Graham on contorsion

This is what Graham says on his web site:

"Whether temperaments with contorsion should even be thought of as
temperaments is a matter of debate. They're really a way of
constructing a scale with a simpler temperament. One problem is that
there's always more than one qualitatively different generator for a
given linear temperament with contorsion. So the temperament isn't
uniquely determined by the mapping by period and generator."

Is there some reason why this is no longer applicable?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> This is what Graham says on his web site:
>
> "Whether temperaments with contorsion should even be thought of as
> temperaments is a matter of debate. They're really a way of
> constructing a scale with a simpler temperament. One problem is that
> there's always more than one qualitatively different generator for a
> given linear temperament with contorsion. So the temperament isn't
> uniquely determined by the mapping by period and generator."
>
> Is there some reason why this is no longer applicable?

Another way of looking at contorsion is that it is sometimes trying to
be a higher-limit temperament, so to speak. 68 et in the 3-limit is
<68 108|, divisible by 4, and in the 5-limit is <68 108 158|,
divisible by 2. So far it is a miserable mess, but in the 7-limit we
get <68 108 158 191| and suddenly there is a point to it all. Graham
gives the example of 5&19 for contorsion, but really 5&19 makes more
sense if you up the ante to the 7-limit and then you have hemifourths.
On the other hand, there is no guarantee that there is a decent higher
limit temperament lurking out there, if only it can be found.

Gene Ward Smith wrote:
> This is what Graham says on his web site:
> > "Whether temperaments with contorsion should even be thought of as
> temperaments is a matter of debate. They're really a way of
> constructing a scale with a simpler temperament. One problem is that
> there's always more than one qualitatively different generator for a
> given linear temperament with contorsion. So the temperament isn't
> uniquely determined by the mapping by period and generator."
> > Is there some reason why this is no longer applicable? It looks good to me, and shows why wedgies are inadequate for generating such scales.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
> > This is what Graham says on his web site:
> >
> > "Whether temperaments with contorsion should even be thought of as
> > temperaments is a matter of debate. They're really a way of
> > constructing a scale with a simpler temperament. One problem is that
> > there's always more than one qualitatively different generator for a
> > given linear temperament with contorsion. So the temperament isn't
> > uniquely determined by the mapping by period and generator."
> >
> > Is there some reason why this is no longer applicable?
>
> It looks good to me, and shows why wedgies are inadequate for
generating
> such scales.

It doesn't mention wedgies, so it can hardly do that. A wedgie is a
wedge product which has deliberately had the common factor taken out
if the gcd is greater than one; if you wanted to leave it in and
proceed on that basis you could, of course.
>
>
> Graham

Gene Ward Smith wrote:

> It doesn't mention wedgies, so it can hardly do that. A wedgie is a
> wedge product which has deliberately had the common factor taken out
> if the gcd is greater than one; if you wanted to leave it in and
> proceed on that basis you could, of course.

It doesn't mention wedgies, but does mention what wedgies are used to derive -- the mapping by (period and) generator.

But if you want wedge products, you can have wedge products. the 5&19 example comes from these two vals:

<5 8 12]
<19 30 44]

whose wedge product is

<<-2 -8 -8]]

The "scale with tempering" (if you don't want to call it a temperament) is:

5/24, 251.7 cent generator

basis:
(1.0, 0.20975898813907931)

mapping by period and generator:
[(1, 0), (2, -2), (4, -8)]

mapping by steps:
[(19, 5), (30, 8), (44, 12)]

highest interval width: 8
complexity measure: 8 (9 for smallest MOS)
highest error: 0.004480 (5.377 cents)
unique

You get a different MOS from the same family if you do 19&24 or 24&43.

A very similar scale can be found by combining 24 and 31:
16/55, 348.3 cent generator

basis:
(1.0, 0.29024101186092066)

mapping by period and generator:
[(1, 0), (1, 2), (0, 8)]

mapping by steps:
[(31, 24), (49, 38), (72, 56)]

highest interval width: 8
complexity measure: 8 (10 for smallest MOS)
highest error: 0.004480 (5.377 cents)
unique

Count back and this is the same as 17&7 with best 5-limit approximations:

7/24, 348.3 cent generator

basis:
(1.0, 0.29024101186092066)

mapping by period and generator:
[(1, 0), (1, 2), (0, 8)]

mapping by steps:
[(17, 7), (27, 11), (40, 16)]

highest interval width: 8
complexity measure: 8 (10 for smallest MOS)
highest error: 0.004480 (5.377 cents)
unique

That's generator of 7/24, which is different to the 5/24 of the previous scale. The original vals are

<24 38 56]
<31 49 72]

and their wedge product is

<<-2 -8 -8]]

which is exactly the same as the 5/24 scale above. So given a wedge product of <<-2 -8 -8]], how do you know if it should give the 5/24 family or the 7/24 family?

