Fokker. It's up to you if
you don't
> want the paper to cover that.

Fokker didn't run into any cases of torsion, but we have! The paper
can cover Fokker's methods but doesn't need to be restricted to them.

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

> Well, the idea of this paper that Gene, Dave, Graham, and I are
> working on, at least it seems to me, is to start with the assumption
> that notes are connected to one another via simple-ratio intervals,
> explain periodicity blocks, show that an MOS results when you temper
> out all but one of the unison vectors, show that MOSs are linear, and
> present the "best" linear temperaments from this point of view.

How much of this is already published?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Well, the idea of this paper that Gene, Dave, Graham, and I are
> > working on, at least it seems to me, is to start with the
assumption
> > that notes are connected to one another via simple-ratio
intervals,
> > explain periodicity blocks, show that an MOS results when you
temper
> > out all but one of the unison vectors, show that MOSs are linear,
and
> > present the "best" linear temperaments from this point of view.
>
> How much of this is already published?

The proof that MOSs are linear might be said to be published. The
periodicity block concept was of course published by Fokker, though
the explanation of periodicity blocks might better take off from this
starting point, which you are all welcome to suggest changes to:

http://www.ixpres.com/interval/td/erlich/intropblock1.htm

As for the rest, I'm fairly certain it's entirely new work.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> #1
>
> 2^-90 3^-15 5^49
>
> This is not only the the one with lowest badness on the list, it is
the smallest comma, which suggests we are not tapering off, and is
evidence for flatness.
>
> Map:
>
> [ 0 1]
> [49 -6]
> [15 0]
>
> Generators: a = 275.99975/1783 = 113.00046/730; b = 1
>
> I suggest the "Woolhouse" as a name for this temperament, because
of the 730. Other ets consistent with this are 84, 323, 407, 1053 and
1460.
>
> badness: 34
> rms: .000763
> g: 35.5
> errors: [-.000234, -.001029, -.000796]
>
> #2 32805/32768 Schismic badness=55
>
> #3 25/24 Neutral thirds badness=82
>
> #4 15625/15552 Kleismic badness=97
>
> #5 81/80 Meantone badness=108
>
> It looks pretty flat so far as this method can show, I think.

How well do these results back up my now-famous (I hope) heuristic,
which involves only the size of the numbers in, and the difference
between numerator and denominator of, the unison vector? How might we
weight the gens and/or cents measures so that the heuristic will work
perfectly?

Me:
> > This has nothing directly to do with unison vectors.

Paul:
> Then what do you mean by "the octave-equivalent mapping [2 8]"?

3:2 is 2 generators and 5:4 is 8 generators, all octave reduced.

Paul:
> > > Huh? Clearly this doesn't work in the Monz sruti 24 case.

Me:
> > No, that can't be expressed in this particular octave equivalent
> system.

Paul:
> Can it be expressed in any?

You could list all notes in the periodicity block as octave-equivalent vectors.
How else were you expecting it to work? Do you have an octave-equivalent algorithm
for getting the periodicity block from the unison vectors?

Me:
> > But if you mean the case where all consonances are specified in
> > terms of fifths, but the generator is a half-fifth, I thought I
> defined
> > those out of existence above.

Paul:
> Defined those out of existence? I thought you were saying this was
> the Vicentino enharmonic case.

Yes, and it can't be unambiguously expressed as an octave-equivalent mapping. It
has torsion. I said we weren't considering such systems yet.

> > If not, you can take the square root.
>
> That's not a just interval.

So?

Paul (on systems where the just and tempered generators octave reduce differently):
> Can you give a specific example?

No, because I haven't coded anything up. If you have code that works, I've been
collecting test cases and I expect some of them will throw up this problem.

If they're allowed, the [2 8] systems are an example, because 350 and 850 give
different octave-specific systems, but optimise to the same meantone. You could
differentiate them by saying that [2 8] means to divide the fifth, and [-2 -8] to
divide the fourth, but that would still break the one to one relationship between
mappings and temperaments.

Me:
> > It should be fairly obvious if you get the mapping right because
> the
> > errors will be small.

Paul:
> Granted, but how can we object to asking the mapping to be octave-
> specific? Wouldn't it be better to do that from the outset than to
> count on the errors being "small"?

If nobody's objecting to the mapping being octave-specific there's no problem.
Even so, there's nothing special about errors needing to be small in an
octave-equivalent system. A period-equivalent system is fully defined by it's
mapping and the period. Different generators will give different octave-specific
mappings, but that's a relationship between the systems, not an inherent problem
with one system.

The problem with period-equivalent systems (what the octave-equivalent route tends
to lead to) is that they're harder to optimise. When a particular interval
approximates to an exact number of periods, you'll get a local maximum so steepest
descent methods won't work. The RMS error by generator isn't a quadratic equation,
so that optimisation won't work. A number of different generator sizes can make
the same interval just, so minimax is harder. I'm sure all these problems can be
overcome, but they are problems.

Paul:
> I think Gene has convinced be that they won't work. The only way you
> can possibly distinguish cases of torsion correctly is with the
> octave-specific mapping.

I haven't seen that proven yet. Let's get an algorithm first, and see if it
doesn't work. Where do you think torsion is a problem? An octave-equivalent
mapping can do everything a wedge product can. You can add a parameter if you want
to distinguish torsion from equal divisions of the octave. In going from unison
vectors to a mapping, torsion might show up as a common factor in the adjoint where
it's a problem. I haven't even got round to checking yet. Pairs of ETs with
torsion don't work with wedge products either. It may be that the sign of the
mapping can be used to disambiguate them. Otherwise, give the range of generators
as part of the definition.

> Fokker didn't run into any cases of torsion, but we have! The paper
> can cover Fokker's methods but doesn't need to be restricted to them.

