Family trees of temperaments

Gene's page http://66.98.148.43/~xenharmo/commaseq.htm defines family relationships of temperaments, which tickled my fancy, so I've been matching up the temperaments in "A Middle Path" with their relatives. Here's what I have so far (dashed arrows for "illegitimate" children):

Bug -> Beep
Meantone -> Dominant
Meantone - - - -> Semaphore
Meantone -> Meantone Jr.
Meantone Jr. -> Huygens
Meantone Jr. -> Meanpop
Meantone -> Flattone
Meantone - - - -> Liese
Meantone - - - -> Mothra
Mothra -> Cynder
Meantone - - - -> Squares
Augmented -> August
Augmented -> Augene
Porcupine -> Porcupine Jr.
Porcupine - - - -> Nautilus
Blackwood -> Blacksmith
Dimipent -> Dimisept
Srutal -> Pajara
Magic -> Magic Jr.
Hanson -> Keemun
Negripent -> Negrisept
Superpyth -> Superpyth Jr.
Helmholtz -> Garibaldi
Sensipent -> Sensisept
Orson -> Orwell

There's also the unique case of Catler, which is kinda like a test-tube baby born of 12-EDO.

The rest of the seven-limit temperaments are "orphaned" in that I can't find their parents (probably because they're not particularly good five-limit temperaments):

? -> Injera
? -> Hedgehog
? -> Lemba
? -> Doublewide
? -> Beatles
? -> Myna
? -> Miracle

Going even further up the tree, they are all legitimate or illegitimate descendents of Grandfather Pythagorean, which is very satisfying in a way.

Corrections and additions would be much appreciated.

Keenan

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:
>
> Gene's page http://66.98.148.43/~xenharmo/commaseq.htm defines
family
> relationships of temperaments, which tickled my fancy, so I've been
matching up
> the temperaments in "A Middle Path" with their relatives. Here's
what I have so
> far (dashed arrows for "illegitimate" children):
>
> Bug -> Beep
> Meantone -> Dominant
> Meantone - - - -> Semaphore
> Meantone -> Meantone Jr.

I call Meantone Jr. "Huygens" if we're referring to temperament
classes and not merely tunings/horagrams.

> Meantone Jr. -> Huygens

Is Gene still using "Huygens" to refer to an 11-limit system? Absurd.

> Meantone Jr. -> Meanpop
> Meantone -> Flattone
> Meantone - - - -> Liese
> Meantone - - - -> Mothra
> Mothra -> Cynder

I don't understand. Shouldn't this be

> Meantone - - - -> Cynder
> Cynder -> Mothra

or something? Don't we go from 5-limit to 7-limit, and then from 7-
limit to 11-limit?

> Meantone - - - -> Squares
> Augmented -> August
> Augmented -> Augene
> Porcupine -> Porcupine Jr.
> Porcupine - - - -> Nautilus
> Blackwood -> Blacksmith
> Dimipent -> Dimisept
> Srutal -> Pajara
> Magic -> Magic Jr.
> Hanson -> Keemun
> Negripent -> Negrisept
> Superpyth -> Superpyth Jr.
> Helmholtz -> Garibaldi
> Sensipent -> Sensisept
> Orson -> Orwell
>
> There's also the unique case of Catler, which is kinda like a test-
tube baby
> born of 12-EDO.

This is exactly why there are problems with the 'family'
system/definition. George Secor also has a different definition
of 'family' that's being published. Despite all this, Monz has taken
the 'family' ball and run with it -- way too far, in my opinion.

> The rest of the seven-limit temperaments are "orphaned" in that I
can't find
> their parents (probably because they're not particularly good five-
limit
> temperaments):
>
> ? -> Injera
> ? -> Hedgehog
> ? -> Lemba
> ? -> Doublewide
> ? -> Beatles
> ? -> Myna
> ? -> Miracle

Right; it's easy to calculate the 5-limit commas of each using the
wedgie; and you can glean both error and complexity right from the
comma in the 2D, 5-limit case.

> Going even further up the tree, they are all legitimate or
illegitimate
> descendents of Grandfather Pythagorean, which is very satisfying in
a way.

Shouldn't there be different notions of "illegitimate" depending on
whether the period, the generator, or both are being divided into
several equal parts?

> Corrections and additions would be much appreciated.
>
> Keenan

Gene has discussed "null" temperaments on tuning-math; they may make
the whole family business a bit better-defined. Maybe we should move
the discussion there . . .

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

> I call Meantone Jr. "Huygens" if we're referring to temperament
> classes and not merely tunings/horagrams.

Septimal meantone is another name for it.

> > Meantone Jr. -> Huygens
>
> Is Gene still using "Huygens" to refer to an 11-limit system? Absurd.

I had to quit because you complained too much. Now, of course, it;s
been left without a name but gets discussed quite a bit anyway.

wallyesterpaulrus wrote:
[snip]
>>Meantone -> Meantone Jr.
> > > I call Meantone Jr. "Huygens" if we're referring to temperament > classes and not merely tunings/horagrams.

OK by me.

