Ratio complexity question

Hi.

Ratio complexity can be expressed as a combination of prime numbers
and exponents.

Let's say that any power of two gives an octave identity and for
this reason is not to be considered in ratio complexity.

So 5/3 = 5^1 * 3^-1
and 15/8 = 5^1 * 3^1

A sixth and a seventh intervals seems to have the same compexity.
But in my experience this is wrong.

What was wrong?

Lorenzo

>Hi.

Hello Lorenzo!

>Ratio complexity can be expressed as a combination of prime numbers
>and exponents.

There are also other ways to express complexity, such as with the
sum of the numerator and denominator...

5/3 = 5 + 3 = 8.

Or their product...

5/3 = 5 * 3 = 15.

This last one (the product) has been a favorite around here.

>Let's say that any power of two gives an octave identity and for
>this reason is not to be considered in ratio complexity.

We would say you are looking for an "octave-equivalent" complexity
measure. Neither of the measures I mention above qualify.

>So 5/3 = 5^1 * 3^-1
>and 15/8 = 5^1 * 3^1
>
>A sixth and a seventh intervals seems to have the same compexity.
>But in my experience this is wrong.
>
>What was wrong?

The prime-factors measure may be wrong. However, all octave-
equivalent measures (that ignore 2) will have some problems.
The most common one is "odd-limit", which is simply the largest
odd number in the ratio...

5/3 = 5
6/5 = 5
15/8 = 15
16/15 = 15

...does 5/3 = 6/5 seem like a problem?

16/5 = 15/8 is a problem, but this is only because 16/15 is so small.
Most toy complexity measures run into this problem, which we call
SPAN.

Another problem is TOLERANCE, for example...

3/2 = low
30001/20001 = high

...though clearly 30001/20001 approximates 3/2.

I hope this is helpful. I'm sorry I don't speak a word of Italian!

-Carl

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
wrote:
> Hi.
>
> Ratio complexity can be expressed as a combination of prime numbers
> and exponents.
>
> Let's say that any power of two gives an octave identity and for
> this reason is not to be considered in ratio complexity.
>
> So 5/3 = 5^1 * 3^-1
> and 15/8 = 5^1 * 3^1
>
> A sixth and a seventh intervals seems to have the same compexity.
> But in my experience this is wrong.
>
> What was wrong?

There are at least two possible ways of answering this question.

The most straightforward is that powers of two are *not* to be
ignored when calculating ratio complexity. They may represent
equivalence along a chroma dimension but certainly can matter when it
comes to smoothness. If you compare 12/7 or 24/7 vs. 7/3 I think
you'll hear it.

So the correct comparison would be between

5/3 = 5^1 * 3^-1
and
15/8 = 5^1 * 3^1 * 2^-3

and 15/8 looks more complex, as it should.

The second way is to follow Harry Partch and Kees van Prooijen --
I've explained this method in detail before. Here, octave equivalence
is indeed considered to apply fully -- admittedly, a simplification,
which is made to aid the composer's conceptualizing. But the
complexity is then given by the largest *odd* factor in either the
numerator or denominator. So 5/3 is considered a "ratio of 5" by
Partch, while 15/8 is considered a "ratio of 15". And again, 15/8
looks more complex, as it should.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> 16/5 = 15/8 is a problem, but this is only because 16/15 is so
small.
> Most toy complexity measures run into this problem, which we call
> SPAN.

Once again, I totally disagree with this assessment/definition of
SPAN. But we discussed this fairly recently so I won't rehash this
now. You might want to look back at the last exchange on this,
though, Carl.

>> 16/5 = 15/8 is a problem, but this is only because 16/15 is so
>> small. Most toy complexity measures run into this problem,
>> which we call SPAN.
>
>Once again, I totally disagree with this assessment/definition of
>SPAN. But we discussed this fairly recently so I won't rehash this
>now. You might want to look back at the last exchange on this,
>though, Carl.

I still totally disagree with your disagreement.

-Carl

Hi Carl!

Don't worry about your Italian... :-)

> We would say you are looking for an "octave-equivalent" complexity
> measure. The most common one is "odd-limit", which is simply the
> largest odd number in the ratio...

This kind of measure seems related to the fact that in the first ten
odd numbers eight are primes. But with 9 and 15 this don't works.
To me is difficult accept that 9/4 is more complex than 7/5.
Or that 15/12 is more complex than 13/11.

Instead I prefer to consider the circle of fifths as a measure of
ratio complexity.
It begins with 2-limits note: 1/1 or 2/1, C.
Then, from each side, go further with 3-limits notes: 3/2, 9/8 and
4/3, 16/9.
Then 5-limits notes: 5/3, 5/4, 15/8 and 6/5, 8/5, 16/15.
Finally with 7-limits: 7/5 (or 10/7).
In this way it's possible going ahead until 13-limits notes (if I
remeber well).
I think harmony's history is slowly going down on this circle.

Each passage to a new prime produces a "comma". Pitagoric between 2
and 3, syntonic between 3 and 5 and eptatonic between 5 and 7 (the
first comma between 1 and 2 is an octave). Commas apart any new ratio
is obtained just with the powers of three (and the rearrangement in
the same octave with powers of two).

A passage to a new prime happens when ratio complexity with previous
primes is too large to be substained. For example 5/3 sound more
natural than 27/16, and 7/5 more than 45/32.
Note that any passage to a new prime produces the ratio of two
primes: 2/1, 3/2, 5/3, 7/5.
These are the simplest ratios between two primes: p^1 * q^-1.

What do you think about it?
Too romantic...?

Ciao

Lorenzo

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
wrote:
> Hi Carl!
>
> Don't worry about your Italian... :-)
>
>
> > We would say you are looking for an "octave-equivalent" complexity
> > measure. The most common one is "odd-limit", which is simply the
> > largest odd number in the ratio...
>
>
> This kind of measure seems related to the fact that in the first
ten
> odd numbers eight are primes.

¡Hello from España!
No, I don´t think the number of primes has anything to do with it.

> But with 9 and 15 this don't works.
> To me is difficult accept that 9/4 is more complex than 7/5.
> Or that 15/12 is more complex than 13/11.

15/12 is not a real ratio -- it reduces to 5/4. So yes, "15/12" is
much simpler, as the numbers 5 and 4 are much smaller than 13 and 11.

As for the other comparison, I´d say it´s close, because 9·4=36, and
7·5=35. And indeed, after much experience with _pure_ 7/5 ratios,
they don´t sound highly discordant at all -- certainly comparable
with 9/4 to my ears.