Wouldn't it be nice to say whether or not Fokker's methods would have worked if he
had run into torsion?

Graham

Paul wrote:

> The proof that MOSs are linear might be said to be published. The
> periodicity block concept was of course published by Fokker, though
> the explanation of periodicity blocks might better take off from this
> starting point, which you are all welcome to suggest changes to:
>
> http://www.ixpres.com/interval/td/erlich/intropblock1.htm
>
> As for the rest, I'm fairly certain it's entirely new work.

C Karp's "Analyzing Musical Tuning Systems" from Acustica Vo.54 (1984)
should be considered. He uses octave-specific, 5-limit matrices,
including some inverses. He does say, p.212, "... the temperament vector
of any interval (a, b, c)_t, is associated with the c/b comma division
temperament" and works through examples for fractional meantones.

Brian McLaren sent me a copy, in the days when he deigned to recognize
mathematical theory. It acknowledges one "Bob Marvin, who devised the
matrix representation of tuning systems used here, and introduced it to
the author."

Graham

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

> Me:
> > > But if you mean the case where all consonances are specified
in
> > > terms of fifths, but the generator is a half-fifth, I thought
I
> > defined
> > > those out of existence above.
>
> Paul:
> > Defined those out of existence? I thought you were saying this
was
> > the Vicentino enharmonic case.
>
> Yes, and it can't be unambiguously expressed as an
octave-equivalent mapping. It
> has torsion.

No, I don't think this is torsion at all! It's a different
phenomenon altogether, for which I gave the name "contortion".

> > That's not a just interval.
>
> So?

You said "just interval".
>
> Paul:
> > I think Gene has convinced be that they won't work. The only way
you
> > can possibly distinguish cases of torsion correctly is with the
> > octave-specific mapping.
>
> I haven't seen that proven yet.

It should be quite straightforward to prove. How could you tell
whether 50:49 produces torsion or not in an octave-invariant
formulation?

> Let's get an algorithm first, and see if it
> doesn't work. Where do you think torsion is a problem? An
octave-equivalent
> mapping can do everything a wedge product can. You can add a
parameter if you want
> to distinguish torsion from equal divisions of the octave. In
going from unison
> vectors to a mapping, torsion might show up as a common factor in
the adjoint where
> it's a problem. I haven't even got round to checking yet.

I thought Gene showed that the common-factor rule only works in the
octave-specific case.

> Pairs of ETs with
> torsion don't work with wedge products either. It may be that the
sign of the
> mapping can be used to disambiguate them. Otherwise, give the
range of generators
> as part of the definition.

You've lost me. Gene, any comments?
>
> > Fokker didn't run into any cases of torsion, but we have! The
paper
> > can cover Fokker's methods but doesn't need to be restricted to
them.
>
> Wouldn't it be nice to say whether or not Fokker's methods would
have worked if he
> had run into torsion?

I'm pretty sure the answer is no. Gene?

--- In tuning-math@y..., graham@m... wrote:
> Paul wrote:
>
> > The proof that MOSs are linear might be said to be published.
The
> > periodicity block concept was of course published by Fokker,
though
> > the explanation of periodicity blocks might better take off from
this
> > starting point, which you are all welcome to suggest changes to:
> >
> > http://www.ixpres.com/interval/td/erlich/intropblock1.htm
> >
> > As for the rest, I'm fairly certain it's entirely new work.
>
> C Karp's "Analyzing Musical Tuning Systems" from Acustica Vo.54
(1984)
> should be considered. He uses octave-specific, 5-limit matrices,
> including some inverses. He does say, p.212, "... the temperament
vector
> of any interval (a, b, c)_t, is associated with the c/b comma
division
> temperament" and works through examples for fractional meantones.
>
> Brian McLaren sent me a copy, in the days when he deigned to
recognize
> mathematical theory. It acknowledges one "Bob Marvin, who devised
the
> matrix representation of tuning systems used here, and introduced
it to
> the author."
>
>
> Graham

Would you send me a copy?

Paul Erlich
57 Grove St.
Somerville, MA 02144

>>>No. It just assumes that the overtones are pretty close to
>>>harmonic, because they will then lead to the same ratio-
>>>intepretations for the fundamentals as the fundamentals by
>>>themselves. If they're 50 cents from harmonic, they will
>>>lead to a larger s value for the resulting harmonic entropy
>>>curve, but that's about it.
>>
>>s represents the blur of the spectral components coming in.
>>How could an inharmonic timbre change that?
>
>When we're dealing with a dyad consisting of complex tones, and
>trying to apply harmonic entropy to that dyad, s is decreased
>below the value that sine waves in place of the complex tones
>would imply. The more inharmonic the timbre, the less s is
>decreased below the sine-wave case.

Still doesn't explain how. You need a way for data from the
combination-sensitive stuff to improve the spectral stuff coming
off the cochlea. I don't think it works that way. The "accuracy"
of the "fundamental" is improved as the spectral components get
closer to just, as the harmonic entropy calc. itself correctly
models. But to change s in this way is a fudge, in my opinion.
With harmonic timbres, h.e. on the fundamentals is a good
approximation of things, but with inharmonic timbres, all spectral
components need to be put in to the h.e. calculation. Jacking up
s may approximate this, but it would be a fudge.

Anyway, there is now psychoacoustic evidence for harmonic entropy.
In fact, it looks like it perfectly models what happens in
populations of "combination-sensitive" neurons in the inferior
colliculus. At least in bats. I plan on posting to
harmonic_entropy on this as soon as I can get the citations
together.

>>>>Yes, to me, pelog sounds like a I and a III with a 4th in
>>>>the middle. But the music seems to use a fixed tonic, with
>>>>not much in the way of triadic structure.
>>>
>>>How about 5-limit intervals?
>>
>>Not sure what you're asking.
>
>Not much in the way of 5-limit intervals?