>> Meantone Jr. -> Huygens
> > > Is Gene still using "Huygens" to refer to an 11-limit system? Absurd.

You mean, it's absurd because Christiaan Huygens would never have used 11 limit intervals? I don't know anything about his work other than that he used 31-EDO.

What would be a good name for it, then?

>> Meantone Jr. -> Meanpop
>>Meantone -> Flattone
>>Meantone - - - -> Liese
>>Meantone - - - -> Mothra
>> Mothra -> Cynder
> > > I don't understand. Shouldn't this be
> > >>Meantone - - - -> Cynder
>> Cynder -> Mothra
> > > or something? Don't we go from 5-limit to 7-limit, and then from 7-
> limit to 11-limit?

Maybe I got it backwards. Did Gene get it backwards on his page? This is the first time I've heard of either of these.

[snip]
>>There's also the unique case of Catler, which is kinda like a test-
> > tube baby > >>born of 12-EDO.
> > > This is exactly why there are problems with the 'family' > system/definition. George Secor also has a different definition > of 'family' that's being published. Despite all this, Monz has taken > the 'family' ball and run with it -- way too far, in my opinion.

It's all in fun. Anyway it doesn't seem like a problem to me, just a special case. Catler has two commas which are both 5-limit, so it's descended from 12-EDO (basically 12-EDO with septimal "blue notes").

>>The rest of the seven-limit temperaments are "orphaned" in that I > > can't find > >>their parents (probably because they're not particularly good five-
> > limit > >>temperaments):
>>
>>? -> Injera
>>? -> Hedgehog
>>? -> Lemba
>>? -> Doublewide
>>? -> Beatles
>>? -> Myna
>>? -> Miracle
> > > Right; it's easy to calculate the 5-limit commas of each using the > wedgie; and you can glean both error and complexity right from the > comma in the 2D, 5-limit case.
> > >>Going even further up the tree, they are all legitimate or > > illegitimate > >>descendents of Grandfather Pythagorean, which is very satisfying in > > a way.
> > Shouldn't there be different notions of "illegitimate" depending on > whether the period, the generator, or both are being divided into > several equal parts?

Could you give an example of the period being divided into equal parts to get a higher limit? I can't think of any.

I think a good definition of a "legitimate" child temperament is one which has the same period and generator, but some interval is identified with a higher prime and the temperament is readjusted accordingly.

>>Corrections and additions would be much appreciated.
>>
>>Keenan
> > > Gene has discussed "null" temperaments on tuning-math; they may make > the whole family business a bit better-defined. Maybe we should move > the discussion there . . .

What's a null temperament? JI? 2-limit? =P

Keenan

Hi Paul and Keenan,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> This is exactly why there are problems with the 'family'
> system/definition. George Secor also has a different
> definition of 'family' that's being published. Despite
> all this, Monz has taken the 'family' ball and run with
> it -- way too far, in my opinion.

The Tonalsoft Encyclopedia has always been an "open" project,
with contributions from all members of the tuning community
encouraged.

If you have a problem with anything in it (and i know you
do, Paul), the best thing to do is simply change my HTML
code to eliminate errors, or to append your own comment
at the bottom, and send it to me. The less work i have to
do on any given webpage, the more likely that the new version
will appear quickly in the Encyclopedia.

-monz
http://tonalsoft.com
Tonescape microtonal music software

monz wrote:
> Hi Paul and Keenan,
> > > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >>This is exactly why there are problems with the 'family' >>system/definition. George Secor also has a different
>>definition of 'family' that's being published. Despite
>>all this, Monz has taken the 'family' ball and run with
>>it -- way too far, in my opinion.
> > > > > The Tonalsoft Encyclopedia has always been an "open" project,
> with contributions from all members of the tuning community
> encouraged.

Well, it says "Copyright... All rights reserved." at the bottom of every page. To me, that doesn't qualify as "open", but let's not quibble over language.

> If you have a problem with anything in it (and i know you
> do, Paul), the best thing to do is simply change my HTML
> code to eliminate errors, or to append your own comment
> at the bottom, and send it to me. The less work i have to
> do on any given webpage, the more likely that the new version
> will appear quickly in the Encyclopedia.
> > > > -monz
> http://tonalsoft.com
> Tonescape microtonal music software

Keenan

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

> > Meantone Jr. -> Meanpop
> > Meantone -> Flattone
> > Meantone - - - -> Liese
> > Meantone - - - -> Mothra
> > Mothra -> Cynder
>
> I don't understand. Shouldn't this be
>
> > Meantone - - - -> Cynder
> > Cynder -> Mothra
>
> or something? Don't we go from 5-limit to 7-limit, and then from 7-
> limit to 11-limit?

I have been taking cynder and mothra to be synonymous.

> > There's also the unique case of Catler, which is kinda like a test-
> tube baby
> > born of 12-EDO.

It's hardly unique; jamesbond, for example, is very much the same kind
of animal.

> This is exactly why there are problems with the 'family'
> system/definition.

But there isn't a problem. If you take the Hermite comma sequence as
your basis, catler is a decendent of compton, and the family
relationships are well-defined.

