Natural commas

Ciao

>> 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?

ehm... I've just invented... sorry.
The right name seems to be "septimal comma" right?

>> >> 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.

Exceptions confirm the rules...

>> 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.

Anyway, reading the definition of comma in Joe Monzo site I've
founded as principal commas exactly the three I consider:
pythagorean, syntonic and septimal. It's this just a coincidence?
What are the most relevant commas in tuning history?

>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.

I suspect that there are a lot of things in the world.
But I'm just trying to understand the main path of tuning history.

> >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.

It would be interesting to me to have a mathemathical explanation of
what a comma is.
It is maybe possible to say that a comma appears always in relation
with two primes?
It is possible to say that a progression on circle of fifths defines
a series of commas?
Follows this series some rule?
Example: considering 3 and 2 the commas would rise in this
progression: 4:3, 9:8, 256:243, pythagorean, (...).

>> >> 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.

Any of the three commas I consider derives naturally from circle of
fifths. You have to shift 27:16 to 5:3 (syntonic) and go ahead until
you reach 45:32 which is shifted to 7:5 (septimal). Pythagoeran
appears when you try to close the circle. But if you consider that a
comma rises as the difference between two primes, the first comma
between 3 and 2 would be an interval of fourth.

Ciao

Lorenzo

Ciao

>> 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?

ehm... I've just invented... sorry.
The right name seems to be "septimal comma" right?

>> >> 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.

Exceptions confirm the rules...

>> 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.

Anyway, reading the definition of comma in Joe Monzo site I've
founded as principal commas exactly the three I consider:
pythagorean, syntonic and septimal. It's this just a coincidence?
What are the most relevant commas in tuning history?

>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.

I suspect that there are a lot of things in the world.
But I'm just trying to understand the main path of tuning history.

> >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.

It would be interesting to me to have a mathemathical explanation of
what a comma is.
It is maybe possible to say that a comma appears always in relation
with two primes?
It is possible to say that a progression on circle of fifths defines
a series of commas?
Follows this series some rule?
Example: considering 3 and 2 the commas would rise in this
progression: 4:3, 9:8, 256:243, pythagorean, (...).

>> >> 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.

Any of the three commas I consider derives naturally from circle of
fifths. You have to shift 27:16 to 5:3 (syntonic) and go ahead until
you reach 45:32 which is shifted to 7:5 (septimal). Pythagoeran
appears when you try to close the circle. But if you consider that a
comma rises as the difference between two primes, the first comma
between 3 and 2 would be an interval of fourth.

Ciao

Lorenzo

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
wrote:
> Ciao
>
> >> 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?
>
> ehm... I've just invented... sorry.
> The right name seems to be "septimal comma" right?

The "septimal comma" usually refers to 64:63.

> >> 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.
>
> Anyway, reading the definition of comma in Joe Monzo site I've
> founded as principal commas exactly the three I consider:
> pythagorean, syntonic and septimal.

225:224 or 64:63?

> It's this just a coincidence?

Where exactly does Monz declare his "three principal commas"?

> What are the most relevant commas in tuning history?

Well, when you temper out one comma (81:80), a whole bunch of others
become equivalent to one another (such as 648:625, 128:125,
2048:2025, 32805:32768, 531441:524288), so in a sense they were all
very important. As for prime limit 7, that's more controversial. Some
don't believe that simple-ratio concordance tells the whole story
when it comes to deliberately discordant sounds, while other are
eager to slap this or that higher prime number onto the picture.

> I suspect that there are a lot of things in the world.
> But I'm just trying to understand the main path of tuning history.

I think the main path in the West has revolved around the diatonic
scale. But Berlioz and Schubert started exploiting 12-equal's more
non-diatonic possibilities, and many followed -- are they still on
the "main path"?

> It would be interesting to me to have a mathemathical explanation
of
> what a comma is.

To me, it's any small ratio formed from the same primes as the
harmonies are, and which is not a power of a simpler ratio.

> It is maybe possible to say that a comma appears always in relation
> with two primes?

Not sure what that means, precisely.

> It is possible to say that a progression on circle of fifths
>defines
> a series of commas?

Sure!

> Follows this series some rule?
> Example: considering 3 and 2 the commas would rise in this
> progression: 4:3, 9:8, 256:243, pythagorean, (...).

Yes . . .

> >> >> 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.
>
> Any of the three commas I consider derives naturally from circle of
> fifths. You have to shift 27:16 to 5:3 (syntonic) and go ahead
until
> you reach 45:32 which is shifted to 7:5 (septimal). Pythagoeran
> appears when you try to close the circle. But if you consider that
a
> comma rises as the difference between two primes, the first comma
> between 3 and 2 would be an interval of fourth.

Why not a fifth?

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...> wrote:

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

> Exceptions confirm the rules...

Exceptions do the exact opposite, they invalidate the rule. In math we
would call it a counterexample. If we look at the actual range of
possibilities, fifths are not what we use in the majority of
interesting cases. Of course, nothing prevents us from concerning
ourselves only with temperaments (such as meantone) which do use the
fifth, but we should be aware this is a personal decision, not a law
of nature.

> It would be interesting to me to have a mathemathical explanation of
> what a comma is.

If we are discussing temperaments, a comma for a temperament is a
small rational interval which the temperament reduces to a unison.

> It is maybe possible to say that a comma appears always in relation
> with two primes?

It always requires at least two primes; in fact, it is impossible to
get a small rational interval at all without at least two primes.

> It is possible to say that a progression on circle of fifths defines
> a series of commas?

> Follows this series some rule?
> Example: considering 3 and 2 the commas would rise in this
> progression: 4:3, 9:8, 256:243, pythagorean, (...).

That's one way to get commas, but not the most important. One way to
get examples of such commas is via continued fractions; the
convergents to the continued fraction for log base 2 of 3 give us
3/2, 4/3, 9/8, 256/243, 3^12/2^19, 2^65/3^41, 3^53/2^84... which is an
infinite list of commas, using only the primes 2 and 3, which decrease
monotonically in size. This sequence clearly follows a rule.

> Any of the three commas I consider derives naturally from circle of
> fifths. You have to shift 27:16 to 5:3 (syntonic) and go ahead until
> you reach 45:32 which is shifted to 7:5 (septimal).

From the point of view of music, more primes are important than just 2
and 3. I think at minimum 2,3,5, and 7 are clearly musical.

hi Lorenzo,

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
wrote:

> Ciao
>
> >> 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?
>
> ehm... I've just invented... sorry.
> The right name seems to be "septimal comma" right?

i called it the "septimal kleisma", because it is
~7.711522991 cents, close to the size of the original
kleisma, Tanaka's pental (5-limit) kleisma of
ratio 15625/15552, 3,5-monzo [ -5, 6 >, ~8.10728 cents.

> >> 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.

this is a really good point. a lot of the tuning history
i'm interested in traces exactly this path: 5-limit
periodicity-blocks of various shapes, and how they
evolved over time, according to the usages that can be
gleaned from composers's works and theorists's descriptions.

> Anyway, reading the definition of comma in Joe Monzo site I've
> founded as principal commas exactly the three I consider:
> pythagorean, syntonic and septimal. It's this just a coincidence?
> What are the most relevant commas in tuning history?

the reason why those three have their own entries is
because i personally perceived them to be historically
the most important. i do give some creedence to the
idea that our comprehension of harmony has progressed
upwards thru the prime-factors, as illustrated in
Partch's book "Genesis of a Music".

on the other hand, one point of my own book
"JustMusic: A New Harmony", was to show the rich
variety of prime-factors that have been a part of
the history of JI theory, culminating in Boethius's
advocacy of 499 in his Enharmonic Genus.

> > > 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.

~cents . . . . . . . 2,3;5,7,11-monzo . . . . . ratio

. 1.95372079 . . . [-15 8, 1 0 0 > . . . 32805 : 32768
. 27.26409180 . . . [ 6 -2, 0 -1 0 > . . . . 64 : 63
. 13.79476661 . . . [ 1 2, -3 1 0 > . . . . 126 : 125
. 7.71152299 . . . [ -5 2, 2 -1 0 > . . . . 225 : 224
. 0.72119728 . . . [ -5 -1, -2 4 0 > . . . 2401 : 2400
. 0.39575587 . . . [ -1 -7, 4 1 0 > . . . 4375 : 4374

all these and many, many more up to the 11-limit
can be found in the Tonalsoft Encyclopaedia
"big list of intervals" page:

http://tonalsoft.com/enc/interval-list.htm

> > 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.
>
> It would be interesting to me to have a mathemathical
> explanation of what a comma is. It is maybe possible to
> say that a comma appears always in relation with two primes?

that is what i always meant by the term "xenharmonic bridge".

Paul Erlich pointed out that Fokker used the term
"unison-vector", which works similarly to my concept,
to refer to commas which appear within certain prescribed
prime-limits, whereas my "xenharmonic bridge" is meant
to include the whole infinite variety of commas which
connect any two primes.

the word "comma" has two distinct meanings: one which
describes intervals around ~20 to ~30 cents (Dave Keenan
has defined some specific boundaries ... please chime in,
Dave), and another which covers not only that category
but also all others up to approximately a "semitone".

see
http://tonalsoft.com/enc/comma.htm

> It is possible to say that a progression on circle
> of fifths defines a series of commas?
> Follows this series some rule?
> Example: considering 3 and 2 the commas would rise in this
> progression: 4:3, 9:8, 256:243, pythagorean, (...).

no, tuning theorists only consider the last of those
to be a "comma".

