Yahoo Tuning Groups Ultimate Backup tuning Comma sequences

> Working away on my web site still:
>
> http://66.98.148.43/~xenharmo/commaseq.htm

Great work, Gene. A few comments...

""
Subsequent commas of the comma sequence are such that the
wedge product of these commas is reduced, in the sense that
the greatest common divisor of its coefficients is 1 ...""

I think this could be fleshed out a bit more. What does
the wedge product of a comma sequence give (a map?) and
why would one want the GCD of its coefficients to be 1
(no torsion?)?

""
...we may call a sequence [c1_1, ..., c_n, c_{n+1}]...""

I'm a little unclear on this notation. Did you mean:

[c_1, ..., c_n, c_{n+1}]

Anyway, this notion of families is great. I was starting
to worry that the variety of temperaments was too great to
manage...

-Carl

🔗Jacob <jbarton@rice.edu>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> Working away on my web site still:
>
> http://66.98.148.43/~xenharmo/commaseq.htm

That's delightful! Is a full family tree forthcoming?

And, I was wondering, would it be constructive for someone to make a big picture of
these "middle path" linear temperaments? Like, an axis representing the size of a
generator interval labelled with the most-agreed upon names. One graph for octave
period, another for half-octave, etc. Related question: what determines a boundary
between two temperaments with similar generators (and the same period)?

(I'm wary to making suggestions around here because they are so often bounced back
to the suggestor! But hey, it might make for a good opportunity to understand these
things more.)

Oh well,

Jacob

🔗monz <monz@tonalsoft.com>

hi Gene,

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

> Working away on my web site still:
>
> http://66.98.148.43/~xenharmo/commaseq.htm

wow, i love it!

you've suddenly gone all the way round from
the abstract mathematical jargon to the most
colorful everyday analogy. this is great.

-monz

🔗monz <monz@tonalsoft.com>

hi Jacob,

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > Working away on my web site still:
> >
> > http://66.98.148.43/~xenharmo/commaseq.htm
>
> That's delightful! Is a full family tree forthcoming?

i don't know how "full" it is ... i'm sure Gene and
the others will keep discovering new long-lost relatives!

but anyway, i've made a family-tree of his meantone
example, and put it on my "family" page:

http://tonalsoft.com/enc/family.htm

let's have more of these! they help to make a lot
more sense out of the jumble of names.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

Hi Jacob!

>> http://66.98.148.43/~xenharmo/commaseq.htm
>
>That's delightful! Is a full family tree forthcoming?
>
>And, I was wondering, would it be constructive for someone to
>make a big picture of these "middle path" linear temperaments?
>Like, an axis representing the size of a generator interval
>labelled with the most-agreed upon names. One graph for octave
>period, another for half-octave, etc.

Erv Wilson's "straight line patterns of the scale tree" shows
something like this, except IIRC he only shows octave periods
and he isn't talking about temperaments per se, but rather
MOS points (which of course apply to temperaments).

>(I'm wary to making suggestions around here because they are
>so often bounced back to the suggestor!

That's happened to me. :)

>Related question: what determines a boundary between two
>temperaments with similar generators (and the same period)?

Good question. Gene's the person to answer it. But note that
a generator/period does not uniquely define a temperament.

In some sense it depends on the scale chosen and the mapping
insisted upon in composition, as all temperaments will eventually
give better approximations than provided by their map if carried
out far enough.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> i don't know how "full" it is ... i'm sure Gene and
> the others will keep discovering new long-lost relatives!

There were some which didn't get mentioned; there's always this
question of when to stop, and what limit to go to.

> but anyway, i've made a family-tree of his meantone
> example, and put it on my "family" page:
>
> http://tonalsoft.com/enc/family.htm

Neat! I'd at least add Uncle Injera, a temperament of which Paul is
very fond, and maybe Uncle Godzilla as well.

It might also be noted that a few oddball temperaments like James Bond
don't have proper comma sequences, in the sense that its comma
sequence is [25/24, 81/80], which is two 5-limit commas. Hence James
Bond shares the 81/80 comma but has been given up for adoption, since
its primary 5-limit comma is 25/24.

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> >Related question: what determines a boundary between two
> >temperaments with similar generators (and the same period)?
>
> Good question. Gene's the person to answer it. But note that
> a generator/period does not uniquely define a temperament.