George Secor also has a different definition
> of 'family' that's being published.

What is it?

> Shouldn't there be different notions of "illegitimate" depending on
> whether the period, the generator, or both are being divided into
> several equal parts?

Sounds good.

> Gene has discussed "null" temperaments on tuning-math; they may make
> the whole family business a bit better-defined. Maybe we should move
> the discussion there . . .

That can be used to show another kind of relationship between
jamesbond and catler, which isn't a familty tree.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> ...
> This is exactly why there are problems with the 'family'
> system/definition. George Secor also has a different definition
> of 'family' that's being published. Despite all this, Monz has taken
> the 'family' ball and run with it -- way too far, in my opinion.

In the Miracle Temperament/Decimal Keyboard article you're referring
to, I was not attempting to establish a formal definition for the
term "family" as applied to temperaments, so I have no problem with
anyone doing this in a different way.

--George

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:
>
> wallyesterpaulrus wrote:
> [snip]
> >>Meantone -> Meantone Jr.
> >
> >
> > I call Meantone Jr. "Huygens" if we're referring to temperament
> > classes and not merely tunings/horagrams.
>
> OK by me.
>
> >> Meantone Jr. -> Huygens
> >
> >
> > Is Gene still using "Huygens" to refer to an 11-limit system?
Absurd.
>
> You mean, it's absurd because Christiaan Huygens would never have
>used

or recognized

> 11 limit
> intervals?

For one thing. As far as I know, he was the first to identify certain
augmented and diminished intervals in the then-current meantone
system with ratios of 7. So his name should go with "Meantone Jr."
or "Septimal Meantone" AFAIC.

>I don't know anything about his work other than that he used 31-EDO.
>
> What would be a good name for it, then?

Nothing jumps out; use your imagination.

> >> Meantone Jr. -> Meanpop
> >>Meantone -> Flattone
> >>Meantone - - - -> Liese
> >>Meantone - - - -> Mothra
> >> Mothra -> Cynder
> >
> >
> > I don't understand. Shouldn't this be
> >
> >
> >>Meantone - - - -> Cynder
> >> Cynder -> Mothra
> >
> >
> > or something? Don't we go from 5-limit to 7-limit, and then from
7-
> > limit to 11-limit?
>
> Maybe I got it backwards. Did Gene get it backwards on his page?
This is the
> first time I've heard of either of these.

I introduced the name 'Cynder' (inpired by the already extant names
Slendric and Wonder for compatible systems) for the 7-limit system in
an early version of my 'Middle Path' paper (it's still there), and
IIRC, Gene introduced 'Mothra' later. So I would have guessed the
latter would have to be 11-limit . . . (?)

> [snip]
> >>There's also the unique case of Catler, which is kinda like a
test-
> >
> > tube baby
> >
> >>born of 12-EDO.
> >
> >
> > This is exactly why there are problems with the 'family'
> > system/definition. George Secor also has a different definition
> > of 'family' that's being published. Despite all this, Monz has
taken
> > the 'family' ball and run with it -- way too far, in my opinion.
>
> It's all in fun. Anyway it doesn't seem like a problem to me, just
a special
> case. Catler has two commas which are both 5-limit, so it's
descended from
> 12-EDO (basically 12-EDO with septimal "blue notes").

Things get hairier in higher limits, and . . . (I'm sure I'll have
the opportunity to continue elsewhere)

> Could you give an example of the period being divided into equal
parts to get a
> higher limit? I can't think of any.

Well, if all 5-limit 2D temperaments are "children" of Pythagorean,
you already know quite a few examples, don't you?

> > Gene has discussed "null" temperaments on tuning-math; they may
make
> > the whole family business a bit better-defined. Maybe we should
move
> > the discussion there . . .
>
> What's a null temperament? JI? 2-limit? =P
>
> Keenan

Ask on tuning-math.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> Hi Paul and Keenan,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > This is exactly why there are problems with the 'family'
> > system/definition. George Secor also has a different
> > definition of 'family' that's being published. Despite
> > all this, Monz has taken the 'family' ball and run with
> > it -- way too far, in my opinion.
>
>
>
> The Tonalsoft Encyclopedia has always been an "open" project,
> with contributions from all members of the tuning community
> encouraged.
>
> If you have a problem with anything in it (and i know you
> do, Paul), the best thing to do is simply change my HTML
> code to eliminate errors, or to append your own comment
> at the bottom, and send it to me. The less work i have to
> do on any given webpage, the more likely that the new version
> will appear quickly in the Encyclopedia.
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

Thanks Monz. I appreciate your offer to accept corrections from me
again as you once did. But for this particular issue, just editing
individual webpages won't really amount to a solution, since there's
already quite a 'castle' of webpages erected on this 'family'
premise. I tried to speak up at the time they were being created, but
I didn't want to just come in and knock your whole castle down.

--- In tuning@yahoogroups.com, "paulerlich" <wallyesterpaulrus@y...>
wrote:

> Thanks Monz. I appreciate your offer to accept corrections from me
> again as you once did. But for this particular issue, just editing
> individual webpages won't really amount to a solution, since there's
> already quite a 'castle' of webpages erected on this 'family'
> premise. I tried to speak up at the time they were being created, but
> I didn't want to just come in and knock your whole castle down.