> >> >> 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.
>
> Any of the three commas I consider derives naturally from circle of
> fifths. You have to shift 27:16 to 5:3 (syntonic) and go ahead until
> you reach 45:32 which is shifted to 7:5 (septimal). Pythagoeran
> appears when you try to close the circle. But if you consider that a
> comma rises as the difference between two primes, the first comma
> between 3 and 2 would be an interval of fourth.
>
> Ciao
>
> Lorenzo

historically, the Pythagorean comma (~23.46001038 cents,
2,3-monzo [-19 12 > , ratio 531441 : 524288) was probably
the first one that was noticed. this derives from a Pythagorean
(3-limit) chain of 13 "5ths" (or "4ths"), in which the 13th
"5th" is approximately an "1/8-tone" higher (or lower,
respectively) than 7 "8ves" above (or below) the origin note.
my suspicion is that this was discovered originally by the
Sumerians, c. 3000 BC. it later was credited to Pythagoras,
c. 400 BC.

later, when 5-limit ratios were recognized in theory
treatises in ancient Greece (c. 300 BC by Archytas, then
later by Eratosthenes, Didymus, and Ptolemy) and then
re-recognized again in Europe (c. 1300-1500, after the
"Dark Ages"), it was noticed that there was another comma
(the "syntonic") which bridged the gap between the 3-limit
"ditone" (~407.8200035 cents, 2,3-monzo [ -6 4 > , ratio 81:64)
and the 5-limit JI "major-3rd" (~386.3137139 cents,
2,3;5-monzo [ -2 0, 1 > , ratio 5:4), and which is very
close in size to the Pythagorean comma: ~21.5062896 cents,
2,3;5-monzo [ -4 4, -1 > , ratio 81:80 .

many important "common-practice" theorists from Rameau
to Schoenberg all used a 5-limit JI diatonic scale as
their basis, which is what it seems to me you are doing.
in modern times, Ben Johnston and many of his students
(notably among them, Kyle Gann) do the same.

but today, most tuning theorists (including me) favor
the Pythagorean (3-limit) tuning as the best basis for
nomenclature.

-monz

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

> hi Lorenzo,
>
>
> --- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
> wrote:
>
> > Ciao
> >
> > >> 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?
> >
> > ehm... I've just invented... sorry.
> > The right name seems to be "septimal comma" right?
>
>
> i called it the "septimal kleisma", because it is
> ~7.711522991 cents, close to the size of the original
> kleisma, Tanaka's pental (5-limit) kleisma of
> ratio 15625/15552, 3,5-monzo [ -5, 6 >, ~8.10728 cents.

upon re-reading, that sounds like i'm trying to take
credit for naming that interval the "septimal kleisma",
but in fact, it was already current on the tuning list
in 1998 when i put it into my lexicon.

-monz

Ciao.

> The "septimal comma" usually refers to 64:63.

You are right. I was wrong.

I've readed in www.tonalsoft.com this:
"The three commas which are most commonly encountered are the
syntonic, the Pythagorean, and the septimal."

The septimal is 64:63. My thought is that from circle of fifths
becomes a septimal comma of 225:224.

> It would be interesting to me to have a mathemathical explanation
> of
> > what a comma is.
>
> To me, it's any small ratio formed from the same primes as the
> harmonies are, and which is not a power of a simpler ratio.

Can you give me more details?

> Why not a fifth?

I've choosen the smaller interval. As for a pythagorean comma you
choose a 24 cents intervals instead of a 1176 one.

Lorenzo

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Lorenzo,
>
>
> --- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
> wrote:
>
> > Ciao
> >
> > >> 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?
> >
> > ehm... I've just invented... sorry.
> > The right name seems to be "septimal comma" right?
>
>
> i called it the "septimal kleisma", because it is
> ~7.711522991 cents, close to the size of the original
> kleisma, Tanaka's pental (5-limit) kleisma of
> ratio 15625/15552, 3,5-monzo [ -5, 6 >, ~8.10728 cents.
>
>
>
> > >> 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.
>
>
> this is a really good point. a lot of the tuning history
> i'm interested in traces exactly this path: 5-limit
> periodicity-blocks of various shapes,
> and how they
> evolved over time, according to the usages that can be
> gleaned from composers's works and theorists's descriptions.

Well, you can join Mathieu in this game, but ultimately (as even
Mathieu is forced to admit in one case), I feel a choice of a
specific comma pair, let alone an actual periodicity block, is
fundamentally ambiguous for the composers I was referring to above.

> > It would be interesting to me to have a mathemathical
> > explanation of what a comma is. It is maybe possible to
> > say that a comma appears always in relation with two primes?
>
>
> that is what i always meant by the term "xenharmonic bridge".
>
> Paul Erlich pointed out that Fokker used the term
> "unison-vector", which works similarly to my concept,
> to refer to commas which appear within certain prescribed
> prime-limits, whereas my "xenharmonic bridge" is meant
> to include the whole infinite variety of commas which
> connect any two primes.

The unison vector concept is more general. Any xenharmonic bridge is
also a unison vector (in a high enough prime limit). But in the
xenharmonic bridge conception it's a relationship between two pitch-
ratios, while in the unison vector view it's an interval-ratio so it
can connect an infinite number of pairs of pitches.

> the word "comma" has two distinct meanings: one which
> describes intervals around ~20 to ~30 cents (Dave Keenan
> has defined some specific boundaries ... please chime in,
> Dave), and another which covers not only that category
> but also all others up to approximately a "semitone".
>
> see
> http://tonalsoft.com/enc/comma.htm

Aren't there any restrictions on when you would consider a semitone-
sized interval a comma? I pretty much restrict my use of "comma" to
mean something effectively a unison in the tuning system in question.

> > It is possible to say that a progression on circle
> > of fifths defines a series of commas?
> > Follows this series some rule?
> > Example: considering 3 and 2 the commas would rise in this
> > progression: 4:3, 9:8, 256:243, pythagorean, (...).
>
>
> no, tuning theorists only consider the last of those
> to be a "comma".

If a semitone can be a comma, why can't a whole-tone?
The answer to that question is this tuning system (my apologies,
Dave):

/tuning/files/Erlich/nana.gif

>
>
>
> > >> >> 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.
> >
> > Any of the three commas I consider derives naturally from circle
of
> > fifths. You have to shift 27:16 to 5:3 (syntonic) and go ahead
until
> > you reach 45:32 which is shifted to 7:5 (septimal). Pythagoeran
> > appears when you try to close the circle. But if you consider
that a
> > comma rises as the difference between two primes, the first comma
> > between 3 and 2 would be an interval of fourth.
> >
> > Ciao
> >
> > Lorenzo
>
>
>
> historically, the Pythagorean comma (~23.46001038 cents,
> 2,3-monzo [-19 12 > , ratio 531441 : 524288) was probably
> the first one that was noticed. this derives from a Pythagorean
> (3-limit) chain of 13 "5ths" (or "4ths"), in which the 13th
> "5th" is approximately an "1/8-tone" higher (or lower,
> respectively) than 7 "8ves" above (or below) the origin note.
> my suspicion is that this was discovered originally by the
> Sumerians, c. 3000 BC. it later was credited to Pythagoras,
> c. 400 BC.
>
> later, when 5-limit ratios were recognized in theory
> treatises in ancient Greece (c. 300 BC by Archytas, then
> later by Eratosthenes, Didymus, and Ptolemy) and then
> re-recognized again in Europe (c. 1300-1500, after the
> "Dark Ages"), it was noticed that there was another comma
> (the "syntonic") which bridged the gap between the 3-limit
> "ditone" (~407.8200035 cents, 2,3-monzo [ -6 4 > , ratio 81:64)
> and the 5-limit JI "major-3rd" (~386.3137139 cents,
> 2,3;5-monzo [ -2 0, 1 > , ratio 5:4), and which is very
> close in size to the Pythagorean comma: ~21.5062896 cents,
> 2,3;5-monzo [ -4 4, -1 > , ratio 81:80 .
>
> many important "common-practice" theorists from Rameau
> to Schoenberg all used a 5-limit JI diatonic scale as
> their basis, which is what it seems to me you are doing.
> in modern times, Ben Johnston and many of his students
> (notably among them, Kyle Gann) do the same.
>
> but today, most tuning theorists (including me) favor
> the Pythagorean (3-limit) tuning as the best basis for
> nomenclature.
>
>
>
> -monz

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
wrote:
> Ciao.
>
> > The "septimal comma" usually refers to 64:63.
>
> You are right. I was wrong.
>
> I've readed in www.tonalsoft.com this:
> "The three commas which are most commonly encountered are the
> syntonic, the Pythagorean, and the septimal."
>
> The septimal is 64:63. My thought is that from circle of fifths
> becomes a septimal comma of 225:224.

64:63 can "come from the circle of fifths" too. 16/9 -> 7/4 is down a
septimal comma.

> > It would be interesting to me to have a mathemathical explanation
> > of
> > > what a comma is.
> >
> > To me, it's any small ratio formed from the same primes as the
> > harmonies are, and which is not a power of a simpler ratio.
>
> Can you give me more details?

If you say the harmonies you're using are essentially just triads,
then a comma would be any narrow-interval ratio (like 15625:15552)
which has no prime factors higher than five. Also, you wouldn't
normally consider powers of commas to be commas themselves -- for
example, 81:80 squared is 6561:6400, and that vanishes if and only if
81:80 does, so it's not an independent comma.

> > Why not a fifth?
>
> I've choosen the smaller interval. As for a pythagorean comma you
> choose a 24 cents intervals instead of a 1176 one.

Yes indeed, a 24 cent interval vanishing is a very different
situation than a 1176 cent interval vanishing!

>
> Lorenzo

hi Paul,

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

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

> > > > 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.
> >
> >
> > this is a really good point. a lot of the tuning history
> > i'm interested in traces exactly this path: 5-limit
> > periodicity-blocks of various shapes, and how they
> > evolved over time, according to the usages that can be
> > gleaned from composers's works and theorists's descriptions.
>
> Well, you can join Mathieu in this game, but ultimately (as even
> Mathieu is forced to admit in one case), I feel a choice of a
> specific comma pair, let alone an actual periodicity block, is
> fundamentally ambiguous for the composers I was referring to above.

i'm not in disagreement with you about that!

certainly, the best composers have had various different
periodicity-blocks in mind within the same piece, switching
back and forth between them at various moments in the piece.