I think the large 2nd approximates a 5:4, and the perfect
4th a 3:2, with some tempering to reduce the roughness of
these intervals on the instrumentation used (as opposed to
tempering to improve the consonance of these interval in
different modes, to distribute any commas, etc.).

>>Right, it's the chinese pentatonic. I threw it in for
>>completeness.
>
>Completeness of what?

Of the journey from North to South, and of the survey of
pentatonic scales in motivic ethnic music of southeast asia.
And it was informative; nothing about the music changed as
we went from Pelog, to the hybrid, to the chinese pentatonic
_except_ the scale. You could transcribe the notes and wind
up with the same stuff, more or less.

>>Western music uses progressions of four fifths and expects
>>to wind up on a major third.
>
>These don't have to be triadic, harmonic progression.

I guess not. But there's a big difference in how this stuff
is used. The Indonesia music is motivic, not modal. At least,
I follow the pitches and their positions in the scale, not the
intervals of the scale and there relation to one another. The
harmonic motion is used to render some consonance, and some
tension/release action, but that's it. It's a backdrop to the
motivic material.

>>I didn't notice anything like this
>>for the [1 -3] map (right?) on the cited discs.
>
>[3 1]. It's not something you should expect to hear as a triadic
>harmonic progression. It's simply the way the 5-limit intervals
>fit together in the scale. If they didn't, the scale, and the
>music that depends on it, wouldn't work.

[3 1]? I thought these maps expressed each odd identity, from
three to the limit, increasing from left to right, in numbers
of generators. Thus up one 3:2 for the 3:2, and down three 3:2s
for the 5:4.

>>But I think a lot of the other stuff that goes along with
>>harmonic music is missing from this music. Western music
>>requires meantone. The pelog 5-limit map is far more extreme,
>>but what suffers in this music as we change the tuning from
>>5-of- 7, to 23, to 16, all the way to strict JI?
>
>23 and 16 give you the Pelog sound. 7 doesn't.

By gods, you're right! 7-of-5 doesn't sound like pelog at all.

>Give me a strict JI scale to try.

1/1 5/4 4/3 3/2 15/8

Sounds like a fine pelog to me.

>>I think the tuning on these discs is closer to JI than 23-tET,
>>and I don't hear them avoiding a disjoint interval. Do you?
>
>Avoiding a disjoint interval? You mean you hear it as 5-of-7?
>It modulates that much??

Actually, it doesn't. They seem to stick mostly to I, IV, and
III (diatonic) with the bass, if you consider those tonics. But
the melodic stuff does center itself on every degree of the
scale -- it treats the "bad" 4ths the same as the perfect 4ths.

To rephrase the question one more time, in what sense are these
bass notes tonics? Do they change anything about the melody?
That is, what used to be scale degree 4 is now 1? I say they
don't. What I hear is a fixed 1. The melody is a very slow
series of scale degrees above that 1. On each note of the melody,
a bunch of ornamentation is hung, which is made of scale arpeggio
bits. The bass starts and ends on 1, and goes to 3, 2, and
sometimes 4 (I-IV-III-V diatonic), to provide a sense of
tension/resolution.

>>Incidentally, I think Wilson agrees with your point of view
>>here. While he does caution against eager interps. of his
>>ethno music theory, I think he thinks that harmonic mapping
>>is inevitable, and atomic in music. I'm not sure I agree.
>>Not sure I disagree.
>
>Well, the idea of this paper that Gene, Dave, Graham, and I
>are working on, at least it seems to me, is to start with the
>assumption that notes are connected to one another via simple-
>ratio intervals, explain periodicity blocks, show that an MOS
>results when you temper out all but one of the unison vectors,
>show that MOSs are linear, and present the "best" linear
>temperaments from this point of view. It's just a paper, not a
>manifesto, so there's nothing wrong with starting with a very
>simple and strong set of assumptions, and seeing where they lead.

Of course! (I already can't wait!)

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>
> >>s represents the blur of the spectral components coming in.
> >>How could an inharmonic timbre change that?
> >
> >When we're dealing with a dyad consisting of complex tones, and
> >trying to apply harmonic entropy to that dyad, s is decreased
> >below the value that sine waves in place of the complex tones
> >would imply. The more inharmonic the timbre, the less s is
> >decreased below the sine-wave case.
>
> Still doesn't explain how. You need a way for data from the
> combination-sensitive stuff to improve the spectral stuff coming
> off the cochlea.

There's less ambiguity as to the possible ratio-intepretations.

> I don't think it works that way.

It does. Devise a baby mathematical model for it and you'll see.

> The "accuracy"
> of the "fundamental" is improved as the spectral components get
> closer to just, as the harmonic entropy calc. itself correctly
> models. But to change s in this way is a fudge, in my opinion.

It would be nice to derive the change in s mathematically, and any
baby model will do so. However, until the model is more fully
developed, I'm hesitant to put forward any exact formulas about how
s changes.

> With harmonic timbres, h.e. on the fundamentals is a good
> approximation of things, but with inharmonic timbres, all spectral
> components need to be put in to the h.e. calculation. Jacking up
> s may approximate this, but it would be a fudge.

Again, if the inharmonicities are only about 50 cents, I thing it's
a pretty darn good one.

> Anyway, there is now psychoacoustic evidence for harmonic entropy.
> In fact, it looks like it perfectly models what happens in
> populations of "combination-sensitive" neurons in the inferior
> colliculus. At least in bats. I plan on posting to
> harmonic_entropy on this as soon as I can get the citations
> together.

You might want to mention this to J on the main tuning list, who
seems to think that Terhardt and I are the only two people in the
world who believe there is such a thing as "virtual pitch".