Just to add to the confusion, Graham wants to define the different
tunings of a single logical temperament to be a "family". Hence, there
isn't a meantone temperament, just a meantone family. Of course I
believe in one wedgie, one temperament.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> But there isn't a problem. If you take the Hermite comma sequence as
> your basis, catler is a decendent of compton, and the family
> relationships are well-defined.

I don't understand. In Compton, prime 5 is reached via -1 generator
(and 28 periods). In Catler, prime 5 is reached via 0 generators (and
28 periods). If the generator mappings to primes are different, isn't
this a total violation of what this "descendant" and "family"
business is supposed to be about? Based on Keenan Pepper's recent
discussion, it would seem so.

> George Secor also has a different definition
> > of 'family' that's being published.
>
> What is it?

I'll defer to George on that.

> > Gene has discussed "null" temperaments on tuning-math; they may
make
> > the whole family business a bit better-defined. Maybe we should
move
> > the discussion there . . .
>
> That can be used to show another kind of relationship between
> jamesbond and catler, which isn't a familty tree.

Hmm . . . so what is one, in your view?

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > ...
> > This is exactly why there are problems with the 'family'
> > system/definition. George Secor also has a different definition
> > of 'family' that's being published. Despite all this, Monz has
taken
> > the 'family' ball and run with it -- way too far, in my opinion.
>
> In the Miracle Temperament/Decimal Keyboard article you're
referring
> to, I was not attempting to establish a formal definition for the
> term "family" as applied to temperaments, so I have no problem with
> anyone doing this in a different way.
>
> --George

I was *not* referring to this article, George, but rather to your
statements posted to these (yahoo) groups. You made a specific
statement about this but I guess you've forgotten . . .

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "paulerlich" <wallyesterpaulrus@y...>
> wrote:
>
> > Thanks Monz. I appreciate your offer to accept corrections from me
> > again as you once did. But for this particular issue, just editing
> > individual webpages won't really amount to a solution, since
there's
> > already quite a 'castle' of webpages erected on this 'family'
> > premise. I tried to speak up at the time they were being created,
but
> > I didn't want to just come in and knock your whole castle down.
>
> Just to add to the confusion, Graham wants to define the different
> tunings of a single logical temperament to be a "family". Hence, there
> isn't a meantone temperament, just a meantone family. Of course I
> believe in one wedgie, one temperament.

I use the term "temperament class" to avoid confusion with a
fixed "tuning". But there are relationships among temperament classes,
which is what, for the rest of us, this "family" business is supposed
to be . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> > But there isn't a problem. If you take the Hermite comma sequence as
> > your basis, catler is a decendent of compton, and the family
> > relationships are well-defined.
>
> I don't understand. In Compton, prime 5 is reached via -1 generator
> (and 28 periods). In Catler, prime 5 is reached via 0 generators (and
> 28 periods).

The Hermite comma sequence in this case starts with a 3-limit comma,
the Pythagoren comma, instead of the usual 5-limit comma. Certainly,
all the rank two temperaments tempering out the Pythagorean comma
share strong similarities, so I don't see why it would be wrong to
consider it a family relationship.

> > That can be used to show another kind of relationship between
> > jamesbond and catler, which isn't a familty tree.
>
> Hmm . . . so what is one, in your view?

They are adjustment siblings, so to speak. In this case, they are both
adjusted versions of the null "temperament".

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

> I use the term "temperament class" to avoid confusion with a
> fixed "tuning".

My problem with that is that if the temperament is defined by the
wedgie, or by what the temperament tempers out, it isn't a class.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> >
> > > But there isn't a problem. If you take the Hermite comma
sequence as
> > > your basis, catler is a decendent of compton, and the family
> > > relationships are well-defined.
> >
> > I don't understand. In Compton, prime 5 is reached via -1
generator
> > (and 28 periods). In Catler, prime 5 is reached via 0 generators
(and
> > 28 periods).
>
> The Hermite comma sequence in this case starts with a 3-limit comma,
> the Pythagoren comma, instead of the usual 5-limit comma.

That doesn't clarify anything for me, nor do I have any idea why this
is the case.

> Certainly,
> all the rank two temperaments tempering out the Pythagorean comma
> share strong similarities, so I don't see why it would be wrong to
> consider it a family relationship.

It would be wrong to consider Catler a descendant of Compton as you
stated above, if (as a number of people have implied, probably
including you) that the descendant must reduce to the ancestor if the
highest prime is taken out of the mapping of the descendant. That
seemed to be the governing principle until now.

> > > That can be used to show another kind of relationship between
> > > jamesbond and catler, which isn't a familty tree.
> >
> > Hmm . . . so what is one, in your view?
>
> They are adjustment siblings, so to speak. In this case, they are
both
> adjusted versions of the null "temperament".

By "what is one" I meant "what is a family tree".

I guess we'll have to discuss null temperaments and the like on
tuning-math again at some point . . .