> > > It would be interesting to me to have a mathemathical
> > > explanation of what a comma is. It is maybe possible to
> > > say that a comma appears always in relation with two primes?
> >
> >
> > that is what i always meant by the term "xenharmonic bridge".
> >
> > Paul Erlich pointed out that Fokker used the term
> > "unison-vector", which works similarly to my concept,
> > to refer to commas which appear within certain prescribed
> > prime-limits, whereas my "xenharmonic bridge" is meant
> > to include the whole infinite variety of commas which
> > connect any two primes.
>
> The unison vector concept is more general. Any xenharmonic
> bridge is also a unison vector (in a high enough prime limit).
> But in the xenharmonic bridge conception it's a relationship
> between two pitch-ratios, while in the unison vector view it's
> an interval-ratio so it can connect an infinite number of pairs
> of pitches.

it looks like we're back to confusion over the definition
of "xenharmonic bridge" again. i've always meant it to be
an interval-ratio. please explain further why you say it's
"a relationship between two pitch-ratios", and apparently
only that.

>
> > the word "comma" has two distinct meanings: one which
> > describes intervals around ~20 to ~30 cents (Dave Keenan
> > has defined some specific boundaries ... please chime in,
> > Dave), and another which covers not only that category
> > but also all others up to approximately a "semitone".
> >
> > see
> > http://tonalsoft.com/enc/comma.htm
>
> Aren't there any restrictions on when you would consider
> a semitone-sized interval a comma? I pretty much restrict
> my use of "comma" to mean something effectively a unison in
> the tuning system in question.

i can go along with that. i personally only like to use
"comma" in the narrow sense, preferring "anomaly" for the
general sense.

to answer your question: as you know, there *are* some
temperaments where intervals are large as a semitone
(and even larger) act as a unison ... especially in
the lower cardinalities. for a few examples:

unison-vectors in some EDOs:

edo .... 3,5-monzo ..... ~cents

10, 15 ... [-5, 0 > .... 90.22499567
11, 16 ... [ 3, 1 > .... 92.17871646
13, 17 ... [-1, 2 > .... 70.67242686
14 ....... [ 7, 0 > ... 113.6850061
18 ....... [-6,-1 > ... 201.9562809

these can be seen on my bingo-card lattices
http://tonalsoft.com/enc/bingo.htm

>
> > > It is possible to say that a progression on circle
> > > of fifths defines a series of commas?
> > > Follows this series some rule?
> > > Example: considering 3 and 2 the commas would rise in this
> > > progression: 4:3, 9:8, 256:243, pythagorean, (...).
> >
> >
> > no, tuning theorists only consider the last of those
> > to be a "comma".
>
> If a semitone can be a comma, why can't a whole-tone?
> The answer to that question is this tuning system
> (my apologies, Dave):
>
> /tuning/files/Erlich/nana.gif

yep, i have to concede that. this graphic says essentially
the same thing i'm saying in the last row of my table above,
regarding 18edo.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Paul,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
>
> > > > > 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.
> > >
> > >
> > > this is a really good point. a lot of the tuning history
> > > i'm interested in traces exactly this path: 5-limit
> > > periodicity-blocks of various shapes, and how they
> > > evolved over time, according to the usages that can be
> > > gleaned from composers's works and theorists's descriptions.
> >
> > Well, you can join Mathieu in this game, but ultimately (as even
> > Mathieu is forced to admit in one case), I feel a choice of a
> > specific comma pair, let alone an actual periodicity block, is
> > fundamentally ambiguous for the composers I was referring to
above.
>
>
>
> i'm not in disagreement with you about that!
>
> certainly, the best composers have had various different
> periodicity-blocks in mind within the same piece, switching
> back and forth between them at various moments in the piece.

I don't agree with that. I don't think they had any periodicity
blocks in mind. To use the analogy provided by the block, the
meantone (1480-1780) composers were working within a
periodicity "strip" (which can be bent around into a cylinder), while
the 12-tone composers (1780-1980) were working within a
periodicity "sheet" (which can be bent into a torus).

> it looks like we're back to confusion over the definition
> of "xenharmonic bridge" again. i've always meant it to be
> an interval-ratio. please explain further why you say it's
> "a relationship between two pitch-ratios", and apparently
> only that.

In order to take you from one prime limit to another prime limit, it
has to be conceived so that one of the pitch-ratios it separates
belongs to the lower prime limit. Whereas if it's simply a unison
vector, it doesn't have that restriction.

> to answer your question: as you know, there *are* some
> temperaments where intervals are large as a semitone
> (and even larger) act as a unison ... especially in
> the lower cardinalities. for a few examples:
>
> unison-vectors in some EDOs:
>
> edo .... 3,5-monzo ..... ~cents
>
> 10, 15 ... [-5, 0 > .... 90.22499567
> 11, 16 ... [ 3, 1 > .... 92.17871646
> 13, 17 ... [-1, 2 > .... 70.67242686
> 14 ....... [ 7, 0 > ... 113.6850061
> 18 ....... [-6,-1 > ... 201.9562809

Yes. (For 11, 13, and 14, though, a case could be made otherwise. It
depends on which "flavor" of 11, 13, or 14 you pick.)

hi Paul,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Paul,
> >
> >
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> >
> > > > > > 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.
> > > >
> > > >
> > > > this is a really good point. a lot of the tuning history
> > > > i'm interested in traces exactly this path: 5-limit
> > > > periodicity-blocks of various shapes, and how they
> > > > evolved over time, according to the usages that can be
> > > > gleaned from composers's works and theorists's descriptions.
> > >
> > > Well, you can join Mathieu in this game, but ultimately (as even
> > > Mathieu is forced to admit in one case), I feel a choice of a
> > > specific comma pair, let alone an actual periodicity block, is
> > > fundamentally ambiguous for the composers I was referring to
> above.
> >
> >
> >
> > i'm not in disagreement with you about that!
> >
> > certainly, the best composers have had various different
> > periodicity-blocks in mind within the same piece, switching
> > back and forth between them at various moments in the piece.
>
> I don't agree with that. I don't think they had any
> periodicity blocks in mind.

i knew you were going to say that. i should have added
something about it to cover my butt.

i'm not saying that they *consciously* had periodicity-blocks
in mind, because certainly none of them (Beethoven, Liszt,
et al) knew any of this theory. but i do think that
the periodicity-block formality models what they did
in their music.

but anyway, to address your main point ...

> To use the analogy provided by the block, the
> meantone (1480-1780) composers were working within a
> periodicity "strip" (which can be bent around into a cylinder),
> while the 12-tone composers (1780-1980) were working within
> a periodicity "sheet" (which can be bent into a torus).

again, i understand what you're saying and agree with you.

but ... i don't think it's wrong to say that there's a
JI periodicity-block basis behind these other geometrical
periodicities. any of these geometries can be created
by using the right unison-vectors, which ultimately
implies the infinite JI lattice in the background.

(i know, i know ... it's my "rational implications
rearing its ugly head again"...)

> > it looks like we're back to confusion over the definition
> > of "xenharmonic bridge" again. i've always meant it to be
> > an interval-ratio. please explain further why you say it's
> > "a relationship between two pitch-ratios", and apparently
> > only that.
>
> In order to take you from one prime limit to another prime limit, it
> has to be conceived so that one of the pitch-ratios it separates
> belongs to the lower prime limit. Whereas if it's simply a unison
> vector, it doesn't have that restriction.

wow, thanks for that! that finally clarifies the
distinction for me.

... now what about the xenharmonic bridges i've posited
which connect rational pitches to irrational ones?
anything you can say on that would be appreciated.

> > to answer your question: as you know, there *are* some
> > temperaments where intervals are large as a semitone
> > (and even larger) act as a unison ... especially in
> > the lower cardinalities. for a few examples:
> >
> > unison-vectors in some EDOs:
> >
> > edo .... 3,5-monzo ..... ~cents
> >
> > 10, 15 ... [-5, 0 > .... 90.22499567
> > 11, 16 ... [ 3, 1 > .... 92.17871646
> > 13, 17 ... [-1, 2 > .... 70.67242686
> > 14 ....... [ 7, 0 > ... 113.6850061
> > 18 ....... [-6,-1 > ... 201.9562809
>
> Yes. (For 11, 13, and 14, though, a case could be made otherwise. It
> depends on which "flavor" of 11, 13, or 14 you pick.)

right you are ... i didn't think to mention that.
i was using the "flavors" which are depicted on my
bingo-card lattices.

-monz

Are not all constant structures periodicity -blocks? Fundamentally
instead of tempering out a comma, it chooses between the two.

monz wrote:

>
> but ... i don't think it's wrong to say that there's a
> JI periodicity-block basis behind these other geometrical
> periodicities. any of these geometries can be created
> by using the right unison-vectors, which ultimately
> implies the infinite JI lattice in the background.
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

hi kraig,

i haven't studied the concept of constant structures
enough to comment, and will have to defer to others.

i'm looking forward to their answers, because i want
to know more about how PBs and CSs relate to each other
anyway. Carl? Paul? Graham?

-monz

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> Are not all constant structures periodicity -blocks?
> Fundamentally instead of tempering out a comma, it chooses
> between the two.
>
> monz wrote:
>
> >
> > but ... i don't think it's wrong to say that there's a
> > JI periodicity-block basis behind these other geometrical
> > periodicities. any of these geometries can be created
> > by using the right unison-vectors, which ultimately
> > implies the infinite JI lattice in the background.

>i haven't studied the concept of constant structures
>enough to comment, and will have to defer to others.
>
>i'm looking forward to their answers, because i want
>to know more about how PBs and CSs relate to each other
>anyway. Carl? Paul? Graham?

My understanding when we last left this topic was that
not all CS are PB, but all PB with unison vectors smaller
than their smallest step are CS.

-Carl

what is PB?

Carl Lumma wrote:

> >i haven't studied the concept of constant structures
> >enough to comment, and will have to defer to others.
> >
> >i'm looking forward to their answers, because i want
> >to know more about how PBs and CSs relate to each other
> >anyway. Carl? Paul? Graham?
>
> My understanding when we last left this topic was that
> not all CS are PB, but all PB with unison vectors smaller
> than their smallest step are CS.
>
> -Carl
>
>
> You can configure your subscription by sending an empty email to one
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>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> what is PB?

Periodicity Blocks?

(from the last person you'd expect to answer)

Cheers,
Jon

you think i would have gotten that from the context!. thanks!