> >>>>Yes, to me, pelog sounds like a I and a III with a 4th in
> >>>>the middle. But the music seems to use a fixed tonic, with
> >>>>not much in the way of triadic structure.
> >>>
> >>>How about 5-limit intervals?
> >>
> >>Not sure what you're asking.
> >
> >Not much in the way of 5-limit intervals?
>
> I think the large 2nd approximates a 5:4, and the perfect
> 4th a 3:2, with some tempering to reduce the roughness of
> these intervals on the instrumentation used (as opposed to
> tempering to improve the consonance of these interval in
> different modes, to distribute any commas, etc.).

Why don't you experiment with this with a number of timbres. I know
you can't, but I can, and believe me, reducing roughness does
absolutely nothing to capture an authentic Gamelan sound.

> >>Right, it's the chinese pentatonic. I threw it in for
> >>completeness.
> >
> >Completeness of what?
>
> Of the journey from North to South, and of the survey of
> pentatonic scales in motivic ethnic music of southeast asia.

You listen to three selections and call that a survey?

> And it was informative; nothing about the music changed as
> we went from Pelog, to the hybrid, to the chinese pentatonic
> _except_ the scale. You could transcribe the notes and wind
> up with the same stuff, more or less.

Probably to people in that part of the world Beethoven and the blues
sound the same. Go live in Southeast Asia for twelve years and then
get back to me.

> >>Western music uses progressions of four fifths and expects
> >>to wind up on a major third.
> >
> >These don't have to be triadic, harmonic progression.
>
> I guess not. But there's a big difference in how this stuff
> is used. The Indonesia music is motivic, not modal. At least,
> I follow the pitches and their positions in the scale, not the
> intervals of the scale and there relation to one another.

Right, those, you might say, are "incidental". Still, they give the
sound a certain texture.

> The
> harmonic motion is used to render some consonance, and some
> tension/release action, but that's it. It's a backdrop to the
> motivic material.

The same is true in Western music, in a certain sense.

> >>I didn't notice anything like this
> >>for the [1 -3] map (right?) on the cited discs.
> >
> >[3 1]. It's not something you should expect to hear as a triadic
> >harmonic progression. It's simply the way the 5-limit intervals
> >fit together in the scale. If they didn't, the scale, and the
> >music that depends on it, wouldn't work.
>
> [3 1]? I thought these maps expressed each odd identity, from
> three to the limit, increasing from left to right, in numbers
> of generators. Thus up one 3:2 for the 3:2, and down three 3:2s
> for the 5:4.

I though you meant the unison vector. OK ,[1 -3] map looks right.

> >>But I think a lot of the other stuff that goes along with
> >>harmonic music is missing from this music. Western music
> >>requires meantone. The pelog 5-limit map is far more extreme,
> >>but what suffers in this music as we change the tuning from
> >>5-of- 7, to 23, to 16, all the way to strict JI?
> >
> >23 and 16 give you the Pelog sound. 7 doesn't.
>
> By gods, you're right! 7-of-5 doesn't sound like pelog at all.

5-of-7?
>
> >Give me a strict JI scale to try.
>
> 1/1 5/4 4/3 3/2 15/8
>
> Sounds like a fine pelog to me.

Yuck -- reminds me of Charles Lucy. Totally inauthentic.

> >>I think the tuning on these discs is closer to JI than 23-tET,
> >>and I don't hear them avoiding a disjoint interval. Do you?
> >
> >Avoiding a disjoint interval? You mean you hear it as 5-of-7?
> >It modulates that much??
>
> Actually, it doesn't. They seem to stick mostly to I, IV, and
> III (diatonic) with the bass, if you consider those tonics. But
> the melodic stuff does center itself on every degree of the
> scale -- it treats the "bad" 4ths the same as the perfect 4ths.

What "bad" 4th are you referring to??

> To rephrase the question one more time, in what sense are these
> bass notes tonics? Do they change anything about the melody?
> That is, what used to be scale degree 4 is now 1? I say they
> don't.

Fine -- I don't see how this is related to anything I'm saying.

> What I hear is a fixed 1. The melody is a very slow
> series of scale degrees above that 1. On each note of the melody,
> a bunch of ornamentation is hung, which is made of scale arpeggio
> bits. The bass starts and ends on 1, and goes to 3, 2, and
> sometimes 4 (I-IV-III-V diatonic), to provide a sense of
> tension/resolution.

How do you get I-IV-III-V out of a 5-tone scale? And these chords
sound major or minor to you?

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

> > Wouldn't it be nice to say whether or not Fokker's methods would
> have worked if he
> > had run into torsion?

> I'm pretty sure the answer is no. Gene?

I don't know they are. What would he have done in the case of the 24-note business which was our first example?

>>>Right, it's the chinese pentatonic. I threw it in for
>>>completeness.
>>>
>>>Completeness of what?
>>
>>Of the journey from North to South, and of the survey of
>>pentatonic scales in motivic ethnic music of southeast asia.
>
>You listen to three selections and call that a survey?

Uh-huh. I didn't make any statistical claims about it. In
fact the only claims I make about it are:

() it's better than a 'survey' of the scala scl archive
() it's better than a 'survey' without the Thai recording.

>>>Give me a strict JI scale to try.
>>
>>1/1 5/4 4/3 3/2 15/8
>>
>>Sounds like a fine pelog to me.
>
>Yuck -- reminds me of Charles Lucy.

Me too.

>Totally inauthentic.

But much closer than 5-of-7.

>>>>I think the tuning on these discs is closer to JI than 23-tET,
>>>>and I don't hear them avoiding a disjoint interval. Do you?
>>>
>>>Avoiding a disjoint interval? You mean you hear it as 5-of-7?
>>>It modulates that much??
>>
>>Actually, it doesn't. They seem to stick mostly to I, IV, and
>>III (diatonic) with the bass, if you consider those tonics. But
>>the melodic stuff does center itself on every degree of the
>>scale -- it treats the "bad" 4ths the same as the perfect 4ths.
>
>What "bad" 4th are you referring to??