The most obvious place to draw the line is at the pivot. From two
sister linear temperaments you can define an equal temperament for the
next lower prime limit--so for instance, two 5-limit sisters determine
a 3-limit et, two 7-limit sisters a 5-limit et, and so forth. I was
calling this the "nexus", but Monz got me to change that to "node."
This node can be uniquely extended to a p-limit temperament in which
the sisters have the same mapping and tuning, and so define the pivot
tuning which can be seen as the boundry between them. Hence the pivot
between meantone and schismic would be 12, between septimal meantone
and dominant also 12, between meantone and flattone 19, between
huygens and meanpop, 31.

🔗Carl Lumma <ekin@lumma.org>

>>> Related question: what determines a boundary between two
>>> temperaments with similar generators (and the same period)?
>>
>> Good question. Gene's the person to answer it. But note that
>> a generator/period does not uniquely define a temperament.
>
>The most obvious place to draw the line is at the pivot. From two
>sister linear temperaments you can define an equal temperament
>for the next lower prime limit--so for instance, two 5-limit
>sisters determine a 3-limit et,

Why an et? I would have thought they defined a temperament of
the same rank (or codimension or whatever we're calling it) as
the sisters themselves. For example:

(c1, c2, c3a)
(c1, c2, c3b)
gives node (c1, c2)

No?

>two 7-limit sisters a 5-limit et, and so forth. I was
>calling this the "nexus", but Monz got me to change that to
>"node." This node can be uniquely extended to a p-limit
>temperament in which the sisters have the same mapping and
>tuning, and so define the pivot tuning which can be seen
>as the boundry between them.

Great. But how does this extension work?

Note that you haven't defined p-limit here. Do you mean to
any limit, or just to the next higher limit from the node?

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >It might also be noted that a few oddball temperaments like
> >James Bond don't have proper comma sequences, in the sense
> >that its comma sequence is [25/24, 81/80],
>
> This violates one of your rules for comma sequences. I'm
> guessing you did that to keep from breaking one of the other
> rules? What's going on here?

It doesn't violate any of the rules, it violates the claim that there
is a unique 5-limit comma for any linear temperament. This is only
usually true; I've changed the page.

🔗Carl Lumma <ekin@lumma.org>

>>>It might also be noted that a few oddball temperaments like
>>>James Bond don't have proper comma sequences, in the sense
>>>that its comma sequence is [25/24, 81/80],
>>
>>This violates one of your rules for comma sequences. I'm
>>guessing you did that to keep from breaking one of the other
>>rules? What's going on here?
>
>It doesn't violate any of the rules, it violates the claim that
>there is a unique 5-limit comma for any linear temperament. This
>is only usually true; I've changed the page.

It seems to be this entire paragraph is violated...

""
A comma sequence defines a temperament by means of a sequence
of uniquely determined commas at increasing prime limits. If p
is the prime limit and n is the number of generators (two for
a linear temperament, and so forth) then the first comma of the
comma sequence will be in the q-limit, where q is the nth odd
prime, and the subsequent commas in prime limits each one prime
beyond the last. Hence a linear temperament will have a 5-limit
comma as the first comma of the comma sequence, a 7-limit comma
as the second comma, and so forth.""

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Why an et?

Because we need to draw a boundry.

I would have thought they defined a temperament of
> the same rank (or codimension or whatever we're calling it) as
> the sisters themselves. For example:
>
> (c1, c2, c3a)
> (c1, c2, c3b)
> gives node (c1, c2)
>
> No?

That gives the mother of the two sisters.

> >two 7-limit sisters a 5-limit et, and so forth. I was
> >calling this the "nexus", but Monz got me to change that to
> >"node." This node can be uniquely extended to a p-limit
> >temperament in which the sisters have the same mapping and
> >tuning, and so define the pivot tuning which can be seen
> >as the boundry between them.
>
> Great. But how does this extension work?

You could for instance find it as the unique linear combination of the
prime mapping vals giving the node. In practice, its normally obvious.

Here's an example. Suppose we have various versions of augmented:

aug1: <<3 0 9 -7 6 21||, comma sequence [128/125, 28/27]
aug2: <<3 0 -6 -7 -18 -14||, comma sequence [128/125, 64/63]
aug3:= <<3 0 6 -7 1 14||, comma sequence [128/125, 36/35]
aug4:= <<3 0 -3 -7 -13 -7||, comma sequence [128/125, 21/20]

We can find the node between any two of these by subtracting the two
wedgies and taking the third, fifth, and sixth coefficent. We get

It's not hard to guess what the 7-limit extension is even without
computing the answer, but in fact to simply draw the boundry we don't
need to--the node already tells us where that lies.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> This might-should be updated...
>
> http://tonalsoft.com/enc/pivot-et.htm
>
> ...given the recent thread, it doesn't seem the
> originating temperaments must be linear, and
> "mappings" can probably be deleted, and more
> detail is needed.