Jon Szanto wrote:

> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> > what is PB?
>
> Periodicity Blocks?
>
> (from the last person you'd expect to answer)
>
> Cheers,
> Jon
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

I guess i would need an example of a Periodic block that was not a
contant structure.
Albeit if one had a temperment where each interval was non repeating, i
guess the term would not be very applicable although still possible

Carl Lumma wrote:

> >i haven't studied the concept of constant structures
> >enough to comment, and will have to defer to others.
> >
> >i'm looking forward to their answers, because i want
> >to know more about how PBs and CSs relate to each other
> >anyway. Carl? Paul? Graham?
>
> My understanding when we last left this topic was that
> not all CS are PB, but all PB with unison vectors smaller
> than their smallest step are CS.
>
> -Carl
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> Are not all constant structures periodicity -blocks?

All the ones that were presented specifically as constant structures,
yes.

> Fundamentally
> instead of tempering out a comma, it chooses between the two.

If you mean "between the two pitches", then yes, you're right, except
often there are three or more pitches a note is "choosing from".

Ciao!

> 64:63 can "come from the circle of fifths" too. 16/9 -> 7/4 is
down a
> septimal comma.

My thought is that a comma is "natural" when it derives from the
passage to the next prime.
This passage happens more naturally on the ratio which includes the
former and the next prime (2:1, 3:2, 5:3, 7:5, 11:7, 13:11).
The passage is necessary to decrease the complexity of the previous
ratio going through the circle of fifths with 3:2 ratios (from 27:16
to 5:3, from 45:32 to 7:5, from 63:40 to 11:7, from 33:28 to 13:11).

In this context I feel there is a sort of equivalence between
exponents of primes of the former ratio and primes of the next one.
I think that the simpler ratio between two primes is that one with
exponents 1 and -1 for these primes (3:2, 5:3, 7:5, etc). Then you
can put other exponents increasing complexity until you need to pass
to the next prime.

So I think that the passage to 7-limit (from 16:9 to 7:4) is too
anticipated here.

> > > It would be interesting to me to have a mathemathical
explanation
> > > of
> > > > what a comma is.
> > >
> > > To me, it's any small ratio formed from the same primes as the
> > > harmonies are, and which is not a power of a simpler ratio.
> >

Is correct to say that a comma exist always as a bridge (using monzo
terminology) between two primes?
Or exists also some comma between two ratio of the same prime limit?

Lorenzo

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
wrote:

> So I think that the passage to 7-limit (from 16:9 to 7:4) is too
> anticipated here.

Some of the most popular tunings and scales in this "alternative
tuning" world include those that are based on primes 2, 3, and 7, but
omit 5 altogether. These are favored by LaMonte Young, for example.

> Is correct to say that a comma exist always as a bridge (using
monzo
> terminology) between two primes?
> Or exists also some comma between two ratio of the same prime limit?

The latter is frequently true. For example, the Pythagorean comma,
531244:524288, in practice, arises between the two ratios 1024/729
and 729:512. 2048:2025 arises most often as the difference between
135/128 and 16/15, or between 64/45 and 45/32. 50:49 arises as the
difference between 10/7 and 7/5. 128:125 as the difference between
8/5 and 25/16. And so on.

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

> > In order to take you from one prime limit to another prime limit,
it
> > has to be conceived so that one of the pitch-ratios it separates
> > belongs to the lower prime limit. Whereas if it's simply a unison
> > vector, it doesn't have that restriction.
>
>
>
> wow, thanks for that! that finally clarifies the
> distinction for me.

I thought of something else. An additional distinction. Unless I'm
mistaken, the vector (or "Monzo") of a xenharmonic bridge must be 1
(or -1) for the highest prime, while a unison vector has no such
restriction. With a xenharmonic bridge, you're never bridging from
the lower prime limit up to a pitch-ratio that features a *higher
power* of the new, higher prime, rather than just the new, higher
prime itself, right?

Periodicity Block(s).

-C.

>what is PB?
>
>Carl Lumma wrote:
>
>> >i haven't studied the concept of constant structures
>> >enough to comment, and will have to defer to others.
>> >
>> >i'm looking forward to their answers, because i want
>> >to know more about how PBs and CSs relate to each other
>> >anyway. Carl? Paul? Graham?
>>
>> My understanding when we last left this topic was that
>> not all CS are PB, but all PB with unison vectors smaller
>> than their smallest step are CS.
>>
>> -Carl

>I guess i would need an example of a Periodic block that was not a
>contant structure.

Here's one:

1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8

-Carl

How is this a periodic block?

Carl Lumma wrote:

> >I guess i would need an example of a Periodic block that was not a
> >contant structure.
>
> Here's one:
>
> 1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8
>
> -Carl
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

hi Lorenzo,

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...> wrote:

> Ciao!
>
> > 64:63 can "come from the circle of fifths" too. 16/9 -> 7/4 is
> down a
> > septimal comma.
>
> My thought is that a comma is "natural" when it derives from the
> passage to the next prime.
> This passage happens more naturally on the ratio which includes the
> former and the next prime (2:1, 3:2, 5:3, 7:5, 11:7, 13:11).
> The passage is necessary to decrease the complexity of the previous
> ratio going through the circle of fifths with 3:2 ratios (from 27:16
> to 5:3, from 45:32 to 7:5, from 63:40 to 11:7, from 33:28 to 13:11).
>
> In this context I feel there is a sort of equivalence between
> exponents of primes of the former ratio and primes of the next one.
> I think that the simpler ratio between two primes is that one with
> exponents 1 and -1 for these primes (3:2, 5:3, 7:5, etc). Then you
> can put other exponents increasing complexity until you need to pass
> to the next prime.

this is my "xenharmonic bridge" concept.
http://tonalsoft.com/enc/xenharmonic-bridge.htm

>
> So I think that the passage to 7-limit (from 16:9 to 7:4) is too
> anticipated here.
>
>
> > > > It would be interesting to me to have a mathemathical
> explanation
> > > > of
> > > > > what a comma is.
> > > >
> > > > To me, it's any small ratio formed from the same primes as the
> > > > harmonies are, and which is not a power of a simpler ratio.
> > >
>
> Is correct to say that a comma exist always as a bridge (using monzo
> terminology) between two primes?
> Or exists also some comma between two ratio of the same prime limit?
>
> Lorenzo

"comma" can be either. (in either of Dave Keenan's definitions)

re: your first question:
the "equal-temperament" and "bingo-card" entries in the Encyclopaedia
have lattices showing several commas in the 5-limit.

re: your second question:
"xenharmonic bridge" is the term that always means a bridge
between two different kinds of tunings, whether they are
different-prime-limit JIs, temperaments, or combinations of both.

-monz

hi Paul,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > In order to take you from one prime limit to another prime limit,
> it
> > > has to be conceived so that one of the pitch-ratios it separates
> > > belongs to the lower prime limit. Whereas if it's simply a unison
> > > vector, it doesn't have that restriction.
> >
> >
> >
> > wow, thanks for that! that finally clarifies the
> > distinction for me.
>
> I thought of something else. An additional distinction. Unless I'm
> mistaken, the vector (or "Monzo") of a xenharmonic bridge must be 1
> (or -1) for the highest prime, while a unison vector has no such
> restriction. With a xenharmonic bridge, you're never bridging from
> the lower prime limit up to a pitch-ratio that features a *higher
> power* of the new, higher prime, rather than just the new, higher
> prime itself, right?

yes !! very good of you to think of adding that!
it's absolutely correct.

my whole idea of "xenharmonic bridging" is based on the
fact that certain lower-prime-limit pitches emulate those
which *define* higher-prime-limit pitches.

-monz

hi kraig,

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I guess i would need an example of a Periodic block that was not a
> >contant structure.
>
> Here's one:
>
> 1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8
>
> -Carl

hopefully this helps you: i've added a lattice-diagram
of Carl's periodicity-block to the bottom of my
"constant structure" page:

http://tonalsoft.com/enc/constant-structure.htm

-monz

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

> my whole idea of "xenharmonic bridging" is based on the
> fact that certain lower-prime-limit pitches emulate those
> which *define* higher-prime-limit pitches.

Here are some xenharmonic bridges:

5-limit

16/15, 135/128, 81/80, 32805/32768

7-limit

200/189, 28/27, 36/35, 525/512, 64/63, 875/864, 126/125,
225/224, 5120/5103, 65625/65536, 4375/4374

11-limit

77/75, 45/44, 55/54, 56/55, 99/98, 100/99, 176/175, 896/891,
385/384, 441/440, 1375/1372, 6250/6237, 540/539, 5632/5625

We might call a temperament "brigable" if it can be defined in terms
of xenharmonic bridges. The classic example would be meantone, where
81/80 bridges 3 to 5, 126/125 or 225/224 bridges 5 to 7, and 99/98 or
385/384 (depending on which version we want) bridges 7 to 11.

5 ~ 3^4/2^4

7 ~ 5^3/2 3^2

11 ~ 2^7 3 / 5 7 (385/384) or

11 ~ 2 7^2 / 3^2 (99/98)

A bridgable temperament has a fifth as a generator and an octave as a
period, which makes it of a rather particular kind.

This does not seem to be correct or i am missing something

In Pauls article of a gentle introduction we have

This property is
respected not only for the two unison vectors, but also for an additional
vector, (3 1), a "greater limma" or 92
cents. This vector is the sum of the two unison vectors: (4 -1) + (-1 2) =
(3 1)

this is equal to 135/128 from 1/1

I do not think you have 8 tone periodic blocks based on 5 limit intervals

Otherwise i think the term has ventured too far from what Fokker meant

monz wrote:

> hi kraig,
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >I guess i would need an example of a Periodic block that was not a
> > >contant structure.
> >
> > Here's one:
> >
> > 1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8
> >
> > -Carl
>
> hopefully this helps you: i've added a lattice-diagram
> of Carl's periodicity-block to the bottom of my
> "constant structure" page:
>
> http://tonalsoft.com/enc/constant-structure.htm
>
> -monz
>
>
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>>>I guess i would need an example of a Periodic block
>>>that was not a contant structure.
>>
>>Here's one:
>>
>>1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8
>
>How is this a periodic block?

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

(scroll down to the graphic)

-Carl

>We might call a temperament "brigable" if it can be defined in terms
>of xenharmonic bridges. The classic example would be meantone, where
>81/80 bridges 3 to 5, 126/125 or 225/224 bridges 5 to 7, and 99/98 or
>385/384 (depending on which version we want) bridges 7 to 11.
//
>A bridgable temperament has a fifth as a generator and an octave as a
>period, which makes it of a rather particular kind.