> Instead I prefer to consider the circle of fifths as a measure of
> ratio complexity.
> It begins with 2-limits note: 1/1 or 2/1, C.
> Then, from each side, go further with 3-limits notes: 3/2, 9/8 and
> 4/3, 16/9.

¿What prevents you from going on to 32/27 and 27/16?

> Then 5-limits notes: 5/3, 5/4, 15/8 and 6/5, 8/5, 16/15.

¿What prevents you from going on to 25/16, 25/24, 45/32, etc.?

If I compare _pure_ (not 12-equal) 5/3 or 5/4 vs. 9/8 or 16/9,
there´s no question that the ratios of 5 sound simpler.

> Finally with 7-limits: 7/5 (or 10/7).

¿What about 7/4, 7/6, 9/7, 21/16, etc.?

> In this way it's possible going ahead until 13-limits notes (if I
> remeber well).

It doesn´t look like there´s a clear rule or pattern to what you´re
putting forth here.

> I think harmony's history is slowly going down on this circle.

In 12-equal, you can´t even distinguish 9/5 from 16/9, for example --
so whatever ´circle´ you´re referring to with JI ratios can´t really
apply to the Western musical system.

And as far as the circle of fifths, two steps (i.e., major seconds
and minor sevenths) were certainly not accepted as stable harmonies
for centuries while major and minor thirds and sixths were. So even
that part of your theory doesn´t seem to hold up.

> Each passage to a new prime produces a "comma". Pitagoric between 2
> and 3, syntonic between 3 and 5 and eptatonic between 5 and 7

¿What´s that? ¿50/49? ¿In what sense is that "eptatonic"?

> (the
> first comma between 1 and 2 is an octave).

There are far more commas than this, even if you don´t go to higher
primes. 32805/32768 is one of the more important ones with all primes
5 and below, while prime 7 introduces such commas as 64/63, 126/125,
225/224, 2401/2400, and 4375/4374, to name but a few.

> Commas apart any new ratio
> is obtained just with the powers of three (and the rearrangement in
> the same octave with powers of two).

Not sure what this means.

>
> A passage to a new prime happens when ratio complexity with
previous
> primes is too large to be substained. For example 5/3 sound more
> natural than 27/16, and 7/5 more than 45/32.

Oh, I guess this explains what I was confused about above. ¿But here,
you´re basically assuming a 12-equal grid that everything snaps to,
yes? ¿Since this is the alternative tuning list, I wonder if you
actually might have some other tuning system than 12-equal in mind?
It isn´t quite clear from what you write.

Ciao.

>As for the other comparison, I´d say it´s close, because 9·4=36, and
>7·5=35. And indeed, after much experience with _pure_ 7/5 ratios,
>they don´t sound highly discordant at all -- certainly comparable
>with 9/4 to my ears.

This is not true to my ears but I understand that this could be
subjective. I think is less subjective that the chord: 1/1, 3/2, 9/4
is much less complex than any other chords which includes 7/5.

In any case is strange that 9/4 has to be considered more complex
than 7/4. This contraddicts the "limit" concept which is normally
assumed in tuning discussion. This implies a clear choice between
prime or odd numbers as measure of complexity.

>15/12 is not a real ratio -- it reduces to 5/4. So yes, "15/12" is
>much simpler, as the numbers 5 and 4 are much smaller than 13 and
11.

This means that odd numbers measure does'nt work with unreducted
ratios. This is strange since a ratio manteins his characteristics
even if not reducted. I think a good complexity measure would have
to work in these cases too.

This happens when you measure complexity as a combination of primes
and exponents:

9/4 = 2^-2 x 3^2
7/5 = 2^0 x 3^0 x 5^-1 x 7^1

15/12 = 2^-2 x 3^(1-1) x 5^1
13/11 = 2^0 x 3^0 x 5^0 x 7^0 x 11^-1 x 13^1

Complexity has to do with the "digestibility" of prime numbers.
To understand you can think to the growing rhythmic complexity
between a couple of notes, a triplet of them, a quintlet (not
sure this is the right word...), a septlet (?)....

Is not possible to subdivide a prime numbers in simpler parts. So
the ear has to "eat it" in one bite only. To me this is the base of
ratio complexity.

>If I compare _pure_ (not 12-equal) 5/3 or 5/4 vs. 9/8 or 16/9,
>there´s no question that the ratios of 5 sound simpler.

This is true but the cause is the ear perceptive model which implies
the critical bandwith: any intervals smaller than a minor third is
not well received by the ear. This don't have to do with
digestibility of prime numbers and ratio complexity.
This is not relative to the object of perception but to the subject.
Not to what you ear but to your ear.

>¿What prevents you from going on to 32/27 and 27/16?
>¿What prevents you from going on to 25/16, 25/24, 45/32, etc.?
>¿What about 7/4, 7/6, 9/7, 21/16, etc.?

I'm not saying that the intervals I purpose are the only possible.
I just think that these derives one from the other in a natural way
and that this has, at the same time, an harmonic and melodic scope.

>> I think harmony's history is slowly going down on this circle.
>
>In 12-equal, you can´t even distinguish 9/5 from 16/9, for example
>so whatever ´circle´ you´re referring to with JI ratios can´t really
>apply to the Western musical system.

12-equal is the end of an historic path. Tunings born for an
harmonic use are developed on circle of fifths.
Pitagoric tuning (which includes also some of the intervals you
wrote) is based on 3-limit; then some of these intervals were
substituted by 5-limits intervals through a syntonic comma shift:
zarlino tuning for example. Meantone tunings are attempts to build a
good circle of fifths around a fundamental note respecting 5-limits
harmony. Circle of fifths represents the model that the ear has used
to reach 12-equal which is the base of our western system.

Is possible to go ahead with this cicle from the bottom, introducing
other ratios as 21/20 and 40/21, 11/7 and 14/11, 13/11 and 22/13.
Naturally this prosecution determines the use of other tempered
systems as 19-et for example.

>And as far as the circle of fifths, two steps (i.e., major seconds
>and minor sevenths) were certainly not accepted as stable harmonies
>for centuries while major and minor thirds and sixths were. So even
>that part of your theory doesn´t seem to hold up.

This is due to critical bandwith. Triads are the only chords without
seconds and sevenths (assuming octave doubles). From this derives
the development of thirds harmony. But, for example, a sixth with a
fifth is something more recent, even if these intervals are both
consonant in triadic harmony.