Let's not hurry into this.

***Math warning***

One way to understand this pivot business is that the pivot is
determined by the nullspace of the union of the kernels, which means
the et which annihilates the commas on both lists. For two sister
planar temperaments, the nullspace would have dimension two and the
pivot would be a linear temperament. Two planar temperaments which
were cousins would determine a pivot et.

I suspect we are better off leaving it be and assuming the two
temperaments are linear and sisters.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>One way to understand this pivot business is that the pivot is
>determined by the nullspace of the union of the kernels, which
>means the et which annihilates the commas on both lists. For two
>sister planar temperaments, the nullspace would have dimension
>two and the pivot would be a linear temperament.

So it doesn't have to be an ET!!

>I suspect we are better off leaving it be and assuming the two
>temperaments are linear and sisters.

I find this sort of big-picture-hiding confusing.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

hi Carl,

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

> This might-should be updated...
>
> http://tonalsoft.com/enc/pivot-et.htm
>
> ...given the recent thread, it doesn't seem the
> originating temperaments must be linear, and
> "mappings" can probably be deleted, and more
> detail is needed.

ok, well ... i'm not one of the experts on linear temperaments
around here (except for 5-limit meantone). so i defer
to others to give me the info i need for the definition.

> By the way Joe, the tonalsoft front page isn't
> spelling Encyclopaedia with the ae.
>
> -Carl

wow, thanks for that! it's been fixed. i'm surprised
that it slipped past me despite the fact that i see it
several times every day!

maybe i should mention that we (at Tonalsoft) deliberately
made the decision to go with British spelling rather
than American (which is my personal preference) ...
feeling that the British spelling is perceived as
more international.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

hi Gene and Graham,

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

> > > This node can be uniquely extended to a p-limit
> > > temperament in which the sisters have the same mapping and
> > > tuning, and so define the pivot tuning which can be seen
> > > as the boundry between them.
> >
> > Great. But how does this extension work?
>
> You could for instance find it as the unique linear combination
> of the prime mapping vals giving the node. In practice, its
> normally obvious.
>
> Here's an example. Suppose we have various versions of augmented:
>
> aug1: <<3 0 9 -7 6 21||, comma sequence [128/125, 28/27]
> aug2: <<3 0 -6 -7 -18 -14||, comma sequence [128/125, 64/63]
> aug3:= <<3 0 6 -7 1 14||, comma sequence [128/125, 36/35]
> aug4:= <<3 0 -3 -7 -13 -7||, comma sequence [128/125, 21/20]
>
> We can find the node between any two of these by subtracting
> the two wedgies and taking the third, fifth, and sixth
> coefficent. We get
>
> aug1, aug2: <15 24 35|
> aug1, aug3: <3 5 7|
> aug1, aug4: <12 19 28|
> aug2, aug3: <12 19 28|
> aug2, aug4: <3 5 7|
> aug3, aug4: <9 14 21|
>
> It's not hard to guess what the 7-limit extension is
> even without computing the answer, but in fact to simply
> draw the boundry we don't need to--the node already tells
> us where that lies.

excellent. i've made a new Encyclopaedia page for "node"
and so far have adapted that post for its content.

please write for me:

1)
a newbie definition explaining what a "node" is,
without using math

2)
an impeccably logical total math jargon definition
for the techie crowd

for the top of the page. i'll keep that post after
the new definitions, since it gives a nice example.

more importantly, i already have a page for "pivot-ET".
how does node differ from that, if it does?

Graham defined "pivot-ET" specifically as the
boundary between two families of linear temperaments
... but i suspect that perhaps now he would make
that more general.

if he does want the specific meaning, then "node"
is more general as it describes the origin-points
of *all* branches on the temperament family-tree.

so "node" defines not only boundaries between families
of temperaments, but also all the relations between
different family members belonging to the same family tree.

-monz

🔗monz <monz@tonalsoft.com>

hi Gene,

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

> ***Math warning***

ha ... that's funny. i remember in the days just before
the tuning-math list split off from this larger amoeba,
i started putting "THEORY WARNING" at the beginning of
my more technical posts. :)

> One way to understand this pivot business is that the
> pivot is determined by the nullspace of the union of the
> kernels, which means the et which annihilates the commas
> on both lists.

i want to be clear about what you say here ...
is "pivot" exactly the same thing as "node"?

it seems so, by what follows ...