The one between 15/8 and 4/3 in the scale above.

>>To rephrase the question one more time, in what sense are these
>>bass notes tonics? Do they change anything about the melody?
>>That is, what used to be scale degree 4 is now 1? I say they
>>don't.
>
> Fine -- I don't see how this is related to anything I'm saying.

I thought somebody was claiming there was an impetus for 5-limit
temperament in Javanese music.

>>What I hear is a fixed 1. The melody is a very slow
>>series of scale degrees above that 1. On each note of the melody,
>>a bunch of ornamentation is hung, which is made of scale arpeggio
>>bits. The bass starts and ends on 1, and goes to 3, 2, and
>>sometimes 4 (I-IV-III-V diatonic), to provide a sense of
>>tension/resolution.
>
>How do you get I-IV-III-V out of a 5-tone scale? And these chords
>sound major or minor to you?

I-IV-III-V as if pelog was a 5-out-of diatonic scale. Am I
forgetting the case of me numerals?

Would be 1-3-2-4 in pelog. 1 and 3 sound major, 2 minor. I
can't tell 4.

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > Wouldn't it be nice to say whether or not Fokker's methods
would
> > have worked if he
> > > had run into torsion?
>
> > I'm pretty sure the answer is no. Gene?
>
> I don't know they are. What would he have done in the case of the >
24-note business which was our first example?

I'd guess he would just leave it as a 24-tone JI scale, but we'll
never know, as he only ever published 12-, 19-, 22-, 31-, 41-, and 53-
tone PBs (and left them all as JI scales).

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>>Right, it's the chinese pentatonic. I threw it in for
> >>>completeness.
> >>>
> >>>Completeness of what?
> >>
> >>Of the journey from North to South, and of the survey of
> >>pentatonic scales in motivic ethnic music of southeast asia.
> >
> >You listen to three selections and call that a survey?
>
> Uh-huh. I didn't make any statistical claims about it. In
> fact the only claims I make about it are:
>
> () it's better than a 'survey' of the scala scl archive

Maybe. We still haven't heard which scales Dave used, and where they
came from. There's lots of other research -- on the internet alone.

> >>>Give me a strict JI scale to try.
> >>
> >>1/1 5/4 4/3 3/2 15/8
> >>
> >>Sounds like a fine pelog to me.
> >
> >Yuck -- reminds me of Charles Lucy.
>
> Me too.
>
> >Totally inauthentic.
>
> But much closer than 5-of-7.

5-of-7 gives you neutral thirds.

> >>>>I think the tuning on these discs is closer to JI than 23-tET,
> >>>>and I don't hear them avoiding a disjoint interval. Do you?
> >>>
> >>>Avoiding a disjoint interval? You mean you hear it as 5-of-7?
> >>>It modulates that much??
> >>
> >>Actually, it doesn't. They seem to stick mostly to I, IV, and
> >>III (diatonic) with the bass, if you consider those tonics. But
> >>the melodic stuff does center itself on every degree of the
> >>scale -- it treats the "bad" 4ths the same as the perfect 4ths.
> >
> >What "bad" 4th are you referring to??
>
> The one between 15/8 and 4/3 in the scale above.

So the fact that they're not avoiding it helps prove I'm right! It's
a GOOD fourth in Pelog, since 135:128 vanishes! It's a bad fourth in
JI, so _if_ the tuning were JI, _then_ there might be a tendency to
avoid that interval.

> >>To rephrase the question one more time, in what sense are these
> >>bass notes tonics? Do they change anything about the melody?
> >>That is, what used to be scale degree 4 is now 1? I say they
> >>don't.
> >
> > Fine -- I don't see how this is related to anything I'm saying.
>
> I thought somebody was claiming there was an impetus for 5-limit
> temperament in Javanese music.

Still don't see any relationship.

> >>What I hear is a fixed 1. The melody is a very slow
> >>series of scale degrees above that 1. On each note of the melody,
> >>a bunch of ornamentation is hung, which is made of scale arpeggio
> >>bits. The bass starts and ends on 1, and goes to 3, 2, and
> >>sometimes 4 (I-IV-III-V diatonic), to provide a sense of
> >>tension/resolution.
> >
> >How do you get I-IV-III-V out of a 5-tone scale? And these chords
> >sound major or minor to you?
>
> I-IV-III-V as if pelog was a 5-out-of diatonic scale. Am I
> forgetting the case of me numerals?

I think so.

> Would be 1-3-2-4 in pelog. 1 and 3 sound major,

So C major and E major ???

> 2 minor. I
> can't tell 4.

>>I thought somebody was claiming there was an impetus for 5-limit
>>temperament in Javanese music.
>
>Still don't see any relationship.

I'm trying to show that the things in Western music that led
to temperament are absent in Indonesian music.

>>>How do you get I-IV-III-V out of a 5-tone scale? And these chords
>>>sound major or minor to you?
>>
>>I-IV-III-V as if pelog was a 5-out-of diatonic scale. Am I
>>forgetting the case of me numerals?
>
>I think so.
>
>>Would be 1-3-2-4 in pelog. 1 and 3 sound major,
>
> So C major and E major ???

C and F major.

>> 2 minor.

E minor.

>>I can't tell 4.

G something.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>I thought somebody was claiming there was an impetus for 5-limit
> >>temperament in Javanese music.
> >
> >Still don't see any relationship.
>
> I'm trying to show that the things in Western music that led
> to temperament are absent in Indonesian music.