From what does this follow?

-Carl

Kraig wrote...

>This does not seem to be correct or i am missing something
>
>In Pauls article of a gentle introduction we have
>
>This property is
>respected not only for the two unison vectors, but also for an
>additional vector, (3 1), a "greater limma" or 92 cents. This
>vector is the sum of the two unison vectors:
>(4 -1) + (-1 2) = (3 1)
>
>this is equal to 135/128 from 1/1
>
>I do not think you have 8 tone periodic blocks based on 5 limit
>intervals
>
>Otherwise i think the term has ventured too far from what
>Fokker meant

Perhaps this would be a good time for Gene to give his definition
of PB.**

How is it related to "epimorphic"?** A google search for
'epimorphic site:www.xenharmony.org' returns nothing. Gene,
does "epimorphic" occur on your site?**

Over at tonalsoft, we have:

"A scale has the epimorphic property, or is epimorphic, if there
is a val h such that if qn is the nth scale degree, then h(qn)=n.
The val h is the characterizing val of the scale."

What data type is qn here?**

** QUESTIONS FOR GENE MARKED WITH **

-Carl

I wrote...

>>We might call a temperament "brigable" if it can be defined in terms
>>of xenharmonic bridges. The classic example would be meantone, where
>>81/80 bridges 3 to 5, 126/125 or 225/224 bridges 5 to 7, and 99/98 or
>>385/384 (depending on which version we want) bridges 7 to 11.
>//
>>A bridgable temperament has a fifth as a generator and an octave as a
>>period, which makes it of a rather particular kind.
>
>From what does this follow?

That is, why do all brigable temperaments have fifth generators?

-Carl

what prevent one from drawing a parallelogram of any size and calling it a
periodic block?
this seems to be completely arbitrary

Carl Lumma wrote:

> >>>I guess i would need an example of a Periodic block
> >>>that was not a contant structure.
> >>
> >>Here's one:
> >>
> >>1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8
> >
> >How is this a periodic block?
>
> http://lumma.org/music/theory/tctmo/
>
> (scroll down to the graphic)
>
> -Carl
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> I wrote...
>
> >>We might call a temperament "brigable" if it can be defined in terms
> >>of xenharmonic bridges. The classic example would be meantone, where
> >>81/80 bridges 3 to 5, 126/125 or 225/224 bridges 5 to 7, and 99/98 or
> >>385/384 (depending on which version we want) bridges 7 to 11.
> >//
> >>A bridgable temperament has a fifth as a generator and an octave as a
> >>period, which makes it of a rather particular kind.
> >
> >From what does this follow?
>
> That is, why do all brigable temperaments have fifth generators?

Brigable <==> octave, fitth generators proof

Octave-fifth ==> brigable

If an octave and a fifth are the period and generator, then an octave
and a twelvth (3) are also. The mapping to primes matrix then tells
how to express 5 and 7 in terms of 2 and 3, which means they give
xenharmonic bridges from {2,3} to 5 and 7

Brigable ==> octave-fifth

If the temperament is brigable, we have commas giving 5 in terms of 2
and 3, 7 in terms of 2,3, and 5, and so forth. We can then solve this
linear system and eliminate everything but 2 and 3; in other words, in
the comma bridging {2,3,5} to 7, replace the 5 with its equivalent in
terms of 2 and 3, and get 7 in terms of 2 and 3. Then replace both the
5 and 7 in the 11 comma with their equivalents in terms of 2 and 3,
and get 11 in terms of 2 and 3. And so forth. Now you have a set of
commas giving the primes in terms of 2 and 3; in other words, a
mapping matrix for 2 and 3 as a pair of generators. But if 2 and 3 are
generators, so are 2 and 3/2.

A third equivalent condition is that the first element of the wedgie
for the temperament is 1. If some other element is 1, we can find two
other primes which work as generators, as I was remarking about
ennealimmal and 3 and 5 a little while back.

>what prevent one from drawing a parallelogram of any size
>and calling it a periodic block?

Nothing, so far as I know. Paul or Gene may have something
to say here. The most 'musically interesting' blocks,
defined by small commas, will be CS according to Paul.

-Carl

>>We might call a temperament "brigable" if it can be defined in terms
>>of xenharmonic bridges. The classic example would be meantone, where
>>81/80 bridges 3 to 5, 126/125 or 225/224 bridges 5 to 7, and 99/98 or
>>385/384 (depending on which version we want) bridges 7 to 11.
>//
>>A bridgable temperament has a fifth as a generator and an octave as a
>>period, which makes it of a rather particular kind.
>
>why do all brigable temperaments have fifth generators?
>
>Brigable <==> octave, fitth generators proof
>
>Octave-fifth ==> brigable

I guess I don't understand xenharmonic bridges. For a comma p/q,
how would I know if it's a bridge?

>If an octave and a fifth are the period and generator, then an octave
>and a twelvth (3) are also.

Yes, ok.

>The mapping to primes matrix then tells
>how to express 5 and 7 in terms of 2 and 3, which means they give
>xenharmonic bridges from {2,3} to 5 and 7

Doesn't *any* map tell us how to express one prime in terms of
another?

>If the temperament is brigable, we have commas giving 5 in terms
>of 2 and 3, 7 in terms of 2,3, and 5, and so forth. We can then
>solve this linear system and eliminate everything but 2 and 3;

Can't a similar chain of substitutions be used to boil everything
down in terms of 5 and 7, or any other pair of primes?

>A third equivalent condition is that the first element of the wedgie
>for the temperament is 1. If some other element is 1, we can find two
>other primes which work as generators, as I was remarking about
>ennealimmal and 3 and 5 a little while back.

I've always thought: to keep complexity low, good temperaments are
likely to use their generator and/or period to hit consonances (not
only primes). How to justify anything stronger than this, I still
have no idea.

-Carl

>"A scale has the epimorphic property, or is epimorphic, if there
>is a val h such that if qn is the nth scale degree, then h(qn)=n.
>The val h is the characterizing val of the scale."
>
>What data type is qn here?**

A ratio, I take it.

-Carl

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> This does not seem to be correct or i am missing something
>
> In Pauls article of a gentle introduction we have
>
> This property is
> respected not only for the two unison vectors, but also for an
additional
> vector, (3 1), a "greater limma" or 92
> cents. This vector is the sum of the two unison vectors: (4 -1) + (-
1 2) =
> (3 1)
>
> this is equal to 135/128 from 1/1

That's a different case -- different unison vectors -- than the one
Carl is pointing to.

> I do not think you have 8 tone periodic blocks based on 5 limit
>intervals

Carl is using 32:25 as a unison vector. That is an interval of 427.37
cents! Not too distant in the lattice either. And you have to,
somehow, think of it as becoming a unison . . .

There are much more reasonable ones than this. For example, the
periodicity block formed by using 648:625 and 250:243 as the unison
vectors has 8 notes. You get

cents ratio

0 1/1
133.24 27/25
315.64 6/5
448.88 162/125
568.72 25/18
701.96 3/2
884.36 5/3
1017.6 9/5

and

0 1/1
182.4 10/9
315.64 6/5
498.04 4/3
568.72 25/18
751.12 125/81
884.36 5/3
1066.8 50/27

for example. Do you see anything "wrong" with these scales, Kraig?

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> what prevent one from drawing a parallelogram of any size and
calling it a
> periodic block?
> this seems to be completely arbitrary

The edges of the parallelogram have to connect functional "unisons"
in the lattice. For purposes of making JI scales, you want these to
be *small* intervals. Carl's construction was a deliberately perverse
one, using a 427-cent interval as a unison vector.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > my whole idea of "xenharmonic bridging" is based on the
> > fact that certain lower-prime-limit pitches emulate those
> > which *define* higher-prime-limit pitches.
>
> Here are some xenharmonic bridges:
>
> 5-limit
>
> 16/15, 135/128, 81/80, 32805/32768
>
> 7-limit
>
> 200/189, 28/27, 36/35, 525/512, 64/63, 875/864, 126/125,
> 225/224, 5120/5103, 65625/65536, 4375/4374
>
> 11-limit
>
> 77/75, 45/44, 55/54, 56/55, 99/98, 100/99, 176/175, 896/891,
> 385/384, 441/440, 1375/1372, 6250/6237, 540/539, 5632/5625
>
> We might call a temperament "brigable" if it can be defined in terms
> of xenharmonic bridges. The classic example would be meantone,
where
> 81/80 bridges 3 to 5, 126/125 or 225/224 bridges 5 to 7, and 99/98
or
> 385/384 (depending on which version we want) bridges 7 to 11.
>
> 5 ~ 3^4/2^4
>
> 7 ~ 5^3/2 3^2
>
> 11 ~ 2^7 3 / 5 7 (385/384) or
>
> 11 ~ 2 7^2 / 3^2 (99/98)

So pajara, defined in terms of 64/63 and 225/224, would
be "bridgable", right? Or do you need to modify your definition?

> A bridgable temperament has a fifth as a generator and an octave as
a
> period, which makes it of a rather particular kind.

Pajara has a half-octave period.

>> >I guess i would need an example of a Periodic block that
>> >was not a contant structure.
>>
>> Here's one:
>>
>> 1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8
>>
>> -Carl
>
>I don't believe that this is a periodicity block.

Since no one has furnished a definition of PB, why not
call it one?

>Are you sure it's not a torsional block?

You accepted it as one several years ago, before torsion was
in the vocabulary. I don't know how to test for it, though.

-Carl

>There are much more reasonable ones than this. For example, the
>periodicity block formed by using 648:625 and 250:243 as the unison
>vectors has 8 notes. You get
>
> cents ratio
>
> 0 1/1
> 133.24 27/25
> 315.64 6/5
> 448.88 162/125
> 568.72 25/18
> 701.96 3/2
> 884.36 5/3
> 1017.6 9/5
>
>and
>
> 0 1/1
> 182.4 10/9
> 315.64 6/5
> 498.04 4/3
> 568.72 25/18
> 751.12 125/81
> 884.36 5/3
> 1066.8 50/27
>
>for example. Do you see anything "wrong" with these scales, Kraig?