In any case I think that, so as 3/2 is the more ancient harmonic
principle after octave, 9/8 is the more ancient melodic principle.

>> Each passage to a new prime produces a "comma". Pitagoric between
>>2 and 3, syntonic between 3 and 5 and eptatonic between 5 and 7
>
>¿What´s that? ¿50/49? ¿In what sense is that "eptatonic"?

between 45/32 and 7/5 = 225/224: an eptatonic comma.

>There are far more commas than this, even if you don´t go to higher
>primes. 32805/32768 is one of the more important ones with all
>primes 5 and below, while prime 7 introduces such commas as 64/63,
>126/125, 225/224, 2401/2400, and 4375/4374, to name but a few.

I'm not saying that the commas I've showed are the only possible.
I just think that these derives one from the other in a natural way
and that this has, at the same time, an harmonic and melodic scope.

>> Commas apart any new ratio
>> is obtained just with the powers of three (and the rearrangement
>> in the same octave with powers of two).
>
>Not sure what this means.

Circle of fifths represents the powers of number three shifted in
some points of a comma.

>¿But here, you´re basically assuming a 12-equal grid that
everything snaps to,
>yes? ¿Since this is the alternative tuning list, I wonder if you
>actually might have some other tuning system than 12-equal in mind?
>It isn´t quite clear from what you write.

I'm trying to understand causes of 12-equal to better undestand how
to use other et systems as 19, 5, 31, 7....

Ciao!

Lorenzo

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
wrote:
> Ciao.
>
> >As for the other comparison, I´d say it´s close, because 9·4=36,
and
> >7·5=35. And indeed, after much experience with _pure_ 7/5 ratios,
> >they don´t sound highly discordant at all -- certainly comparable
> >with 9/4 to my ears.
>
> This is not true to my ears but I understand that this could be
> subjective. I think is less subjective that the chord: 1/1, 3/2,
9/4
> is much less complex than any other chords which includes 7/5.

Hi Lorenzo!

But that's not a fair comparison.

Any such chord will have *two* 3:2s in it!!! So of course it will
show up as much simpler than any chord which includes 7/5! But chord
complexity and interval complexity are two different things. Each
interval, and their combined effect, must be included in an
evaluation of chord complexity.

> In any case is strange that 9/4 has to be considered more complex
> than 7/4. This contraddicts the "limit" concept which is normally
> assumed in tuning discussion. This implies a clear choice between
> prime or odd numbers as measure of complexity.

I disagree with this assessment. The "prime limit" concept is
extremely useful and fruitful within considerations of all kinds,
without impinging on the validity of a product measure of dyadic
complexity.

> >15/12 is not a real ratio -- it reduces to 5/4. So yes, "15/12" is
> >much simpler, as the numbers 5 and 4 are much smaller than 13 and
> 11.
>
> This means that odd numbers measure does'nt work with unreducted
> ratios. This is strange since a ratio manteins his characteristics
> even if not reducted.

I don't know what you mean.

> I think a good complexity measure would have
> to work in these cases too.

It works exactly as it should, since 15/12 and 5/4 are identical.

> This happens when you measure complexity as a combination of primes
> and exponents:
>
> 9/4 = 2^-2 x 3^2
> 7/5 = 2^0 x 3^0 x 5^-1 x 7^1
>
> 15/12 = 2^-2 x 3^(1-1) x 5^1
> 13/11 = 2^0 x 3^0 x 5^0 x 7^0 x 11^-1 x 13^1
>
> Complexity has to do with the "digestibility" of prime numbers.
> To understand you can think to the growing rhythmic complexity
> between a couple of notes, a triplet of them, a quintlet (not
> sure this is the right word...), a septlet (?)....

I don't think the rhythmic analogy holds in the pitch sphere when it
comes to dyadic complexity. Complexity of larger musical units (such
as chords or even entire infinite tuning systems) has to be
considered as a separate and larger problem, and there are indeed
fruitful lines of investigation in these regards on these lists,
especially the tuning-math and harmonic_entropy lists.

> Is not possible to subdivide a prime numbers in simpler parts. So
> the ear has to "eat it" in one bite only. To me this is the base of
> ratio complexity.

A third note in a chord provides some "juice" that helps swallow
complex composite intervals. Without such additional notes, the
factorability is no aid whatsoever, and the interval dry as toast.

> >If I compare _pure_ (not 12-equal) 5/3 or 5/4 vs. 9/8 or 16/9,
> >there´s no question that the ratios of 5 sound simpler.
>
> This is true but the cause is the ear perceptive model which
implies
> the critical bandwith: any intervals smaller than a minor third is
> not well received by the ear.

This is only true for sine waves. And it's certainly not true that
16/9, for example, is afflicted with a special critical band penalty.
Have you performed any actual calculations of critical band roughness
for variout intevals for any given timbre? Or seen Kameoka &
Kuriyagawa or Sethares's results in this area?

> >> I think harmony's history is slowly going down on this circle.
> >
> >In 12-equal, you can´t even distinguish 9/5 from 16/9, for example
> >so whatever ´circle´ you´re referring to with JI ratios can´t
really
> >apply to the Western musical system.
>
> 12-equal is the end of an historic path. Tunings born for an
> harmonic use are developed on circle of fifths.

Not always. Exceptions include Miracle, Pajara, Blackwood, Porcupine,
etc.

> Pitagoric tuning (which includes also some of the intervals you
> wrote) is based on 3-limit; then some of these intervals were
> substituted by 5-limits intervals through a syntonic comma shift:
> zarlino tuning for example.

Zarlino's supposed JI tuning for the diatonic scale is not a true
reflection of much in western musical history.

> Meantone tunings are attempts to build a
> good circle of fifths around a fundamental note respecting 5-limits
> harmony.

Yes, and it seems to be incredibly attractive -- by my calculations,
far more effective harmonically in the 5-limit than any other system
which hides a comma. But there are other, if dimmer, stars in that
sky.

> Circle of fifths represents the model that the ear has used
> to reach 12-equal which is the base of our western system.

One can also derive 12-equal by combining the elimination of, not the
Pythagorean (531441:524288) and syntonic (81:80) commas, but
additionally many other pairs, such as 128:125 and 648:625, for
example. This has much relevance in the music of Schubert, Liszt, etc.

Other temperaments which may not derive entirely from single-cycle-of-
fifths systems are also expressible in terms of the elimination of
simple commas. This opens the door to non-western-sounding music
which is still harmonically based, as Joseph Pehrson, Herman Miller,
Gene Ward Smith and others have been showing with their music.