On 1/19/06, wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:
> > The Hermite comma sequence in this case starts with a 3-limit comma,
> > the Pythagoren comma, instead of the usual 5-limit comma.
>
> That doesn't clarify anything for me, nor do I have any idea why this
> is the case.

It makes sense to me. To get a 5-limit linear temperament from
Pythagorean we have to temper out one 5-limit comma. The Pythagorean
comma is a 5-limit comma which happens to be 3-limit as well, but we
can still temper it out and get Compton.

To get from Compton to a 7-limit linear temperament we have to temper
out one 7-limit comma. The syntonic comma is a 7-limit comma which
happens to be 5-limit as well, but again, we can temper it out and get
Catler.

The family tree goes Pythagorean -> Compton -> Catler.

> > Certainly,
> > all the rank two temperaments tempering out the Pythagorean comma
> > share strong similarities, so I don't see why it would be wrong to
> > consider it a family relationship.
>
> It would be wrong to consider Catler a descendant of Compton as you
> stated above, if (as a number of people have implied, probably
> including you) that the descendant must reduce to the ancestor if the
> highest prime is taken out of the mapping of the descendant. That
> seemed to be the governing principle until now.

But Catler does reduce to a form of Compton if you take out the
sevens. It reduces to 12-EDO, which is a degenerate form of Compton
("degenerate" meaning some of the notes happen to be tuned the same).

The reason it's degenerate is because the 7-limit comma we used
happened to be a 5-limit comma as well.

If you take out the fives from Compton you also get 12-EDO, but this
time it's a degenerate form of Pythagorean.

> > > > That can be used to show another kind of relationship between
> > > > jamesbond and catler, which isn't a familty tree.
> > >
> > > Hmm . . . so what is one, in your view?
> >
> > They are adjustment siblings, so to speak. In this case, they are
> both
> > adjusted versions of the null "temperament".
>
> By "what is one" I meant "what is a family tree".
>
> I guess we'll have to discuss null temperaments and the like on
> tuning-math again at some point . . .

Would someone mind telling me what "jamesbond" is? I can't find a
description of it anywhere.

Keenan

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > I use the term "temperament class" to avoid confusion with a
> > fixed "tuning".
>
> My problem with that is that if the temperament is defined by the
> wedgie, or by what the temperament tempers out, it isn't a class.

There's a class of tuning systems, with rather fuzzy boundaries around
it, that can be said to be examples of a given "temperament" by your
definition. The problem is that many people use "temperament" to mean a
specific tuning system. So, to avoid confusion, I say "temperament
class" when there's more than one possible specific tuning for it.

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:
>
> On 1/19/06, wallyesterpaulrus <wallyesterpaulrus@y...> wrote:
> > > The Hermite comma sequence in this case starts with a 3-limit
comma,
> > > the Pythagoren comma, instead of the usual 5-limit comma.
> >
> > That doesn't clarify anything for me, nor do I have any idea why
this
> > is the case.
>
> It makes sense to me. To get a 5-limit linear temperament from
> Pythagorean we have to temper out one 5-limit comma. The Pythagorean
> comma is a 5-limit comma which happens to be 3-limit as well, but we
> can still temper it out and get Compton.
>
> To get from Compton to a 7-limit linear temperament we have to
temper
> out one 7-limit comma. The syntonic comma is a 7-limit comma which
> happens to be 5-limit as well, but again, we can temper it out and
get
> Catler.
>
> The family tree goes Pythagorean -> Compton -> Catler.

This doesn't make sense to me. Comma sequences seem to be another
issue. Waage makes sense as a 7-limit descendant of Compton. Catler
doesn't, because it changes the way prime 5 is mapped.

> > > Certainly,
> > > all the rank two temperaments tempering out the Pythagorean
comma
> > > share strong similarities, so I don't see why it would be wrong
to
> > > consider it a family relationship.
> >
> > It would be wrong to consider Catler a descendant of Compton as
you
> > stated above, if (as a number of people have implied, probably
> > including you) that the descendant must reduce to the ancestor if
the
> > highest prime is taken out of the mapping of the descendant. That
> > seemed to be the governing principle until now.
>
> But Catler does reduce to a form of Compton if you take out the
> sevens. It reduces to 12-EDO, which is a degenerate form of Compton
> ("degenerate" meaning some of the notes happen to be tuned the
>same).

I don't think that should count.

> The reason it's degenerate is because the 7-limit comma we used
> happened to be a 5-limit comma as well.
>
> If you take out the fives from Compton you also get 12-EDO, but this
> time it's a degenerate form of Pythagorean.
>
> > > > > That can be used to show another kind of relationship
between
> > > > > jamesbond and catler, which isn't a familty tree.
> > > >
> > > > Hmm . . . so what is one, in your view?
> > >
> > > They are adjustment siblings, so to speak. In this case, they
are
> > both
> > > adjusted versions of the null "temperament".
> >
> > By "what is one" I meant "what is a family tree".
> >
> > I guess we'll have to discuss null temperaments and the like on
> > tuning-math again at some point . . .
>
> Would someone mind telling me what "jamesbond" is? I can't find a
> description of it anywhere.
>
> Keenan

Just came up on tuning-math . . .