Graham

My rambling thoughts:

If the wedge product of the monzos (i.e., bimonzo) of two commas lead
to torsion, they lead to torsion; one should make a big deal about
this fact but not sweep it under the rug. Similarly, if the wedge
product of the breeds (i.e., cross-breed) of two ETs lead to
contorsion, they lead to contorsion; one should make a big deal about
this fact but not sweep it under the rug. It's fine to then define
the "wedgie" (why don't we call this the smith) as either of these
wedge products with the (con)torsion removed by dividing through by
gcd, and then insist that true temperaments correspond to a
wedgie/smith.

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

> which is exactly the same as the 5/24 scale above. So given a wedge
> product of <<-2 -8 -8]], how do you know if it should give the 5/24
> family or the 7/24 family?

The question is meaningless. It should give both if you want
contorsion, and neither if (like me) you don't.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> My rambling thoughts:
>
> If the wedge product of the monzos (i.e., bimonzo) of two commas lead
> to torsion, they lead to torsion; one should make a big deal about
> this fact but not sweep it under the rug. Similarly, if the wedge
> product of the breeds (i.e., cross-breed) of two ETs lead to
> contorsion, they lead to contorsion; one should make a big deal about
> this fact but not sweep it under the rug. It's fine to then define
> the "wedgie" (why don't we call this the smith) as either of these
> wedge products with the (con)torsion removed by dividing through by
> gcd, and then insist that true temperaments correspond to a
> wedgie/smith.

Up to a change in language (let's call that erlich) this is my point
of view. The fact that the gcd is not one signals torsion or
contorsion; note that fact and move on. If you want contorsion, you
can always produce it. No biggie.

Me:
>>which is exactly the same as the 5/24 scale above. So given a wedge >>product of <<-2 -8 -8]], how do you know if it should give the 5/24 >>family or the 7/24 family?

Gene:
> The question is meaningless. It should give both if you want
> contorsion, and neither if (like me) you don't.

The question is perfectly meaningful, and if you don't understand it you can't have been paying attention. Go back and re-read the message in which I asked it. I give a particular result for a pair of vals, and a different result for a different pair of vals. It isn't a mattor of "want contorsion" or "don't want contorsion" but wanting a *particular scale* with tempering, which happens to contain contorsion. Being given a list of scales that may be correct isn't good enough.

If you don't want contorsion, then you shouldn't be following this thread, which is about contorsion (and clearly says so in the subject line).

Graham

Paul Erlich wrote:
> My rambling thoughts:
> > If the wedge product of the monzos (i.e., bimonzo) of two commas lead > to torsion, they lead to torsion; one should make a big deal about > this fact but not sweep it under the rug. Similarly, if the wedge > product of the breeds (i.e., cross-breed) of two ETs lead to > contorsion, they lead to contorsion; one should make a big deal about > this fact but not sweep it under the rug. It's fine to then define > the "wedgie" (why don't we call this the smith) as either of these > wedge products with the (con)torsion removed by dividing through by > gcd, and then insist that true temperaments correspond to a > wedgie/smith.

But the two cases are different. There is at least one historically important instance of contorsion: Vicentino's enharmonic of 1555. Although he gives higher limit ratios (sometimes incorrect) the harmony is all 5-limit, but with an equal division of the chromatic semitone. I expect other "quartertone" composers work on similar principles. My program will give this system of any two 7, 24 and 31-equal (7 and 31 are 5-limit consistent, 24 is twice 12). These numbers are important because:

- 7 is the number of notes to the octave in staff notation. The alternative 5-limit contorted-meantone can't be written in staff notation with "quartertones".

- 24 is the number of notes to a quartertone scale. If you think in terms of a 12 note chromatic, this is the simplest subdivision that gives you quartertones, or neutral thirds, or whatever. Vicentino never invokes such a scale, but never gives examples that lie outside it. Also Example 48.3 of Book III (p.205 of the Maniates translation) includes a fourth descending through all quartertones and Example 46.1 (p.199) includes a note that is incorrect given his definition of the enharmonic, but belongs to the correct "quartertone" scale.

- 31 is the number of notes in Vicentino's tuning.

If you asked for 7&24 or 24&31 than you should expect to get Vicentino's system. If the program automatically removed contorsion, you would be rightly surprised to get a result that wasn't consistent with 24-equal.

If you asked for 7&31 in some other context, you may be surprised to get the quartertones coming back. The program can't really be sure which question you meant to ask. It could give you an uncontorted scale if neither of the inputs were contorted, but that means it won't give the right answer if you did want a microtonal system with 7 nominals consistent with 31-equal. So it currently returns the contorted result and leaves you to explicitly remove the contorsion (explicit is better than implicit).