So what? They may have had their own reasons, and inharmonicity makes
the situations rather different. Gamelan intervals are "pastelized",
as Margo Schulter says.

>
> >>>How do you get I-IV-III-V out of a 5-tone scale? And these chords
> >>>sound major or minor to you?
> >>
> >>I-IV-III-V as if pelog was a 5-out-of diatonic scale. Am I
> >>forgetting the case of me numerals?
> >
> >I think so.
> >
> >>Would be 1-3-2-4 in pelog. 1 and 3 sound major,
> >
> > So C major and E major ???
>
> C and F major.

How did the note A get in there?

> >> 2 minor.
>
> E minor.
>
> >>I can't tell 4.
>
> G something.

You mean you can't hear which notes make up the chord?

Carl, it seemed you yourself just gave evidence for the 135:128
vanishing, didn't you?

>>I'm trying to show that the things in Western music that led
>>to temperament are absent in Indonesian music.
>
>So what? They may have had their own reasons,

Sure. What might they be?

>and inharmonicity makes the situations rather different.
>Gamelan intervals are "pastelized", as Margo Schulter says.

?

>>> So C major and E major ???
>>
>> C and F major.
>
> How did the note A get in there?

It didn't; these aren't triads after all. Is
there an Ab? No. But it sounds like if there
was a note, it'd be major.

>>>2 minor.
>>
>>E minor.
>>
>>>I can't tell 4.
>>
>>G something.
>
>You mean you can't hear which notes make up the chord?

I've never heard a voice in the music that was triads, Paul.
Have you? I've heard triads formed between voices, in
between all other kinds of chords made up of degrees of the
scale. Nonetheless, I do admit that it sounds like triadic
motion, to me. It sounds like a progression involving the
chords Cmaj, Emin, Fmaj, and G... something. Obviously, the
note, when it's there, is B natural.

-Carl

>Carl, it seemed you yourself just gave evidence for the 135:128
>vanishing, didn't you?

They treat it like it's vanished, but I don't think it's
vanished. This 4th sounds different to me. I don't think
they've tempered it, I think they just use it, despite it
being out. Just like in Wilson's keyboard layouts. How
does it sound to you, and on what recordings does it sound
that way?

Sometime this week, I may be able to make some mp3s, but
don't hold your breath. It will be busy at work, since
we're starting up after basically a month off, and we're
broke, since nobody's buying serial adapters.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>I'm trying to show that the things in Western music that led
> >>to temperament are absent in Indonesian music.
> >
> >So what? They may have had their own reasons,
>
> Sure. What might they be?

Wanting to play 4/3 together with 15/8 and make them consonant with
one another, as you observed.

>
> >and inharmonicity makes the situations rather different.
> >Gamelan intervals are "pastelized", as Margo Schulter says.
>
> ?

Search for "pastelize".
>
> >>> So C major and E major ???
> >>
> >> C and F major.
> >
> > How did the note A get in there?
>
> It didn't; these aren't triads after all. Is
> there an Ab? No. But it sounds like if there
> was a note, it'd be major.

Maybe the music moves to one of the other two Pelog pitches on the 7-
tone instruments.

> >>>2 minor.
> >>
> >>E minor.
> >>
> >>>I can't tell 4.
> >>
> >>G something.
> >
> >You mean you can't hear which notes make up the chord?
>
> I've never heard a voice

Voice?

> in the music that was triads, Paul.
> Have you?

No, but you used roman numerals, so I thought you did.

> I've heard triads formed between voices,

Ok, what does the voice thing mean in your first statement above?

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >Carl, it seemed you yourself just gave evidence for the 135:128
> >vanishing, didn't you?
>
> They treat it like it's vanished, but I don't think it's
> vanished. This 4th sounds different to me.

I think it's very difficult for ears with Western=trained categorical
perception not to hear it as different. We're talking about 4ths
which average about 523 cents, after all! Texturally, though, it
comes out sounding a lot like the other 4ths, to me.

> I don't think
> they've tempered it, I think they just use it, despite it
> being out. Just like in Wilson's keyboard layouts. How
> does it sound to you, and on what recordings does it sound
> that way?

I used to spend lots of time in libraries listening to this stuff.
Maybe I should start again.

I wrote,

> I think it's very difficult for ears with Western=trained
categorical
> perception not to hear it as different.

That is, because they're the 4th and the maj. 7th, and we're _used_
to hearing these as a characteristic dissonance. 523 is far enough
from 500 that our Western mind can categorize the entire pentatonic
scale as root, M3, p4, p5, M7.

>> I've never heard a voice
>
> Voice?

As in, part or parts in the music sharing the same rhythm.

-Carl

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > #1
> >
> > 2^-90 3^-15 5^49
> >
> > This is not only the the one with lowest badness on the list, it
is
> the smallest comma, which suggests we are not tapering off, and is
> evidence for flatness.
> >
> > Map:
> >
> > [ 0 1]
> > [49 -6]
> > [15 0]
> >
> > Generators: a = 275.99975/1783 = 113.00046/730; b = 1
> >
> > I suggest the "Woolhouse" as a name for this temperament, because
> of the 730. Other ets consistent with this are 84, 323, 407, 1053
and
> 1460.
> >
> > badness: 34
> > rms: .000763
> > g: 35.5
> > errors: [-.000234, -.001029, -.000796]
> >
> > #2 32805/32768 Schismic badness=55
> >
> > #3 25/24 Neutral thirds badness=82
> >
> > #4 15625/15552 Kleismic badness=97
> >
> > #5 81/80 Meantone badness=108
> >
> > It looks pretty flat so far as this method can show, I think.
>
> How well do these results back up my now-famous (I hope) heuristic,
> which involves only the size of the numbers in, and the difference
> between numerator and denominator of, the unison vector? How might
we
> weight the gens and/or cents measures so that the heuristic will
work
> perfectly?