Keenan Pepper wrote:

> Would someone mind telling me what "jamesbond" is? I can't find a
> description of it anywhere.

It _is_ somewhat an unfortunate name in that googling for "jamesbond temperament" returns all sorts of irrelevant hits. :)

Jamesbond is a regular temperament based on generators of approximately 172.78 cents and 86.69 cents (as one possible tuning), practically not much different from an octave stretched 14-note equal temperament. It has a relatively large error for a named temperament, comparable to dicot (a neutral-thirds temperament). It tempers out 25/24 and 81/80, so it could be considered a sort of warped diatonic scale with extra 7-limit notes in between; the tuning map is:

<7, 11, 16, 20], <0, 0, 0, -1]

On 1/19/06, wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:
> This doesn't make sense to me. Comma sequences seem to be another
> issue. Waage makes sense as a 7-limit descendant of Compton. Catler
> doesn't, because it changes the way prime 5 is mapped.

Is "Waage" the one where the intervals of 5 are one step off from
12-EDO and the intervals of 7 are two steps off?

> > But Catler does reduce to a form of Compton if you take out the
> > sevens. It reduces to 12-EDO, which is a degenerate form of Compton
> > ("degenerate" meaning some of the notes happen to be tuned the
> >same).
>
> I don't think that should count.

Well, it does seem pretty silly. But if that doesn't count, then
things like Blackwood and Compton shouldn't be considered descendents
of Pythagorean, because they are degenerate in the same way.

> > Would someone mind telling me what "jamesbond" is? I can't find a
> > description of it anywhere.
> >
> > Keenan
>
> Just came up on tuning-math . . .

Wow, Jamesbond sounds like an awful temperament! =P

Keenan

On 1/19/06, Herman Miller <hmiller@io.com> wrote:
> Keenan Pepper wrote:
>
> > Would someone mind telling me what "jamesbond" is? I can't find a
> > description of it anywhere.
>
> It _is_ somewhat an unfortunate name in that googling for "jamesbond
> temperament" returns all sorts of irrelevant hits. :)
>
> Jamesbond is a regular temperament based on generators of approximately
> 172.78 cents and 86.69 cents (as one possible tuning), practically not
> much different from an octave stretched 14-note equal temperament. It
> has a relatively large error for a named temperament, comparable to
> dicot (a neutral-thirds temperament). It tempers out 25/24 and 81/80, so
> it could be considered a sort of warped diatonic scale with extra
> 7-limit notes in between; the tuning map is:
>
> <7, 11, 16, 20], <0, 0, 0, -1]

Sounds awful. Has anyone actually written anything in this beast? =P

Keenan

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> Keenan Pepper wrote:
>
> > Would someone mind telling me what "jamesbond" is? I can't find a
> > description of it anywhere.
>
> It _is_ somewhat an unfortunate name in that googling
for "jamesbond
> temperament" returns all sorts of irrelevant hits. :)
>
> Jamesbond is a regular temperament based on generators of
approximately
> 172.78 cents and 86.69 cents (as one possible tuning), practically
not
> much different from an octave stretched 14-note equal temperament.
It
> has a relatively large error for a named temperament, comparable to
> dicot (a neutral-thirds temperament). It tempers out 25/24 and
81/80, so
> it could be considered a sort of warped diatonic scale with extra
> 7-limit notes in between; the tuning map is:
>
> <7, 11, 16, 20], <0, 0, 0, -1]

I just saw "tuning map" defined on the tuning-math list. According to
that definition, this isn't a tuning map at all. Instead, it's a
mapping from periods and generators to approximate primes, not
specific as to exact tuning.

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:
>
> On 1/19/06, wallyesterpaulrus <wallyesterpaulrus@y...> wrote:
> > This doesn't make sense to me. Comma sequences seem to be another
> > issue. Waage makes sense as a 7-limit descendant of Compton.
Catler
> > doesn't, because it changes the way prime 5 is mapped.
>
> Is "Waage" the one where the intervals of 5 are one step off from
> 12-EDO and the intervals of 7 are two steps off?

Yes, and the "step" or generator is about 15 cents.

> > > But Catler does reduce to a form of Compton if you take out the
> > > sevens. It reduces to 12-EDO, which is a degenerate form of
Compton
> > > ("degenerate" meaning some of the notes happen to be tuned the
> > >same).
> >
> > I don't think that should count.
>
> Well, it does seem pretty silly. But if that doesn't count, then
> things like Blackwood and Compton shouldn't be considered
descendents
> of Pythagorean, because they are degenerate in the same way.

True.

wallyesterpaulrus wrote:
>>it could be considered a sort of warped diatonic scale with extra >>7-limit notes in between; the tuning map is:
>>
>><7, 11, 16, 20], <0, 0, 0, -1]
> > > I just saw "tuning map" defined on the tuning-math list. According to > that definition, this isn't a tuning map at all. Instead, it's a > mapping from periods and generators to approximate primes, not > specific as to exact tuning.