> >And as far as the circle of fifths, two steps (i.e., major seconds
> >and minor sevenths) were certainly not accepted as stable harmonies
> >for centuries while major and minor thirds and sixths were. So even
> >that part of your theory doesn´t seem to hold up.
>
> This is due to critical bandwith.

This is a misassessment of the critical band phenomenon. One needs to
proceed a little more carefully to determine its effect, and its
result on dyads is, as calculated in many published papers and books,
related to the *size* of the numbers in the reduced ratios, and not
at all to factorability, when one is comparing intervals tunable by
eliminating critical band roughness.

> Triads are the only chords without
> seconds and sevenths (assuming octave doubles). From this derives
> the development of thirds harmony.

If you mean triads, then I agree that you have reached a somewhat
valid conclusion. But timbre, and most of all musical style, are huge
variables in this assessment.

> But, for example, a sixth with a
> fifth is something more recent, even if these intervals are both
> consonant in triadic harmony.

That's largely because of that complex *second* existing in the
chord. It's the complexity of *all* the dyads, not just some of them,
that contributed to chordal complexity.

> In any case I think that, so as 3/2 is the more ancient harmonic
> principle after octave, 9/8 is the more ancient melodic principle.

4/3 more ancient still, I think. Meanwhile, melodic seconds of all
sizes are found throughout the world's musics, and 9/8 are favored in
comparatively few of them.

> >> Each passage to a new prime produces a "comma". Pitagoric
between
> >>2 and 3, syntonic between 3 and 5 and eptatonic between 5 and 7
> >
> >¿What´s that? ¿50/49? ¿In what sense is that "eptatonic"?
>
> between 45/32 and 7/5 = 225/224: an eptatonic comma.

Interesting name. Of course this comma has been the center of a huge
amount of discussion. Where did you get the name?

> >There are far more commas than this, even if you don´t go to higher
> >primes. 32805/32768 is one of the more important ones with all
> >primes 5 and below, while prime 7 introduces such commas as 64/63,
> >126/125, 225/224, 2401/2400, and 4375/4374, to name but a few.
>
> I'm not saying that the commas I've showed are the only possible.
> I just think that these derives one from the other in a natural way
> and that this has, at the same time, an harmonic and melodic scope.

So do many non-diatonic systems, as I'll show you after about 5 days.

> >> Commas apart any new ratio
> >> is obtained just with the powers of three (and the rearrangement
> >> in the same octave with powers of two).
> >
> >Not sure what this means.
>
> Circle of fifths represents the powers of number three shifted in
> some points of a comma.

So one of the terms in what you would consider a "comma" has to be
a "Pythagorean" term? This is similar to Dave Keenan and George
Secor's "notational" commas, but I would argue that those don't have
to be the only relevant ones for music.

> >¿But here, you´re basically assuming a 12-equal grid that
> everything snaps to,
> >yes? ¿Since this is the alternative tuning list, I wonder if you
> >actually might have some other tuning system than 12-equal in mind?
> >It isn´t quite clear from what you write.
>
> I'm trying to understand causes of 12-equal to better undestand how
> to use other et systems as 19, 5, 31, 7....

Great -- I think in 5 days I will be able to help you a lot!

It's a real pleasure talking to you. Thanks for making this list a
much more interesting place!

Cheers,
Paul

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> Yes, and it seems to be incredibly attractive -- by my calculations,
> far more effective harmonically in the 5-limit than any other system
> which hides a comma. But there are other, if dimmer, stars in that
> sky.

I'll bite--how do you compare "harmonic effectiveness", and why does
81/80 come out better than, for instance, 32805/32768, 15625/15552 or
25/24?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > Yes, and it seems to be incredibly attractive -- by my
calculations,
> > far more effective harmonically in the 5-limit than any other
system
> > which hides a comma. But there are other, if dimmer, stars in
that
> > sky.
>
> I'll bite--how do you compare "harmonic effectiveness", and why does
> 81/80 come out better than, for instance, 32805/32768, 15625/15552
or
> 25/24?

It's what we've been discussing on tuning-math. Perhaps I'm biased by
the observation that the great majority of the world's music uses
musical scales/tunings with roughly 4 to 12 notes per octave within
any section (not counting small, highly variable inflections). Seldom
more than 9 unless you count passing tones. Computer music that uses
many more pitches is certainly a valid artistic path but, then, why
not free (possibly harmonizing) glissandi instead of a set of fixed
pitches? For some reason, the human musical art has largely chosen
the "stepping stone" path across the compositional river, to
paraphrase Johnny.

Meanwhile, the 25/24 system looks appealing, but I don't know of
examples in world music where the related scales, such as 3 or 4
notes in a chain of neutral thirds and octave repetition, can be
found. Also, the harmonic accuracy is very poor, at least by Western
standards, since the neutral third is called on to approximate both
5:4 and 6:5.

Really, I was trying to use a definition of "effectiveness" which was
appropriate to the thread in question, one rather closely involved
with the considerations surrounding western common practice music,
but which you know I've been considering in a generalized sense.

Hello...being new to the list, I don't have mastery yet over a good
deal of the discussion below, but was wondering if there is anything
unusual, and whether it might relate to 'limits' and powers', about
the following chord (one of my favorites, which I call a 'power
chord')--c-e-g-a-d'.

Aside from having a nicely ambiguous triadic strucure (regarding the
root, etc), I also like it because it is most simply constructed, not
by thirds, but by combining c-g-d' and e-a-d', which are,
respecitively, 3/2x3/2=9/4, and 4/3x4/3=16/9 -- all powers of 2, 3,
and 4. Doesn't this chord sound unusually bright? (I'll be
performing a difference tone analysis on it...)