Torsion of unison vectors is a different matter. I don't know of any cases, theoretical or otherwise, in which torsion is desired in a tempered MOS. I'm not even sure what it would mean. A periodicity block with torsion certainly doesn't correspond to an MOS with contorsion. Where you don't supply a chromatic unison vector, it's even more likely that you simply wanted a torsion free scale that tempers out all the commatic unison vectors, and that's what you get. Maybe if you supplied the chromatic unison vector, you would want to be warned of torsion. But you aren't. So there.

Graham

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

> The question is perfectly meaningful, and if you don't understand it
you
> can't have been paying attention. Go back and re-read the message in
> which I asked it.

My point was that it shouldn't give either family. If you read the
stuff I've been posting off and on about white and black keys, that
treats the question as one about scales which removes consideration of
temperament altogether. If you want to introduce temperament, then
starting from 5/24 and asking what temperament that seems to be can be
done, and at that point you might also enforce compatibilty with a
pair of vals if you wanted to. I really dislike the idea that a 5-val
and a 19-val in the 5-limit ought to produce a contorted
"temperament", but if you want to go that route you can do it using
wedgies if you like, since the information you deem so crucial is
found simply from the fact that 5 is about 1/4 of 19, and not (like 7
and 17) between 1/2 and 1/3.

> If you don't want contorsion, then you shouldn't be following this
> thread, which is about contorsion (and clearly says so in the
subject line).

I wrote the subject line, so presumably I know what it is about--which
were your remarks suggesting contorted "temperaments" were dubious at
best.

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

> If you asked for 7&24 or 24&31 than you should expect to get
Vicentino's
> system.

How do you figure that? 24+31=55, so you should expect to get a
55-equal system. The penultimate convergent for 31/24 is 22/17, so I
would expect to get (17+22)/(24+31) = 39/55. In the 5-limit that means
contorsion, and in the 7-limit, "Number 101", the temperament with
commas generated by 81/80 and 6144/6125, and which maps 7/64 to -11
half-fifth generator steps. 7&24 is similar, but not identical; the
penultimate convergent to 24/7 is 7/2, and (2+7)/(7+24) = 9/31; this
time instead of 2/"half-fifth" we get the half-fifth itself, and
instead of 55 we get 31; the comments on temperaments being the same.

If the program automatically removed contorsion, you would be
> rightly surprised to get a result that wasn't consistent with 24-equal.

Since 24 equal is contorted in the 5-limit and is a contorted meantone
system, why would you be surprised to get meantone? Anyway, any linear
combination of 12 and 19 should do; if I stick in
17<12 19 28| - 3<19 30 44| = <109 173 253| I would expect to learn it
is a meantone system. If I put it together with <31 49 72| I am happy
to learn it is meantone; I am not so interested to find it has
22-contorsion and 22 generators of 9/140 give me an 8/3, but I can
live with the information. As for scales, I think I am better ignoring
the vals looking at 9/140 directly. That would suggest a
microtemperament with 2401/2400 and 65625/65536 as commas in the 7-limit.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> > My rambling thoughts:
> >
> > If the wedge product of the monzos (i.e., bimonzo) of two commas
lead
> > to torsion, they lead to torsion; one should make a big deal
about
> > this fact but not sweep it under the rug. Similarly, if the wedge
> > product of the breeds (i.e., cross-breed) of two ETs lead to
> > contorsion, they lead to contorsion; one should make a big deal
about
> > this fact but not sweep it under the rug. It's fine to then
define
> > the "wedgie" (why don't we call this the smith) as either of
these
> > wedge products with the (con)torsion removed by dividing through
by
> > gcd, and then insist that true temperaments correspond to a
> > wedgie/smith.
>
> But the two cases are different. There is at least one
historically
> important instance of contorsion: Vicentino's enharmonic of 1555.

It's about equally meaningful to say that there are at least two
historically important instances of torsion: Helmholtz's schismic-24,
and Groven's schismic-36.

> Torsion of unison vectors is a different matter. I don't know of
any
> cases, theoretical or otherwise, in which torsion is desired in a
> tempered MOS. I'm not even sure what it would mean. A periodicity
> block with torsion certainly doesn't correspond to an MOS with
> contorsion.

As we've discussed before, tempering a torsional block results in an
nMOS wherever tempering its non-torsional equivalent would result in
an MOS -- such as the two schismic cases just mentioned.

> Where you don't supply a chromatic unison vector, it's even
> more likely that you simply wanted a torsion free scale that
tempers out
> all the commatic unison vectors, and that's what you get. Maybe if
you
> supplied the chromatic unison vector, you would want to be warned
of
> torsion. But you aren't. So there.

Sounds like you're assuming linear.