"Mapping", then; sorry about the terminology confusion. I can't see someone saying "mapping from periods and generators to approximate primes" every time one of these comes up. I suppose "generator mapping" would be the best way to say this if "mapping" isn't specific enough.

Just to make sure that I understand the terminology: a tuning map of TOP jamesbond, by this definition, would be:

<1.00786, 1.58378, 2.30368, 2.80735]

right?

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> wallyesterpaulrus wrote:
> >>it could be considered a sort of warped diatonic scale with extra
> >>7-limit notes in between; the tuning map is:
> >>
> >><7, 11, 16, 20], <0, 0, 0, -1]
> >
> >
> > I just saw "tuning map" defined on the tuning-math list.
According to
> > that definition, this isn't a tuning map at all. Instead, it's a
> > mapping from periods and generators to approximate primes, not
> > specific as to exact tuning.
>
> "Mapping", then; sorry about the terminology confusion. I can't see
> someone saying "mapping from periods and generators to approximate
> primes" every time one of these comes up. I suppose "generator
mapping"
> would be the best way to say this if "mapping" isn't specific
enough.
>
> Just to make sure that I understand the terminology: a tuning map
of TOP
> jamesbond, by this definition, would be:
>
> <1.00786, 1.58378, 2.30368, 2.80735]
>
> right?

Right; or more commonly, you'd multiply by 1200 and say it's in cents.

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

> > > > But there isn't a problem. If you take the Hermite comma
> sequence as
> > > > your basis, catler is a decendent of compton, and the family
> > > > relationships are well-defined.
> > >
> > > I don't understand. In Compton, prime 5 is reached via -1
> generator
> > > (and 28 periods). In Catler, prime 5 is reached via 0 generators
> (and
> > > 28 periods).
> >
> > The Hermite comma sequence in this case starts with a 3-limit comma,
> > the Pythagoren comma, instead of the usual 5-limit comma.
>
> That doesn't clarify anything for me, nor do I have any idea why this
> is the case.

Because unlike most rank two temperaments, they temper out a 3-limit
comma.

> It would be wrong to consider Catler a descendant of Compton as you
> stated above, if (as a number of people have implied, probably
> including you) that the descendant must reduce to the ancestor if the
> highest prime is taken out of the mapping of the descendant. That
> seemed to be the governing principle until now.

I was thinking about various kinds of family relationships. They both
do reduce to the same thing, <12 19|, in the 3-limit.

> By "what is one" I meant "what is a family tree".

A Hermite comma sequence, together with the projection of the wedgie
down to lower prime limits, gives various kinds of family
relationships, which could be sorted out, as I started to do, with
various terms.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > It would be wrong to consider Catler a descendant of Compton as
you
> > stated above, if (as a number of people have implied, probably
> > including you) that the descendant must reduce to the ancestor if
the
> > highest prime is taken out of the mapping of the descendant. That
> > seemed to be the governing principle until now.
>
> I was thinking about various kinds of family relationships.

What about the kind that makes the patriarch of
the "superpythagorean" temperament the one where only 64:63 is
tempered out, and then various 5-limit R2 temperaments descendants of
this?

> They both
> do reduce to the same thing, <12 19|, in the 3-limit.

But they don't reduce to the same thing in the 5-limit, which seemed
most relevant here.

> > By "what is one" I meant "what is a family tree".
>
> A Hermite comma sequence, together with the projection of the wedgie
> down to lower prime limits, gives various kinds of family
> relationships, which could be sorted out, as I started to do, with
> various terms.

Isn't Compton the projection of the wedgie of Waage, but not of
Catler, down to a lower prime limit?

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...> wrote:

> It makes sense to me. To get a 5-limit linear temperament from
> Pythagorean we have to temper out one 5-limit comma. The Pythagorean
> comma is a 5-limit comma which happens to be 3-limit as well, but we
> can still temper it out and get Compton.
>
> To get from Compton to a 7-limit linear temperament we have to temper
> out one 7-limit comma. The syntonic comma is a 7-limit comma which
> happens to be 5-limit as well, but again, we can temper it out and get
> Catler.
>
> The family tree goes Pythagorean -> Compton -> Catler.

Whatever names we give them, we have a temperament tempering out the
Pythagorean comma in the 3-limit, which is <12 19|, we have the rank
two temperament tempering it out in the 5-limit, which is <<0 12 19||,
and we have various 7-limit temperaments which descend from this, such
as the ones which in addition temper out 81/80 or 225/224. Clearly
this describes some sort of family relationship.

> Would someone mind telling me what "jamesbond" is? I can't find a
> description of it anywhere.

It's a sort of jumped up 5-limit 7-equal. It tempers out the same
commas, 25/24 and 81/80, but has an extra generator of around 80 cents
to deal with the 7-limit. Its octave-equivalent part to the wedgie
goes 007, and hence the name.

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

> I just saw "tuning map" defined on the tuning-math list. According to
> that definition, this isn't a tuning map at all. Instead, it's a
> mapping from periods and generators to approximate primes, not
> specific as to exact tuning.