: --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> --- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
> wrote:
> > Ciao.
> >
> > >As for the other comparison, I´d say it´s close, because 9·4=36,
> and
> > >7·5=35. And indeed, after much experience with _pure_ 7/5 ratios,
> > >they don´t sound highly discordant at all -- certainly comparable
> > >with 9/4 to my ears.
> >
> > This is not true to my ears but I understand that this could be
> > subjective. I think is less subjective that the chord: 1/1, 3/2,
> 9/4
> > is much less complex than any other chords which includes 7/5.
>
> Hi Lorenzo!
>
> But that's not a fair comparison.
>
> Any such chord will have *two* 3:2s in it!!! So of course it will
> show up as much simpler than any chord which includes 7/5! But
chord
> complexity and interval complexity are two different things. Each
> interval, and their combined effect, must be included in an
> evaluation of chord complexity.
>
> > In any case is strange that 9/4 has to be considered more complex
> > than 7/4. This contraddicts the "limit" concept which is normally
> > assumed in tuning discussion. This implies a clear choice between
> > prime or odd numbers as measure of complexity.
>
> I disagree with this assessment. The "prime limit" concept is
> extremely useful and fruitful within considerations of all kinds,
> without impinging on the validity of a product measure of dyadic
> complexity.
>
> > >15/12 is not a real ratio -- it reduces to 5/4. So yes, "15/12"
is
> > >much simpler, as the numbers 5 and 4 are much smaller than 13
and
> > 11.
> >
> > This means that odd numbers measure does'nt work with unreducted
> > ratios. This is strange since a ratio manteins his
characteristics
> > even if not reducted.
>
> I don't know what you mean.
>
> > I think a good complexity measure would have
> > to work in these cases too.
>
> It works exactly as it should, since 15/12 and 5/4 are identical.
>
> > This happens when you measure complexity as a combination of
primes
> > and exponents:
> >
> > 9/4 = 2^-2 x 3^2
> > 7/5 = 2^0 x 3^0 x 5^-1 x 7^1
> >
> > 15/12 = 2^-2 x 3^(1-1) x 5^1
> > 13/11 = 2^0 x 3^0 x 5^0 x 7^0 x 11^-1 x 13^1
> >
> > Complexity has to do with the "digestibility" of prime numbers.
> > To understand you can think to the growing rhythmic complexity
> > between a couple of notes, a triplet of them, a quintlet (not
> > sure this is the right word...), a septlet (?)....
>
> I don't think the rhythmic analogy holds in the pitch sphere when
it
> comes to dyadic complexity. Complexity of larger musical units
(such
> as chords or even entire infinite tuning systems) has to be
> considered as a separate and larger problem, and there are indeed
> fruitful lines of investigation in these regards on these lists,
> especially the tuning-math and harmonic_entropy lists.
>
> > Is not possible to subdivide a prime numbers in simpler parts. So
> > the ear has to "eat it" in one bite only. To me this is the base
of
> > ratio complexity.
>
> A third note in a chord provides some "juice" that helps swallow
> complex composite intervals. Without such additional notes, the
> factorability is no aid whatsoever, and the interval dry as toast.
>
> > >If I compare _pure_ (not 12-equal) 5/3 or 5/4 vs. 9/8 or 16/9,
> > >there´s no question that the ratios of 5 sound simpler.
> >
> > This is true but the cause is the ear perceptive model which
> implies
> > the critical bandwith: any intervals smaller than a minor third
is
> > not well received by the ear.
>
> This is only true for sine waves. And it's certainly not true that
> 16/9, for example, is afflicted with a special critical band
penalty.
> Have you performed any actual calculations of critical band
roughness
> for variout intevals for any given timbre? Or seen Kameoka &
> Kuriyagawa or Sethares's results in this area?
>
> > >> I think harmony's history is slowly going down on this circle.
> > >
> > >In 12-equal, you can´t even distinguish 9/5 from 16/9, for
example
> > >so whatever ´circle´ you´re referring to with JI ratios can´t
> really
> > >apply to the Western musical system.
> >
> > 12-equal is the end of an historic path. Tunings born for an
> > harmonic use are developed on circle of fifths.
>
> Not always. Exceptions include Miracle, Pajara, Blackwood,
Porcupine,
> etc.
>
> > Pitagoric tuning (which includes also some of the intervals you
> > wrote) is based on 3-limit; then some of these intervals were
> > substituted by 5-limits intervals through a syntonic comma shift:
> > zarlino tuning for example.
>
> Zarlino's supposed JI tuning for the diatonic scale is not a true
> reflection of much in western musical history.
>
> > Meantone tunings are attempts to build a
> > good circle of fifths around a fundamental note respecting 5-
limits
> > harmony.
>
> Yes, and it seems to be incredibly attractive -- by my
calculations,
> far more effective harmonically in the 5-limit than any other
system
> which hides a comma. But there are other, if dimmer, stars in that
> sky.
>
> > Circle of fifths represents the model that the ear has used
> > to reach 12-equal which is the base of our western system.
>
> One can also derive 12-equal by combining the elimination of, not
the
> Pythagorean (531441:524288) and syntonic (81:80) commas, but
> additionally many other pairs, such as 128:125 and 648:625, for
> example. This has much relevance in the music of Schubert, Liszt,
etc.
>
> Other temperaments which may not derive entirely from single-cycle-
of-
> fifths systems are also expressible in terms of the elimination of
> simple commas. This opens the door to non-western-sounding music
> which is still harmonically based, as Joseph Pehrson, Herman
Miller,
> Gene Ward Smith and others have been showing with their music.
>
> > >And as far as the circle of fifths, two steps (i.e., major
seconds
> > >and minor sevenths) were certainly not accepted as stable
harmonies
> > >for centuries while major and minor thirds and sixths were. So
even
> > >that part of your theory doesn´t seem to hold up.
> >
> > This is due to critical bandwith.
>
> This is a misassessment of the critical band phenomenon. One needs
to
> proceed a little more carefully to determine its effect, and its
> result on dyads is, as calculated in many published papers and
books,
> related to the *size* of the numbers in the reduced ratios, and not
> at all to factorability, when one is comparing intervals tunable by
> eliminating critical band roughness.
>
> > Triads are the only chords without
> > seconds and sevenths (assuming octave doubles). From this
derives
> > the development of thirds harmony.
>
> If you mean triads, then I agree that you have reached a somewhat
> valid conclusion. But timbre, and most of all musical style, are
huge
> variables in this assessment.
>
> > But, for example, a sixth with a
> > fifth is something more recent, even if these intervals are both
> > consonant in triadic harmony.
>
> That's largely because of that complex *second* existing in the
> chord. It's the complexity of *all* the dyads, not just some of
them,
> that contributed to chordal complexity.
>
> > In any case I think that, so as 3/2 is the more ancient harmonic
> > principle after octave, 9/8 is the more ancient melodic principle.
>
> 4/3 more ancient still, I think. Meanwhile, melodic seconds of all
> sizes are found throughout the world's musics, and 9/8 are favored
in
> comparatively few of them.
>
> > >> Each passage to a new prime produces a "comma". Pitagoric
> between
> > >>2 and 3, syntonic between 3 and 5 and eptatonic between 5 and 7
> > >
> > >¿What´s that? ¿50/49? ¿In what sense is that "eptatonic"?
> >
> > between 45/32 and 7/5 = 225/224: an eptatonic comma.
>
> Interesting name. Of course this comma has been the center of a
huge
> amount of discussion. Where did you get the name?
>
> > >There are far more commas than this, even if you don´t go to
higher
> > >primes. 32805/32768 is one of the more important ones with all
> > >primes 5 and below, while prime 7 introduces such commas as
64/63,
> > >126/125, 225/224, 2401/2400, and 4375/4374, to name but a few.
> >
> > I'm not saying that the commas I've showed are the only possible.
> > I just think that these derives one from the other in a natural
way
> > and that this has, at the same time, an harmonic and melodic
scope.
>
> So do many non-diatonic systems, as I'll show you after about 5
days.
>
> > >> Commas apart any new ratio
> > >> is obtained just with the powers of three (and the
rearrangement
> > >> in the same octave with powers of two).
> > >
> > >Not sure what this means.
> >
> > Circle of fifths represents the powers of number three shifted in
> > some points of a comma.
>
> So one of the terms in what you would consider a "comma" has to be
> a "Pythagorean" term? This is similar to Dave Keenan and George
> Secor's "notational" commas, but I would argue that those don't
have
> to be the only relevant ones for music.
>
> > >¿But here, you´re basically assuming a 12-equal grid that
> > everything snaps to,
> > >yes? ¿Since this is the alternative tuning list, I wonder if you
> > >actually might have some other tuning system than 12-equal in
mind?
> > >It isn´t quite clear from what you write.
> >
> > I'm trying to understand causes of 12-equal to better undestand
how
> > to use other et systems as 19, 5, 31, 7....
>
> Great -- I think in 5 days I will be able to help you a lot!
>
> It's a real pleasure talking to you. Thanks for making this list a
> much more interesting place!
>
> Cheers,
> Paul