Both of these are CS, so they're not the exceptions Kraig was
looking for. Nice scales, though.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >There are much more reasonable ones than this. For example, the
> >periodicity block formed by using 648:625 and 250:243 as the
unison
> >vectors has 8 notes. You get
> >
> > cents ratio
> >
> > 0 1/1
> > 133.24 27/25
> > 315.64 6/5
> > 448.88 162/125
> > 568.72 25/18
> > 701.96 3/2
> > 884.36 5/3
> > 1017.6 9/5
> >
> >and
> >
> > 0 1/1
> > 182.4 10/9
> > 315.64 6/5
> > 498.04 4/3
> > 568.72 25/18
> > 751.12 125/81
> > 884.36 5/3
> > 1066.8 50/27
> >
> >for example. Do you see anything "wrong" with these scales, Kraig?
>
> Both of these are CS, so they're not the exceptions Kraig was
> looking for.

Carl, I was replying to Kraig's specific comment about 5-limit scales
with 8 notes. The fact that these are CS should only make the point
even more strongly to Kraig.

> Nice scales, though.

You can assume I'm paying very close attention to the posts to which
I'm replying. Thanks, though.

>> Nice scales, though.
>
>You can assume I'm paying very close attention to the posts to which
>I'm replying. Thanks, though.

Are you disputing that your post does, in fact, not address Kraig's
original question?

-Carl

Hi Paul---

As far as spacing of scale steps these seem in keeping with an 8 tone
scale.

That one can find any number of ratios of a certain general size within
any limit one chooses to fill in any number of scales steps is nothing
new to me. I have written quite a bit of music where the actual ratios
mattered less than the intervallic size, especially if harmonic concerns
were not a priority.

The normal assumption though with using any limit though is that one
uses uninterrupted chains of connected ratios.

In example one does not use 27/16 until one has used a 9/8, unless of
course it is connected via some other way. I believe that Fokker is quite
consistent with this.

You are then suggesting that periodicity blocks is another unifying
factor that overrides the necessity
of using limits in the most conventional way?

These examples do not uses ratios in this way at all or even treat it as a
concern.
As i state above , i have also used ratios in such fashion within a
particular limit, but this is only is because i have instruments already
built. Not that i objected to resulting scale, but not sure if one did not
have to, why would one even use a limit at all. I have my own answer to
this, but curious about your own)

wallyesterpaulrus wrote:

>
>
> Carl, I was replying to Kraig's specific comment about 5-limit scales
> with 8 notes. The fact that these are CS should only make the point
> even more strongly to Kraig.
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Kraig, I'd really enjoy continuing this communication with you, so
I'm doing the best I can. Please forgive me -- your words, and train
of though, are not always clear to me.

Kraig is replying here to this post:

> >There are much more reasonable ones than this. For example, the
> >periodicity block formed by using 648:625 and 250:243 as the
unison
> >vectors has 8 notes. You get
> >
> > cents ratio
> >
> > 0 1/1
> > 133.24 27/25
> > 315.64 6/5
> > 448.88 162/125
> > 568.72 25/18
> > 701.96 3/2
> > 884.36 5/3
> > 1017.6 9/5
> >
> >and
> >
> > 0 1/1
> > 182.4 10/9
> > 315.64 6/5
> > 498.04 4/3
> > 568.72 25/18
> > 751.12 125/81
> > 884.36 5/3
> > 1066.8 50/27
> >

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> Hi Paul---
>
> As far as spacing of scale steps these seem in keeping with an 8
tone
> scale.
>
> That one can find any number of ratios of a certain general size
within
> any limit one chooses to fill in any number of scales steps is
nothing
> new to me. I have written quite a bit of music where the actual
ratios
> mattered less than the intervallic size, especially if harmonic
concerns
> were not a priority.
>
> The normal assumption though with using any limit though is that
one
> uses uninterrupted chains of connected ratios.
>
> In example one does not use 27/16 until one has used a 9/8, unless
of
> course it is connected via some other way.

Neither of the scales above use 27/16 at all. So I'm not sure why you
bring this up.

I'm going to draw a lattice of the second scale. The first scale is
the same but flipped relative to 1/1.

125/81
....\
.....\
......\
.....50/27---25/18
........\...../.\
.........\.../...\
..........\./.....\
.........10/9-----5/3
............\...../.\
.............\.../...\
..............\./.....\
..............4/3-----1/1
........................\
.........................\
..........................\
.........................6/5

> You are then suggesting that periodicity blocks is another
>unifying
> factor that overrides the necessity
> of using limits in the most conventional way?

I have no idea what you mean.

> These examples do not uses ratios in this way at all or even treat
it as a
> concern.

Don't know what you're getting at.

> As i state above , i have also used ratios in such fashion within a
> particular limit, but this is only is because i have instruments
already
> built. Not that i objected to resulting scale, but not sure if one
did not
> have to, why would one even use a limit at all. I have my own
answer to
> this, but curious about your own)

I don't see your point at all. Neither of the scales above have 27/16
as one of the pitches, nor do they have other features that might be
considered similar, as far as I can see. All the notes are
harmonically contiguous. But yes, Fokker's own blocks did sometimes
have a couple of notes that were harmonically disconnected from the
others -- this was the result of pointy parallelepiped corners.
My "excursion" which you read hints at one way to reduce the
occurrence of such pointy corners.

wallyesterpaulrus wrote:

> Kraig, I'd really enjoy continuing this communication with you, so
> I'm doing the best I can. Please forgive me -- your words, and train
> of though, are not always clear to me.

It is difficult often both ways which is why i prefer examples
which you supply below

>
>
> >
> > The normal assumption though with using any limit though is that
> one
> > uses uninterrupted chains of connected ratios.
> >
> > In example one does not use 27/16 until one has used a 9/8, unless
> of
> > course it is connected via some other way.
>
> Neither of the scales above use 27/16 at all. So I'm not sure why you
> bring this up.

It was an example that lattice space traditionally uses connected ratios
via a grid.
In other words one normally does not jump to remote places of a lattice
grid without
the intermediary tones.

>
>
> I'm going to draw a lattice of the second scale. The first scale is
> the same but flipped relative to 1/1.

Ok now i see how this is a connected chain. for some reason I was having a
difficult time seeing this pattern last night

>
>
> 125/81
> ....\
> .....\
> ......\
> .....50/27---25/18
> ........\...../.\
> .........\.../...\
> ..........\./.....\
> .........10/9-----5/3
> ............\...../.\
> .............\.../...\
> ..............\./.....\
> ..............4/3-----1/1
> ........................\
> .........................\
> ..........................\
> .........................6/5
>
> > You are then suggesting that periodicity blocks is another
> >unifying
> > factor that overrides the necessity
> > of using limits in the most conventional way?
>
> I have no idea what you mean.

the above lattice dispenses with this being a concern . (what i thought
was not correct in the first place)

But now to my original concern which is the relation of constant
structures to periodic blocks and vice versa.
Are they, as far as you have noticed, the same with the latter being a
unique method in which they can be formed?
or to reverse the problem how would you take Centaur and describe it as a
periodic block, if it is possible?
As you might guess i am curious how they overlap and where they do not.

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> the above lattice dispenses with this being a concern . (what i
thought
> was not correct in the first place)

OK, so you agree that there can be "good" 8-note 5-limit scales?

> But now to my original concern which is the relation of constant
> structures to periodic blocks and vice versa.
> Are they, as far as you have noticed, the same with the latter
being a
> unique method in which they can be formed?

They're pretty much the same . . .

I guess there are some far-fetched cases, like the one Carl brought
up, in which the notes of a Fokker periodicity block would have to be
ordered differently than the pitch order, if the CS property is to
hold.

I use "Fokker periodicity block" to mean cases where the
parallelogram/parallelepiped construction is used, and "periodicity
block" more generally -- for example in my excursion, which you read,
I create hexagonal periodicity blocks by rearranging the
parallelograms in a certain way.

> or to reverse the problem how would you take Centaur and describe
it as a
> periodic block, if it is possible?

I'm 99.9% sure it's possible to describe it as a periodicity block,
though it might not be a Fokker periodicity block (Gene could
probably give us a firm yea or nay). Centaur essentially takes 12-
equal and restricts each pitch to only one the harmonic 'functions'
it has in 12-equal, yes? If so, then it's a periodicity block. This
also implies that copies of Centaur can regularly tile the entire 7-
limit lattice with no gaps or overlaps, with the copies separated
from one another by intervals that vanish in 12-equal. This tiling
property is probably the best way to understand where the
phrase "periodic block" comes from. It's as if you're building the
infinite lattice up from these building blocks, each of which is
identical in interval structure, and very close in pitch, to its
neighbors.

> As you might guess i am curious how they overlap and where they do
>not.

All the CSs specifically presented as such have been periodicity
blocks. But it looks like "constant structure" is something you need
to define a little more precisely, because the current definition
contains no implications of JI, of harmonic contiguity, or anything
that would exclude a randomly generated irrational scale.

Fun!

wallyesterpaulrus wrote:

> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
> > the above lattice dispenses with this being a concern . (what i
> thought
> > was not correct in the first place)
>
> OK, so you agree that there can be "good" 8-note 5-limit scales?

It appears so! congratulations.
I imagine that one could possibly discover a large number of other unusual
scales
by applying this method.

> > But now to my original concern which is the relation of constant
> > structures to periodic blocks and vice versa.
> > Are they, as far as you have noticed, the same with the latter
> being a
> > unique method in which they can be formed?
>
> They're pretty much the same . . .
>
> I guess there are some far-fetched cases, like the one Carl brought
> up, in which the notes of a Fokker periodicity block would have to be
> ordered differently than the pitch order, if the CS property is to
> hold.
>
> I use "Fokker periodicity block" to mean cases where the
> parallelogram/parallelepiped construction is used, and "periodicity
> block" more generally -- for example in my excursion, which you read,
> I create hexagonal periodicity blocks by rearranging the
> parallelograms in a certain way.