I tried calling that an "ikon" some years back, but the usual name is
"prime mapping" or "mapping to primes".

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> Just to make sure that I understand the terminology: a tuning map of
TOP
> jamesbond, by this definition, would be:
>
> <1.00786, 1.58378, 2.30368, 2.80735]
>
> right?

Or else that times 1200, if you are using cents.

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

> > I was thinking about various kinds of family relationships.
>
> What about the kind that makes the patriarch of
> the "superpythagorean" temperament the one where only 64:63 is
> tempered out, and then various 5-limit R2 temperaments descendants of
> this?

If you want to consider it, fine. It doesn't arise from Hermite comma
sequences, though.

> Isn't Compton the projection of the wedgie of Waage, but not of
> Catler, down to a lower prime limit?

So it is written, so mote it be. Meaning, yes, so far as I have these
things recorded.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > > I was thinking about various kinds of family relationships.
> >
> > What about the kind that makes the patriarch of
> > the "superpythagorean" temperament the one where only 64:63 is
> > tempered out, and then various 5-limit R2 temperaments
descendants of
> > this?
>
> If you want to consider it, fine. It doesn't arise from Hermite
comma
> sequences, though.
>
> > Isn't Compton the projection of the wedgie of Waage, but not of
> > Catler, down to a lower prime limit?
>
> So it is written, so mote it be. Meaning, yes, so far as I have
these
> things recorded.

So what do you get when you project the wedgie of Waage down to a
lower prime limit (namely 5)?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:
>
> > It makes sense to me. To get a 5-limit linear temperament from
> > Pythagorean we have to temper out one 5-limit comma. The
Pythagorean
> > comma is a 5-limit comma which happens to be 3-limit as well, but
we
> > can still temper it out and get Compton.
> >
> > To get from Compton to a 7-limit linear temperament we have to
temper
> > out one 7-limit comma. The syntonic comma is a 7-limit comma which
> > happens to be 5-limit as well, but again, we can temper it out
and get
> > Catler.
> >
> > The family tree goes Pythagorean -> Compton -> Catler.
>
> Whatever names we give them, we have a temperament tempering out the
> Pythagorean comma in the 3-limit, which is <12 19|,

That's 12-equal, not Pythagorean.

> we have the rank
> two temperament tempering it out in the 5-limit, which is <<0 12
19||,
> and we have various 7-limit temperaments which descend from this,
such
> as the ones which in addition temper out 81/80 or 225/224. Clearly
> this describes some sort of family relationship.

What you've described goes R1 -> R2 -> R2. I thought all the members
of the famlly were supposed to be R2 (rank 2).

In any case, this is only one sort of family relationship that's been
referred to.

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

> So what do you get when you project the wedgie of Waage down to a
> lower prime limit (namely 5)?

That gives <<0 12 19||, with comma 3^19/5^12.

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

> What you've described goes R1 -> R2 -> R2. I thought all the members
> of the famlly were supposed to be R2 (rank 2).

Then stop with <<0 12 19||. This is what we usually do, stop at a
5-limit temperament.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > So what do you get when you project the wedgie of Waage down to a
> > lower prime limit (namely 5)?
>
> That gives <<0 12 19||, with comma 3^19/5^12.

So on that basis, Waage seems like a candidate for a descendent of
Compton. Now, what do you get when you project the wedgie of Catler
down to a lower prime limit (namely 5)? 12-equal? So this would be a
different set of family relationships than what the Hermite comma
sequence gives? Because you said "together with" . . . (?)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > What you've described goes R1 -> R2 -> R2. I thought all the
members
> > of the famlly were supposed to be R2 (rank 2).
>
> Then stop with <<0 12 19||. This is what we usually do, stop at a
> 5-limit temperament.

You mean start? But doesn't it make a lot of sense to start with an R2
3-limit tuning, namely Pythagorean, especially if that bears some
resemblance to the 5-limit temperament you'd otherwise start with? Does
projecting wedgies down to lower prime limits say something different
here?

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

> Now, what do you get when you project the wedgie of Catler
> down to a lower prime limit (namely 5)? 12-equal?

I'm afraid what you get is the infamous null temperament. On the other
hand, the Hermite comma sequence is Pythagorean, Schisma. That tells
us it has some kind of family relationship with Compton.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > Now, what do you get when you project the wedgie of Catler
> > down to a lower prime limit (namely 5)? 12-equal?
>
> I'm afraid what you get is the infamous null temperament.

Aha! Is that considered rank 2, abstractly?

wallyesterpaulrus wrote:

> There's a class of tuning systems, with rather fuzzy boundaries around > it, that can be said to be examples of a given "temperament" by your > definition. The problem is that many people use "temperament" to mean a > specific tuning system. So, to avoid confusion, I say "temperament > class" when there's more than one possible specific tuning for it.

"Temperament class" is good with me.

Graham

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

> > I'm afraid what you get is the infamous null temperament.
>
> Aha! Is that considered rank 2, abstractly?

If temperaments are projective, it's not a temperment; it is the only
wedgie which isn't projective. But yeah, it's the "temperament" for
which the period and the generator are both 1.