> For two sister
> planar temperaments, the nullspace would have dimension
> two and the pivot would be a linear temperament. Two
> planar temperaments which were cousins would determine
> a pivot et.
>
> I suspect we are better off leaving it be and assuming the
> two temperaments are linear and sisters.

whether i change the other definition or not, i sure
would like some examples of what you're saying right here.
thanks.

i want to create a whole series of webpages now for
the different family relationships. so how about some
nice concise definitions for "grandmother", "mother",
"sister", "cousin", "uncle"? ... and maybe a little
more on "legitimacy".

-monz

🔗Graham Breed <graham@microtonal.co.uk>

monz wrote:

> maybe i should mention that we (at Tonalsoft) deliberately
> made the decision to go with British spelling rather
> than American (which is my personal preference) ...
> feeling that the British spelling is perceived as
> more international.

However strange a decision that may be, you've picked the least popular of the three spellings on the web (according to Google):

"encyclopedia": 18,700,000
"encyclopï¿½dia": 2,940,000
"encyclopaedia": 1,410,000

Graham

🔗Graham Breed <graham@microtonal.co.uk>

monz wrote:

> 1)
> a newbie definition explaining what a "node" is,
> without using math

This seems to be Gene's word. I'd take it to mean any ET as a special case of a linear temperament (or planar temperament, or LT as a special case of a PT and so on). But I don't think that's what Gene means.

> 2)
> an impeccably logical total math jargon definition
> for the techie crowd

That shouldn't be needed. You know that 12-equal, 19-equal, 31-equal, 50-equal and so on belong to the meantone family? The "pivot" is an equal (in the simple case) temperament that belongs to two different families. I think that's an intuitive and precise definition.

The question is whether 31-equal is a pivot between meantone and 7&31 (Vicentino's neutral thirds/enharmonic meantone, which has a distinct mapping in higher limits). The difference is that they don't have the same generator.

Also, what about 12&29 schismic (garibaldi) and this 12&7 "dominant" meantone. They both work with fifths sharp of 12-equal, so can 12 really be called the pivot between them? Well, by analogy with levers it can. It's called a second or third class lever here:

http://www.enchantedlearning.com/physics/machines/Levers.shtml

> Graham defined "pivot-ET" specifically as the
> boundary between two families of linear temperaments
> ... but i suspect that perhaps now he would make
> that more general.

It can certainly be made more general. That's why I specified "pivot-ET", so that "pivot" can be used in other contexts.

> if he does want the specific meaning, then "node"
> is more general as it describes the origin-points
> of *all* branches on the temperament family-tree.
> > so "node" defines not only boundaries between families
> of temperaments, but also all the relations between
> different family members belonging to the same family tree.

I'm still not sure what "family" means. The definition page seems inconsistent. You talk about ETs, Gene talks about LTs, and there's no overview. Oh, and if you want to persist in using Britishisms, get rid of the "math"!

Graham

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> monz wrote:
>
> > maybe i should mention that we (at Tonalsoft) deliberately
> > made the decision to go with British spelling rather
> > than American (which is my personal preference) ...
> > feeling that the British spelling is perceived as
> > more international.
>
> However strange a decision that may be, you've picked the least
popular
> of the three spellings on the web (according to Google):
>
> "encyclopedia": 18,700,000
> "encyclopædia": 2,940,000
> "encyclopaedia": 1,410,000
>
>
> Graham

hmm ... i wasn't even go to mention this, but i'm
glad i did. something to think about.

-monz

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >> However strange a decision that may be, you've picked the least
> >> popular of the three spellings on the web (according to Google):
> >>
> >> "encyclopedia": 18,700,000
> >> "encyclopædia": 2,940,000
> >> "encyclopaedia": 1,410,000
> >>
> >> Graham
> >
> >hmm ... i wasn't even go to mention this, but i'm
> >glad i did. something to think about.
>
> I vote for "encyclopedia".
>
> -Carl

hmmm ... could you set up a poll? i'm interested specifically
in knowing which spelling is preferred by the non-Americans
on the list. thanks.

-monz

🔗Carl Lumma <ekin@lumma.org>

>> >> However strange a decision that may be, you've picked the least
>> >> popular of the three spellings on the web (according to Google):
>> >>
>> >> "encyclopedia": 18,700,000
>> >> "encyclopædia": 2,940,000
>> >> "encyclopaedia": 1,410,000
>> >>
>> >> Graham
>> >
>> >hmm ... i wasn't even go to mention this, but i'm
>> >glad i did. something to think about.
>>
>> I vote for "encyclopedia".
>
>hmmm ... could you set up a poll? i'm interested specifically
>in knowing which spelling is preferred by the non-Americans
>on the list. thanks.

Sure, but did you know that you can too? Let me know if you still
want me to do it.

-Carl