I am now curious how far you have taken this in the direction of say using
different ratios or 'generating chords' i might call them (sorry if you
already have a term for this). the examples so far have been around the
4-5-6 or the major chord and its complement. i would leave the notion of
chord out of this altogether but it seems possibly a bit soon to examines
grid with a chain of 7/6 running in one direction and 11/9 in another.
The 6-7-8 is probably one that is often mentioned , so curious if you
have followed this one out using the same technique.

>
>
> > or to reverse the problem how would you take Centaur and describe
> it as a
> > periodic block, if it is possible?
>
> I'm 99.9% sure it's possible to describe it as a periodicity block,
> though it might not be a Fokker periodicity block (Gene could
> probably give us a firm yea or nay). Centaur essentially takes 12-
> equal and restricts each pitch to only one the harmonic 'functions'
> it has in 12-equal, yes? If so, then it's a periodicity block. This
> also implies that copies of Centaur can regularly tile the entire 7-
> limit lattice with no gaps or overlaps, with the copies separated
> from one another by intervals that vanish in 12-equal. This tiling
> property is probably the best way to understand where the
> phrase "periodic block" comes from. It's as if you're building the
> infinite lattice up from these building blocks, each of which is
> identical in interval structure, and very close in pitch, to its
> neighbors.

It would have been a lot to ask to be able to easily throw out such
'reverse engineering' of a scale. Like Viggo Brun' algorithm, one can know
that a scale can be formed by using it, but is often hard to figure out
how starting in reverse. Especially those formed by the use of more than
two variables.

>
> > As you might guess i am curious how they overlap and where they do
> >not.
>
> All the CSs specifically presented as such have been periodicity
> blocks. But it looks like "constant structure" is something you need
> to define a little more precisely, because the current definition
> contains no implications of JI, of harmonic contiguity, or anything
> that would exclude a randomly generated irrational scale.

I do not think that constant structures are limited to JI. One could use
noble numbers for that matter. One could say that the term is usually
applied in cases where a large number of the intervals are repeated and
always involve the same number of scale steps. What constitutes large or
even reasonable number will always be subjective and varied with personal
musical taste.

>
>
> Fun!
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

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

> I'm 99.9% sure it's possible to describe it as a periodicity block,
> though it might not be a Fokker periodicity block (Gene could
> probably give us a firm yea or nay).

Centaur is epimorphic and one of the scales which arise from
(15/14)^5 (16/15)^2 (21/20)^3 (28/27)^2 = 2, the sort of thing you
may recall me discussing some time ago. We could pick one of the four
intervals and then find the ratios of it with the rest, and enumerate
the resultant periodicity blocks; for instance picking 21/20 gives us
81/80, 64/63, and 50/49. A survey of this sort of thing with both the
centaur collection of superparticular intervals and the similar
(15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2 might be interesting.

hi Carl,

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

> I guess I don't understand xenharmonic bridges.
> For a comma p/q, how would I know if it's a bridge?

according to what i mean by the term, every ratio which
contains at least two prime-factors > 3 is a xenharmonic-bridge.

... i guess that should be in my definition.

-monz

>> I guess I don't understand xenharmonic bridges.
>> For a comma p/q, how would I know if it's a bridge?
>
>according to what i mean by the term, every ratio which
>contains at least two prime-factors > 3 is a xenharmonic-bridge.
>
>... i guess that should be in my definition.

Does 7/5 count?

-Carl

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

> according to what i mean by the term, every ratio which
> contains at least two prime-factors > 3 is a xenharmonic-bridge.
>
> ... i guess that should be in my definition.

This definition has no relationship at all with your other one, and I
don't see why you need such a word in any case. What's wrong with
saying "7-limit interval" or "11-limit interval"?

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Carl,
>
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > I guess I don't understand xenharmonic bridges.
> > For a comma p/q, how would I know if it's a bridge?
>
>
>
> according to what i mean by the term, every ratio which
> contains at least two prime-factors > 3 is a xenharmonic-bridge.
>
> ... i guess that should be in my definition.
>
>
>
> -monz

oops ... i really meant any two primes > 2,
or >= 3 if you prefer.

-monz

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I guess I don't understand xenharmonic bridges.
> >> For a comma p/q, how would I know if it's a bridge?
> >
> >according to what i mean by the term, every ratio which
> >contains at least two prime-factors > 3 is a xenharmonic-bridge.
> >
> >... i guess that should be in my definition.
>
> Does 7/5 count?
>
> -Carl

i love the way you deliberately use the most perverse
example you can find, and good for you to do so, and
wake up sleepyheads like me.

of course, there need to be other restrictions, namely
that there is some implication that the two pitches
connected by a xenharmonic-bridge are intended in some
way to be perceived as "the same".

in the case of the ancient Greek genera, notes with
the same name could be as much as a 9/8 ratio apart
(= ~204 cents, a pythagorean "whole-tone"), which i
would consider to be a very large xenharmonic-bridge,
practically stretching the term to meaninglessness.

i'd be hard pressed to find an example larger than that
... perhaps JI ratios approximated by 5-ET.

so basically, a xenharmonic-bridge is smaller than a
"whole-tone", and usually much smaller. when i first
thought of the term, i meant it to mean intervals around
the size of a skhisma (~2 cents) or less.

but as i looked more and more, i realized that
xenharmonic-bridges were everywhere. ancient theorists
often posited certain largish prime-factors which gave
nice superparticular arithmetical proportions of string-lengths
for their genera. upon further analysis, many of these
notes turned out to be clever imposters for ratios which
had already been long-established in lower-prime-limit
tunings, especially pythagorean.

i really should try to set aside some time to put
examples of this into the Encyclopaedia entry.
i've posted on this stuff to this list in the past
... maybe someone can find it.

-monz

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
>
> wallyesterpaulrus wrote:
>
> > --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...>
wrote:
> >
> > > the above lattice dispenses with this being a concern . (what i
> > thought
> > > was not correct in the first place)
> >
> > OK, so you agree that there can be "good" 8-note 5-limit scales?
>
> It appears so! congratulations.

Thanks, but if you think about it, it's no big deal. I've
merely "detempered" the familiar octatonic or diminished scale. Or
equivalently, I could get the same result by "detempering" an 8-note
porcupine scale, which is an MOS whose generator is about 160-170
cents.

> I imagine that one could possibly discover a large number of other
unusual
> scales
> by applying this method.

Well, I posted a bunch here, over three years ago I think, under the
title "What's your favorite number?" . . . and I also remember having
a similar discussion with you in which I presented a 20-note example.

Even the 8-note scale can be reshuffled in many ways, by transposing
one note by 648:625 and/or another by 250:243, to create other,
related 8-note 5-limit JI scales.

> I am now curious how far you have taken this in the direction of
say using
> different ratios or 'generating chords' i might call them (sorry if
you
> already have a term for this). the examples so far have been around
the
> 4-5-6 or the major chord and its complement. i would leave the
notion of
> chord out of this altogether but it seems possibly a bit soon to
examines
> grid with a chain of 7/6 running in one direction and 11/9 in
another.

This could be done, though I personally haven't done much in this
direction. Bill Sethares and I are planning on creating periodicity
blocks where the grid has chains of irrational intervals that come
from inharmonic spectra.

> The 6-7-8 is probably one that is often mentioned , so curious if
you
> have followed this one out using the same technique.

Sure, there's been some work done on the case where primes 2, 3, and
7 form the framework, and 5 is left out. In fact, Gene was just
posting some such periodicity blocks here, about a month or two ago,
in response to Aaron. These were 12-note blocks, but that's far from
being the only number you can get in this context. What's your
favorite number?

> > > or to reverse the problem how would you take Centaur and
describe
> > it as a
> > > periodic block, if it is possible?
> >
> > I'm 99.9% sure it's possible to describe it as a periodicity
block,
> > though it might not be a Fokker periodicity block (Gene could
> > probably give us a firm yea or nay). Centaur essentially takes 12-
> > equal and restricts each pitch to only one the
harmonic 'functions'
> > it has in 12-equal, yes? If so, then it's a periodicity block.
This
> > also implies that copies of Centaur can regularly tile the entire
7-
> > limit lattice with no gaps or overlaps, with the copies separated
> > from one another by intervals that vanish in 12-equal. This tiling
> > property is probably the best way to understand where the
> > phrase "periodic block" comes from. It's as if you're building the
> > infinite lattice up from these building blocks, each of which is
> > identical in interval structure, and very close in pitch, to its
> > neighbors.
>
> It would have been a lot to ask to be able to easily throw out such
> 'reverse engineering' of a scale.

I think it would be asking very little, but maybe we're
misunderstanding one another . . .

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Carl,
>
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > I guess I don't understand xenharmonic bridges.
> > For a comma p/q, how would I know if it's a bridge?
>
>
>
> according to what i mean by the term, every ratio which
> contains at least two prime-factors > 3 is a xenharmonic-bridge.
>
> ... i guess that should be in my definition.
>
>
>
> -monz

Hi Monz,

I thought you just finished agreeing with me that a xenharmonic
bridge has to have an exponent of 1 or -1 on its highest prime.

What happened?

-Paul

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

> So pajara, defined in terms of 64/63 and 225/224, would
> be "bridgable", right? Or do you need to modify your definition?

No, because you don't have a {2,3} to 5 bridge. In fact, pajara is
not bridable from any pair of primes, since none of the coefficients
of the wedgie is +-1.

> > A bridgable temperament has a fifth as a generator and an octave
as
> a
> > period, which makes it of a rather particular kind.
>
> Pajara has a half-octave period.

It's not bridgable, either.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > So pajara, defined in terms of 64/63 and 225/224, would
> > be "bridgable", right? Or do you need to modify your definition?
>
> No, because you don't have a {2,3} to 5 bridge.

So it seems that you do, indeed, need to modify your definition, to
say what you want it to mean.

> In fact, pajara is
> not bridable from any pair of primes, since none of the coefficients
> of the wedgie is +-1.

This is the first reference to "pair of primes" I'm seeing in any
discussion of "xenharmonic bridges". Care to elaborate where this is
popping up from?

--- 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:
> >
> > > So pajara, defined in terms of 64/63 and 225/224, would
> > > be "bridgable", right? Or do you need to modify your definition?
> >
> > No, because you don't have a {2,3} to 5 bridge.
>
> So it seems that you do, indeed, need to modify your definition, to
> say what you want it to mean.