Let's start with #5 and work our way up to #1:

U V W X Y Z
unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y
------------ -------- --------------- ---- ---- ------ ---
80/81 0.003517 0.002809 .799 2.944 4.394 .67
15552/15625 0.0008583 0.0004838 .564 4.546 9.657 .471
24/25 0.024083 0.012427 .516 1.414 3.219 .439
32768/32805 0.00013475 0.00010847 .805 6.976 10.398 .671
[90 15 -49] 6.358e-007 3.44e-007 .541 35.5 78.862 .450

Our current "g" measure is clearly too large when the generator is a
fifth, as I've been trying to complain for quite a while now and only
Dave Keenan has changes his ways accordingly, and the comparison with
the heuristic, though agreeable, suggests that the heuristic is in
fact better than our current measure. What if we weighted the
intervals unequally in both the g and in the "rms" calculations?
Could we get the heuristic to work perfectly? I think that would be
very interesting for our paper.

Moving on to some relatively "bad" examples . . . Gene,
your "Enneadecal" comma should have a power of 2 equal to 14, not 15
as you said, right?

U V W X Y Z
unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y
------------ -------- --------------- ---- ---- ------ ---
[52 17 -34] 2.864e-005 1.5102e-005 .527 24.042 54.721 .439
128/135 0.01508 0.010571 .701 2.94 4.905 .599
[14 19 -19] 8.733e-005 5.314e-005 .608 15.513 30.579 .507
648/625 0.009217 0.005716 .620 3.27 6.438 .508
[8 14 -13] 0.0002305 0.0001463 .635 11.045 20.923 .528

Here the heuristic seems to work even better (and the V/U and X/Y are
well within the range of their values for the top 5).

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >> I've never heard a voice
> >
> > Voice?
>
> As in, part or parts in the music sharing the same rhythm.

So what does the sentence,

"I've never heard a voice in the music that was triads, Paul."

mean? You haven't heard parallel triads? Me either!

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> U V W X Y Z
> unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y
> ------------ -------- --------------- ---- ---- ------ ---
> 80/81 0.003517 0.002809 .799 2.944 4.394 .67
> 15552/15625 0.0008583 0.0004838 .564 4.546 9.657 .471
> 24/25 0.024083 0.012427 .516 1.414 3.219 .439
> 32768/32805 0.00013475 0.00010847 .805 6.976 10.398 .671
> [90 15 -49] 6.358e-007 3.44e-007 .541 35.5 78.862 .450

I was going to ask what this was, but I see when I follow-up the table makes more sense.

> Our current "g" measure is clearly too large when the generator is a
> fifth,

Why?

as I've been trying to complain for quite a while now and only
> Dave Keenan has changes his ways accordingly, and the comparison with
> the heuristic, though agreeable, suggests that the heuristic is in
> fact better than our current measure.

Better for what? If you mean as a badness measure, g isn't one.

What if we weighted the
> intervals unequally in both the g and in the "rms" calculations?

We could, but I was never clear how exactly you wanted this done.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > U V W X Y Z
> > unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d)
X/Y
> > ------------ -------- --------------- ---- ---- ------ ---
> > 80/81 0.003517 0.002809 .799 2.944 4.394 .67
> > 15552/15625 0.0008583 0.0004838 .564 4.546
9.657 .471
> > 24/25 0.024083 0.012427 .516 1.414
3.219 .439
> > 32768/32805 0.00013475 0.00010847 .805 6.976
10.398 .671
> > [90 15 -49] 6.358e-007 3.44e-007 .541 35.5
78.862 .450
>
> I was going to ask what this was, but I see when I follow-up the
table makes more sense.
>
> > Our current "g" measure is clearly too large when the generator
is a
> > fifth,
>
> Why?

Because of two systems with the same "g" measure, the one which has a
generator of a 3:2 is musically simpler. Movements by 3:2 are easier
to hear than movements by 5:3 or 5:4.

> as I've been trying to complain for quite a while now and only
> > Dave Keenan has changes his ways accordingly, and the comparison
with
> > the heuristic, though agreeable, suggests that the heuristic is
in
> > fact better than our current measure.
>
> Better for what? If you mean as a badness measure, g isn't one.

I mean as a measure of the musical complexity of the system.

> > What if we weighted the
> > intervals unequally in both the g and in the "rms" calculations?
>
> We could, but I was never clear how exactly you wanted this done.

The suggestion I gave you in the past was, in the g measure, to
weight the 3:2 by 1/log(3), 5:4 by 1/log(5), and 5:3 by 1/log(5).
This is what Dave Keenan is now using. It's unclear to me that an
root-mean-square calculation for g is exactly what we want, though
it's probably a good approximation.

But my nagging suspicion is that some perfectly musically reasonable
formulations of g, and of 'rms' or the cents-error measure, should
conform to my heuristics _exactly_, or at least way better than the
current measures are. You're the mathematician, can't you figure this
out? Read my original justification for the heuristics again if you
need to (message #1437), and consider the van Prooijen lattice:
http://www.kees.cc/tuning/perbl.html and
http://www.kees.cc/tuning/perbl.html except for the last graph. This
should be doable!

I wrote,

> http://www.kees.cc/tuning/perbl.html and
> http://www.kees.cc/tuning/perbl.html

Oops -- the second link should be

http://www.kees.cc/tuning/lat_perbl.html

In-Reply-To: <a1atde+mhm5@eGroups.com>
Paul wrote:

> No, I don't think this is torsion at all! It's a different
> phenomenon altogether, for which I gave the name "contortion".

It gives the same wedge product as unison vectors with torsion.

Paul:
> > > That's not a just interval.

Me:
> > So?

Paul:
> You said "just interval".

I also said I wasn't considering systems with this contorsion.