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> Hello...being new to the list, I don't have mastery yet over a good
> deal of the discussion below, but was wondering if there is
anything
> unusual, and whether it might relate to 'limits' and powers', about
> the following chord (one of my favorites, which I call a 'power
> chord')--c-e-g-a-d'.

I've talked about this one chord a hell of a lot here (power chords
are usually defined as open fifths, though -- c-e-g-a-d is usually
called a "C 6/9" chord in pop and jazz). Tuning becomes a really
interesting question with this chord because it seems to call out for
temperament. That is, if you want the intervals c-e, e-g, and c-a to
be somewhat of 5-limit consonances, without severely breaking any of
the 3-limit consonances in the chord.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > > Yes, and it seems to be incredibly attractive -- by my
> calculations,
> > > far more effective harmonically in the 5-limit than any other
> system
> > > which hides a comma. But there are other, if dimmer, stars in
> that
> > > sky.
> >
> > I'll bite--how do you compare "harmonic effectiveness", and why does
> > 81/80 come out better than, for instance, 32805/32768, 15625/15552
> or
> > 25/24?
>
> It's what we've been discussing on tuning-math.

I don't recall any discussion about 81/80 uber alles on tuning-math.
Was I asleep?

Perhaps I'm biased by
> the observation that the great majority of the world's music uses
> musical scales/tunings with roughly 4 to 12 notes per octave within
> any section (not counting small, highly variable inflections). Seldom
> more than 9 unless you count passing tones. Computer music that uses
> many more pitches is certainly a valid artistic path but, then, why
> not free (possibly harmonizing) glissandi instead of a set of fixed
> pitches?

If you want to lead the way, have at it. Fixed pitches are valid and
important, however.

For some reason, the human musical art has largely chosen
> the "stepping stone" path across the compositional river, to
> paraphrase Johnny.

Computers are a recent development, and very few people have made use
of them for alternative tunings. Citing world music for the last 3000
years in this connection makes little sense to me.

> Really, I was trying to use a definition of "effectiveness" which was
> appropriate to the thread in question, one rather closely involved
> with the considerations surrounding western common practice music,
> but which you know I've been considering in a generalized sense.

I thought you had a specific, quantifiable definition in mind. Some
ways you might reasonably quantify the issue make schismic or hanson
come out better than meantone for 5-limit purposes, a fact to keep in
mind along with the thought that some people seem to find meantone not
close enough to JI to make them happy. Hanson and schismic are, of
course, far more accurate in their tuning, with 53 the happy medium
able to accomodate both. Hanson is also a 72-et system, which is a
consideration for some people. The question, it seems to me, is vastly
less cut and dried than you made it seem, and I just don't think it is
true that meantone dominates the 5-limit.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> > wrote:
> > >
> > > > Yes, and it seems to be incredibly attractive -- by my
> > calculations,
> > > > far more effective harmonically in the 5-limit than any other
> > system
> > > > which hides a comma. But there are other, if dimmer, stars in
> > that
> > > > sky.
> > >
> > > I'll bite--how do you compare "harmonic effectiveness", and why
does
> > > 81/80 come out better than, for instance, 32805/32768,
15625/15552
> > or
> > > 25/24?
> >
> > It's what we've been discussing on tuning-math.
>
> I don't recall any discussion about 81/80 uber alles on tuning-math.
> Was I asleep?

It's implied by the badness measures Dave and I put forth.

> For some reason, the human musical art has largely chosen
> > the "stepping stone" path across the compositional river, to
> > paraphrase Johnny.
>
> Computers are a recent development, and very few people have made
use
> of them for alternative tunings. Citing world music for the last
3000
> years in this connection makes little sense to me.

Many traditional instruments, such as voices, strings, and trombones,
are as free in pitch capability as computers. So it does make sense.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> It's implied by the badness measures Dave and I put forth.

Which no one but you and Dave have found much use for, so far as I
recall. I have been completely unconvinced, as you know, that those
measures are getting to anything of theoretical importance, and I
don't think Carl or Graham bought into it either, so proclaiming the
preeminance of 81/80 as a fact based on discussions from tuning-math
hardly makes sense.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > It's implied by the badness measures Dave and I put forth.
>
> Which no one but you and Dave have found much use for, so far as I
> recall. I have been completely unconvinced, as you know, that those
> measures are getting to anything of theoretical importance, and I
> don't think Carl or Graham bought into it either, so proclaiming the
> preeminance of 81/80 as a fact based on discussions from tuning-math
> hardly makes sense.

What does Herman think?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > It's implied by the badness measures Dave and I put forth.
>
> Which no one but you and Dave have found much use for, so far as I
> recall. I have been completely unconvinced, as you know, that those
> measures are getting to anything of theoretical importance, and I
> don't think Carl or Graham bought into it either,

Carl seems to be implying much the same thing about 81:80 here:

http://lumma.org/music/theory/tctmo/

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > > It's implied by the badness measures Dave and I put forth.
> >
> > Which no one but you and Dave have found much use for, so far as I
> > recall. I have been completely unconvinced, as you know, that those
> > measures are getting to anything of theoretical importance, and I
> > don't think Carl or Graham bought into it either,
>
> Carl seems to be implying much the same thing about 81:80 here:
>
> http://lumma.org/music/theory/tctmo/

It seems to me that what he is saying is, more or less, that 81/80 is
the smallest superparticular 5-limit comma. If you follow that logic
out to the bitter end, you'd get ennealimmal in the 7-limit. But I
presume Carl can explain what he really meant to say.

wallyesterpaulrus wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> >>--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> > > wrote:
> >>>It's implied by the badness measures Dave and I put forth.
>>
>>Which no one but you and Dave have found much use for, so far as I
>>recall. I have been completely unconvinced, as you know, that those
>>measures are getting to anything of theoretical importance, and I
>>don't think Carl or Graham bought into it either, so proclaiming the
>>preeminance of 81/80 as a fact based on discussions from tuning-math
>>hardly makes sense.
> > > What does Herman think?

There's no question that 81/80 is an important comma. You run across it pretty quickly when doing any kind of triadic harmony involving progressions of fifths, so it can be useful to sort temperaments according to what they do with 81/80 (some temperaments actually make it wider, and one version of 64-ET makes it go negative, for instance). You could also say that about 25/24 and 135/128 (I have a particular interest in 135/128), but their use is more limited, and the harmony produced by tempering out these commas is noticeably less smooth than meantone. 32805/32768 only starts being of interest with relatively large scales of 12 or more notes, and then you have to be careful to avoid progressions that run into the 81/80 (unless of course you're in 12-ET). 15625/15552 is a nice comma, and the 11-note scale that tempers it out has a lot of harmonic and melodic potential (you also tend not to run into the 81/80 problem with it so much, since root movement by fifths is limited when it takes 6 generators of the temperament to reach one fifth). Even some of the more exotic commas like 27/25 or 2187/2048 are not without their uses. Still, it's _really_ hard to beat meantone as a general-purpose 5-limit temperament. 2048/2025 is probably the best alternative; it also works well with root movement by fifths and the complexity of the scales is not too bad for the extra accuracy. Certainly it seems to have the advantage in the 7-limit.

(As far as 7-limit commas go, there are a number of nice choices; it's hard to decide between 50/49, 64/63, 126/125, and 225/224.)

>> Carl seems to be implying much the same thing about 81:80 here:
>>
>> http://lumma.org/music/theory/tctmo/
>
>It seems to me that what he is saying is, more or less, that 81/80 is
>the smallest superparticular 5-limit comma. If you follow that logic
>out to the bitter end, you'd get ennealimmal in the 7-limit. But I
>presume Carl can explain what he really meant to say.

I'm not sure what's being debated here. That 81:80 is the only
worthwhile 5-limit comma, or that it's the best 5-limit comma or...?

I had hoped my page was very clear. The relevant section is...

> 7.1-- You can see that the syntonic comma (81:80), which defines
> the meantone temperaments that have dominated Western music for
> hundreds of years, is one of the simplest 5-limit commas and is
> by far the smallest among the few most simple (on a list which
> is itself the result of searching 5-limit ratio space for commas
> with low badness).
>
> 7.2-- Another comma that looks good is the major diesis
> (648:625), which leads to the "diminished" temperament. If we
> take 8 tones/octave of this temperament, we get the octatonic
> scale of Stravinsky and Messiaen (not to mention jazz).
>
> 7.3-- However, while diminished makes a good showing, you can
> see that "porcupine" is better. Note, however...
//
>8-- This suggests that porcupine is a potentially fertile direction
>for new music.

...how can I make it clearer, or what am I being asked?

In friendship,

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> ...how can I make it clearer, or what am I being asked?

My guess, only that, is that you were saying that 81/80 is the
smallest superparticular 5-limit comma, among other things. Is that
correct?

>> ...how can I make it clearer, or what am I being asked?
>
>My guess, only that, is that you were saying that 81/80 is the
>smallest superparticular 5-limit comma, among other things. Is
>that correct?

I don't think I use the word superparticular, and I don't
know any formal relation between superparticulars and badness.

It would not be a stretch to claim I was saying 81:80 is
in some sense the best 5-limit comma.

-Carl

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> wallyesterpaulrus wrote:
>
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> >
> >>--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> >
> > wrote:
> >
> >>>It's implied by the badness measures Dave and I put forth.
> >>
> >>Which no one but you and Dave have found much use for, so far as I
> >>recall. I have been completely unconvinced, as you know, that
those
> >>measures are getting to anything of theoretical importance, and I
> >>don't think Carl or Graham bought into it either, so proclaiming
the
> >>preeminance of 81/80 as a fact based on discussions from tuning-
math
> >>hardly makes sense.
> >
> >
> > What does Herman think?
>
> There's no question that 81/80 is an important comma. You run
across it
> pretty quickly when doing any kind of triadic harmony involving
> progressions of fifths, so it can be useful to sort temperaments
> according to what they do with 81/80 (some temperaments actually
make it
> wider, and one version of 64-ET makes it go negative, for instance).

As does the usual version of 21-tET.

> Still, it's _really_ hard to beat meantone
> as a general-purpose 5-limit temperament. 2048/2025 is probably the
best
> alternative;

I'm pretty sure I agree with this, and I'm inclining toward calling
the latter temperament "srutal" (instead of "diaschismic"), since its
22-tone omitetrachordal scale is virtually identical with the Indian
Shruti scale.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> ...how can I make it clearer, or what am I being asked?
> >
> >My guess, only that, is that you were saying that 81/80 is the
> >smallest superparticular 5-limit comma, among other things. Is
> >that correct?
>
> I don't think I use the word superparticular, and I don't
> know any formal relation between superparticulars and badness.
>
> It would not be a stretch to claim I was saying 81:80 is
> in some sense the best 5-limit comma.
>
> -Carl

Thanks Carl. This is what Gene originally objected to.

hello all,

just a quick notice to let everyone know that
the Tonalsoft website now has a new look.

http://tonalsoft.com

we really were not ready to make the change yet,
but had to ... so folks surfing around in the
Encyclopaedia will find many pages which still
have broken links etc. ... i'm only just now
finishing the "C"s.

in particular, a lot of my longer essays on tuning
(under the "monzo" directory) are not yet uploaded.
please bear with us.