Paul:
> It should be quite straightforward to prove. How could you tell
> whether 50:49 produces torsion or not in an octave-invariant
> formulation?

Do you care about it being [dis]proven, then? I expect your algorithm for
generating periodicity blocks will solve everything. But I haven't looked
it up because people keep saying they aren't interested, while asking more
and more questions. It won't change anything musically.

Paul:
> I thought Gene showed that the common-factor rule only works in the
> octave-specific case.

I don't remember him considering the adjoint, rather than the wedge
product. But we may not need it anyway.

Me:
> > Pairs of ETs with
> > torsion don't work with wedge products either. It may be that the
> sign of the
> > mapping can be used to disambiguate them. Otherwise, give the
> range of generators
> > as part of the definition.

Paul:
> You've lost me. Gene, any comments?

Meaning contorsion here. The octave-specific wedge product can remove it,
but not use it. An octave-equivalent wedge product (the octave-equivalent
mapping) will treat such systems, wrongly, as requiring a division of the
octave. But starting from ETs it does make more sense to use
octave-specific vectors in the first place. Perhaps we should only ask if
unison vectors can work in an octave-equivalent system, in which case this
problem doesn't apply.

Me:
> > Wouldn't it be nice to say whether or not Fokker's methods would
> have worked if he
> > had run into torsion?

Paul:
> I'm pretty sure the answer is no. Gene?

The main thing we've added to Fokker (after Wilson) is the mapping,
instead of merely counting the number of notes in the periodicity block.
The Monz-shruti example gives a periodicity block with more notes than you
need for the temperament, but the mappings still come out. There are more
insidious examples of torsion where the mappings don't work either. The
problem being that octave-equivalent matrices don't differentiate commatic
torsion from systems requiring a period that isn't the octave.

Graham

>> As in, part or parts in the music sharing the same rhythm.
>
>So what does the sentence,
>
>"I've never heard a voice in the music that was triads, Paul."
>
>mean. You haven't heard parallel triads? Me either!

I've never heard a voice that played triads, one after the
other. -C.

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

> Paul:
> > It should be quite straightforward to prove. How could you tell
> > whether 50:49 produces torsion or not in an octave-invariant
> > formulation?
>
> Do you care about it being [dis]proven, then?

Sure, but this works pretty well as a "proof" for me.

> I expect your algorithm for
> generating periodicity blocks will solve everything.

How so? It doesn't detect torsion . . . I needed Gene's fix, which
takes the powers of 2 into account, to do that.

> But I haven't looked
> it up because people keep saying they aren't interested, while
>asking more
> and more questions.

Looked what up?

> It won't change anything musically.

Not sure what you mean.
>
>
> Paul:
> > I thought Gene showed that the common-factor rule only works in
the
> > octave-specific case.
>
> I don't remember him considering the adjoint, rather than the wedge
> product. But we may not need it anyway.

Gene, any enlightenment?
>
> Me:
> > > Pairs of ETs with
> > > torsion don't work with wedge products either. It may be that
the
> > sign of the
> > > mapping can be used to disambiguate them. Otherwise, give the
> > range of generators
> > > as part of the definition.
>
> Paul:
> > You've lost me. Gene, any comments?
>
> Meaning contorsion here. The octave-specific wedge product can
remove it,
> but not use it. An octave-equivalent wedge product (the octave-
equivalent
> mapping) will treat such systems, wrongly, as requiring a division
of the
> octave. But starting from ETs it does make more sense to use
> octave-specific vectors in the first place. Perhaps we should only
ask if
> unison vectors can work in an octave-equivalent system, in which
case this
> problem doesn't apply.
>
> Me:
> > > Wouldn't it be nice to say whether or not Fokker's methods
would
> > have worked if he
> > > had run into torsion?
>
> Paul:
> > I'm pretty sure the answer is no. Gene?
>
> The main thing we've added to Fokker (after Wilson) is the mapping,
> instead of merely counting the number of notes in the periodicity
block.
> The Monz-shruti example gives a periodicity block with more notes
than you
> need for the temperament, but the mappings still come out.

What do you mean, "the mappings still come out"? The 3:2, for
example, is not always represented by the same number of steps.

> There are more
> insidious examples of torsion where the mappings don't work
either. The
> problem being that octave-equivalent matrices don't differentiate
commatic
> torsion from systems requiring a period that isn't the octave.

Exactly my point.

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >> As in, part or parts in the music sharing the same rhythm.
> >
> >So what does the sentence,
> >
> >"I've never heard a voice in the music that was triads, Paul."
> >
> >mean. You haven't heard parallel triads? Me either!
>
> I've never heard a voice that played triads, one after the
> other. -C.

Fine. But I never mentioned triads in these discussions, only
intervals.

>>>>As in, part or parts in the music sharing the same rhythm.
>>>
>>>So what does the sentence,
>>>
>>>"I've never heard a voice in the music that was triads, Paul."
>>>
>>>mean. You haven't heard parallel triads? Me either!
>>
>>I've never heard a voice that played triads, one after the
>>other. -C.
>
>Fine. But I never mentioned triads in these discussions, only
>intervals.

You asked if it sounded triadic.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> You asked if it sounded triadic.

Well, we're going in circles.

Tune up a scale generated by 677-cent fifths, using only power-of-two
partials and a marimba-like decay, and play some Gamelan-like music
with it. Tell me if it sounds right to your ears.

>>You asked if it sounded triadic.
>
>Well, we're going in circles.
>
>Tune up a scale generated by 677-cent fifths, using only power-
>of-two partials and a marimba-like decay, and play some Gamelan-
>like music with it. Tell me if it sounds right to your ears.

(Midway between 16 and 23?) Okay, I'll do it!

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> (Midway between 16 and 23?)

(A little closer to 23.)