-monz

sorry about the duplication ... the last post had
the wrong subject line.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hello all,
>
>
> just a quick notice to let everyone know that
> the Tonalsoft website now has a new look.
>
> http://tonalsoft.com
>
>
> we really were not ready to make the change yet,
> but had to ... so folks surfing around in the
> Encyclopaedia will find many pages which still
> have broken links etc. ... i'm only just now
> finishing the "C"s.
>
> in particular, a lot of my longer essays on tuning
> (under the "monzo" directory) are not yet uploaded.
> please bear with us.
>
>
>
> -monz

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> > It would not be a stretch to claim I was saying 81:80 is
> > in some sense the best 5-limit comma.

> Thanks Carl. This is what Gene originally objected to.

No, you made a stronger and more controversial claim, and it is that
claim I objected to and object to. You said not just that it was
"incredibly attractive" but that your calculations showed it was "far
more effective harmonically in the 5-limit than any other system."
This claim is so misleading and tendentious you should retract it.
Your calculations were rigged in advance to make systems of that
degree of complexity look good, so you've fed in meantone, in effect,
at one end and had it come out the other. There is a relationship one
might call domination, whereby a comma is dominant if complexity and
error by some measure are such that any other comma must have a higher
figure for at least one of these. Any comma which has the property of
being dominant you can rig a calculation for and prove it is "best".
It would not be hard to "prove" schismic is best by this means, for
example.

Why not simply say what Carl and Herman have said, which is that it is
a really great comma and maybe in practice the best all-around bet? At
least leave off the calculations!

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > > It would not be a stretch to claim I was saying 81:80 is
> > > in some sense the best 5-limit comma.
>
> > Thanks Carl. This is what Gene originally objected to.
>
> No, you made a stronger and more controversial claim, and it is that
> claim I objected to and object to. You said not just that it was
> "incredibly attractive"

Isn't it?

> but that your calculations showed it was "far
> more effective harmonically in the 5-limit than any other system."

"Far" meaning "not slightly".

> This claim is so misleading and tendentious you should retract it.
> Your calculations were rigged in advance

By an understanding of human musical practice, yes.

> to make systems of that
> degree of complexity look good, so you've fed in meantone, in
effect,
> at one end and had it come out the other.
> There is a relationship one
> might call domination, whereby a comma is dominant if complexity and
> error by some measure are such that any other comma must have a
higher
> figure for at least one of these.

I call these "distinguished temperaments".

> Any comma which has the property of
> being dominant you can rig a calculation for and prove it is "best".

And so you have, very often. In your case, with arbitrary constraints.

> It would not be hard to "prove" schismic is best by this means, for
> example.
>
> Why not simply say what Carl and Herman have said, which is that it
is
> a really great comma and maybe in practice the best all-around bet?

That's exactly what I said, in different words.

> At
> least leave off the calculations!

What calculations?? There were none here indicated. All you have to
do is look at the chart (error vs. complexity chart of commas), and
*vaguely* scale the chart in a human-music-meaningful way. You'll see
meantone standing out clearly, no calculations required.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> > This claim is so misleading and tendentious you should retract it.
> > Your calculations were rigged in advance
>
> By an understanding of human musical practice, yes.

There's more to that than you seem to think.

> > Any comma which has the property of
> > being dominant you can rig a calculation for and prove it is "best".
>
> And so you have, very often. In your case, with arbitrary costraints.

I don't play the best comma game; please don't attribute the very
mistake I am objecting to to me. Can you be a little less sloppy with
your wild claims?

> > It would not be hard to "prove" schismic is best by this means, for
> > example.
> >
> > Why not simply say what Carl and Herman have said, which is that it
> is
> > a really great comma and maybe in practice the best all-around bet?
>
> That's exactly what I said, in different words.

No, it is not.

> > At
> > least leave off the calculations!
>
> What calculations?? There were none here indicated.

You *claimed* there were. Is the claim false?

All you have to
> do is look at the chart (error vs. complexity chart of commas), and
> *vaguely* scale the chart in a human-music-meaningful way. You'll see
> meantone standing out clearly, no calculations required.

Very vague. What I see is different, the distinguished commas are,
well, distinguished. Beyond that, what?

Congratulations, monz, on reaching this important
milestone!!

-Carl

>hello all,
>
>just a quick notice to let everyone know that
>the Tonalsoft website now has a new look.
>
>http://tonalsoft.com
>
>we really were not ready to make the change yet,
>but had to ... so folks surfing around in the
>Encyclopaedia will find many pages which still
>have broken links etc. ... i'm only just now
>finishing the "C"s.
>
>in particular, a lot of my longer essays on tuning
>(under the "monzo" directory) are not yet uploaded.
>please bear with us.
>
>-monz

hi Carl,

thanks much for the congrats. it's been a lot of work,
and things are really coming together now.

-monz

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Congratulations, monz, on reaching this important
> milestone!!
>
> -Carl
>
>
> >hello all,
> >
> >just a quick notice to let everyone know that
> >the Tonalsoft website now has a new look.
> >
> >http://tonalsoft.com
> >
> >we really were not ready to make the change yet,
> >but had to ... so folks surfing around in the
> >Encyclopaedia will find many pages which still
> >have broken links etc. ... i'm only just now
> >finishing the "C"s.
> >
> >in particular, a lot of my longer essays on tuning
> >(under the "monzo" directory) are not yet uploaded.
> >please bear with us.
> >
> >-monz

i want to publicly thank Jon Szanto for providing me
(a few years ago) with the "frames" template which i'm
now using on the Tonalsoft Encyclopaedia of Tuning.

http://tonalsoft.com/enc

thanks, Jon. :)

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> i want to publicly thank Jon Szanto for providing me
> (a few years ago) with the "frames" template which i'm
> now using on the Tonalsoft Encyclopaedia of Tuning.

It pains me to see that it was a non-musical contribution, but I
guess it beats getting poked in the eye with a sharp minor third.

Keep at it, Joe, and thanks for the nod.

Cheers,
Jon