I didn't define it to mean "has a wedgie which can be defined in
terms of xenharmonic commas", which is a pointless claim in limits
above 5. I meant it could be defined in terms of xenharmonic bridges,
which means we have a xenharmonic bridge from {2,3} to 7, etc, in
accordance with what Monz said. It is equivalent to saying "the
wedgie has a 1 as its first coefficient" (meaning a bival, of course.)

> > In fact, pajara is
> > not bridable from any pair of primes, since none of the
coefficients
> > of the wedgie is +-1.
>
> This is the first reference to "pair of primes" I'm seeing in any
> discussion of "xenharmonic bridges". Care to elaborate where this
is
> popping up from?

From what Monz said. The bridge bridges from lower limits to the next
prime limit, so the first bridge has to bridge from {2,3} to 5, the
second from {2,3,5} to 7, and so forth. So, of course, you could
start anywhere you liked in terms of primes, not just {2,3}. You
could even extend the question to whether there is a pair of
consonances which we can start the bridge from, I suppose.

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

> From what Monz said. The bridge bridges from lower limits to the
next
> prime limit,

Monz didn't say "next prime limit" -- it can be any higher prime
limit -- but it looks like you've satisfactorily patched up your
definition now.

hi Paul,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Carl,
> >
> >
> > --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >
> > > I guess I don't understand xenharmonic bridges.
> > > For a comma p/q, how would I know if it's a bridge?
> >
> >
> >
> > according to what i mean by the term, every ratio which
> > contains at least two prime-factors > 3 is a xenharmonic-bridge.
> >
> > ... i guess that should be in my definition.
> >
> >
> >
> > -monz
>
> Hi Monz,
>
> I thought you just finished agreeing with me that a xenharmonic
> bridge has to have an exponent of 1 or -1 on its highest prime.
>
> What happened?
>
> -Paul

yes, actually, i did agree with you, and you are right.

i was thinking along these lines: any ratio which has
at least two prime-factors >= 3 will bridge from one
prime-dimension to another. in most cases, the bridge
will not connect to the higher prime at the +/-1 exponent,
but of course all replicants of that interval are identical,
so that the one which *does* connect to the higher prime
at the +/-1 exponent is the one i would use to define the
bridge.

hope that's clear. perhaps you can re-word it better.
if so, i'll put it into the definition.

-monz

hi Paul and Gene,

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

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > From what Monz said. The bridge bridges from lower limits to the
> next
> > prime limit,
>
> Monz didn't say "next prime limit" -- it can be any higher prime
> limit -- but it looks like you've satisfactorily patched up your
> definition now.

when i first came up with the idea of a "xenharmonic-bridge",
back in 1998, it was specifically in connection with
Eratosthenes's use of certain 19-limit pitches which are
audibly indistinguishable frome their 3-limit (Pythagorean)
counterparts, which is how they had always been tuned before
him, at least in theory.

i'm just tossing this into the discussion in hopes that
it helps you guys clarify things. i myself have had a
difficult time definining exactly what i mean by
"xenharmonic-bridge", so i very much welcome this give-and-take.

-monz

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

> hi Paul and Gene,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > > From what Monz said. The bridge bridges from lower limits to the
> > next
> > > prime limit,
> >
> > Monz didn't say "next prime limit" -- it can be any higher prime
> > limit -- but it looks like you've satisfactorily patched up your
> > definition now.
>
>
>
> when i first came up with the idea of a "xenharmonic-bridge",
> back in 1998, it was specifically in connection with
> Eratosthenes's use of certain 19-limit pitches which are
> audibly indistinguishable frome their 3-limit (Pythagorean)
> counterparts, which is how they had always been tuned before
> him, at least in theory.

just to clarify: that xenharmonic-bridge in Eratosthenes
is thus a 3==19 bridge. so it skips 5 primes in between.

-monz

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

> just to clarify: that xenharmonic-bridge in Eratosthenes
> is thus a 3==19 bridge. so it skips 5 primes in between.

These are the kind of xenharmonic bridges used in sagittal, so we
might call them sagittal bridges:

Definition 1: A *xenharmonic bridge* is a small rational interval
such that the largest prime dividing it (ie. dividing either
numerator or denominator in reduced form) has an exponent of +-1.

Definition 2: A *sagittal bridge* is a xenharmonic bridge with only
2, 3 and a single prime p>3 in its factorization; that is, a small
rational interval of the form 2^a 3^b p or 2^a 3^b / p.

Do these definitions capture what you want out of the notion?

I assume that the results would be closely related scales which to me
would be interesting

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
>
> > I'm 99.9% sure it's possible to describe it as a periodicity block,
> > though it might not be a Fokker periodicity block (Gene could
> > probably give us a firm yea or nay).
>
> Centaur is epimorphic and one of the scales which arise from
> (15/14)^5 (16/15)^2 (21/20)^3 (28/27)^2 = 2, the sort of thing you
> may recall me discussing some time ago. We could pick one of the four
> intervals and then find the ratios of it with the rest, and enumerate
> the resultant periodicity blocks; for instance picking 21/20 gives us
> 81/80, 64/63, and 50/49. A survey of this sort of thing with both the
> centaur collection of superparticular intervals and the similar
> (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2 might be interesting.
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > just to clarify: that xenharmonic-bridge in Eratosthenes
> > is thus a 3==19 bridge. so it skips 5 primes in between.
>
> These are the kind of xenharmonic bridges used in sagittal,
> so we might call them sagittal bridges:
>
> Definition 1: A *xenharmonic bridge* is a small rational
> interval such that the largest prime dividing it (ie.
> dividing either numerator or denominator in reduced form)
> has an exponent of +-1.
>
> Definition 2: A *sagittal bridge* is a xenharmonic bridge
> with only 2, 3 and a single prime p>3 in its factorization;
> that is, a small rational interval of the form 2^a 3^b p or
> 2^a 3^b / p.
>
> Do these definitions capture what you want out of the notion?

they look good to me ... but i'm interested in what others
have to say, particularly Paul.

-monz

>> just to clarify: that xenharmonic-bridge in Eratosthenes
>> is thus a 3==19 bridge. so it skips 5 primes in between.
>
>These are the kind of xenharmonic bridges used in sagittal, so we
>might call them sagittal bridges:
>
>Definition 1: A *xenharmonic bridge* is a small rational interval
>such that the largest prime dividing it (ie. dividing either
>numerator or denominator in reduced form) has an exponent of +-1.

Aha!

-Carl

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Paul,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > hi Carl,
> > >
> > >
> > > --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >
> > > > I guess I don't understand xenharmonic bridges.
> > > > For a comma p/q, how would I know if it's a bridge?
> > >
> > >
> > >
> > > according to what i mean by the term, every ratio which
> > > contains at least two prime-factors > 3 is a xenharmonic-bridge.
> > >
> > > ... i guess that should be in my definition.
> > >
> > >
> > >
> > > -monz
> >
> > Hi Monz,
> >
> > I thought you just finished agreeing with me that a xenharmonic
> > bridge has to have an exponent of 1 or -1 on its highest prime.
> >
> > What happened?
> >
> > -Paul
>
>
> yes, actually, i did agree with you, and you are right.
>
> i was thinking along these lines: any ratio which has
> at least two prime-factors >= 3 will bridge from one
> prime-dimension to another. in most cases, the bridge
> will not connect to the higher prime at the +/-1 exponent,
> but of course all replicants of that interval are identical,
> so that the one which *does* connect to the higher prime
> at the +/-1 exponent is the one i would use to define the
> bridge.

Hmmm? I don't understand this. Either the exponent at the final
position the vector is +/-1, or it isn't. If it isn't, the vector
can't be a xenharmonic bridge. Didn't you agree with this?

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > just to clarify: that xenharmonic-bridge in Eratosthenes
> > > is thus a 3==19 bridge. so it skips 5 primes in between.
> >
> > These are the kind of xenharmonic bridges used in sagittal,
> > so we might call them sagittal bridges:
> >
> > Definition 1: A *xenharmonic bridge* is a small rational
> > interval such that the largest prime dividing it (ie.
> > dividing either numerator or denominator in reduced form)
> > has an exponent of +-1.
> >
> > Definition 2: A *sagittal bridge* is a xenharmonic bridge
> > with only 2, 3 and a single prime p>3 in its factorization;
> > that is, a small rational interval of the form 2^a 3^b p or
> > 2^a 3^b / p.
> >
> > Do these definitions capture what you want out of the notion?
>
>
>
> they look good to me ... but i'm interested in what others
> have to say, particularly Paul.

Looks good to me! Dave/George?

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > > just to clarify: that xenharmonic-bridge in Eratosthenes
> > > > is thus a 3==19 bridge. so it skips 5 primes in between.
> > >
> > > These are the kind of xenharmonic bridges used in sagittal,
> > > so we might call them sagittal bridges:
> > >
> > > Definition 1: A *xenharmonic bridge* is a small rational
> > > interval such that the largest prime dividing it (ie.
> > > dividing either numerator or denominator in reduced form)
> > > has an exponent of +-1.
> > >
> > > Definition 2: A *sagittal bridge* is a xenharmonic bridge
> > > with only 2, 3 and a single prime p>3 in its factorization;
> > > that is, a small rational interval of the form 2^a 3^b p or
> > > 2^a 3^b / p.
> > >
> > > Do these definitions capture what you want out of the notion?
> >
> > they look good to me ... but i'm interested in what others
> > have to say, particularly Paul.
>
> Looks good to me! Dave/George?

The linchpin of the Sagittal system is 4095:4096 (3^2*5*7*13:2^12,
~0.423c), as mentioned in our XH18 article, page 5, paragraph 3.
A "preprint" of this is now available on the Sagittal homepage:

http://dkeenan.com/sagittal/

This "tridecimal schismina" enables ratios of 13 to be notated using
symbols that would otherwise represent only complex 7-limit ratios.
Look at Table 1 on page 9: near the bottom you'll find bold entries
for the 13M and 13L dieses grouped with 7-limit ratios of
approximately the same size. Other Sagittal "schisminas" may be
determined from this table by looking for the 11:13 kleisma, 7:13 S-
diesis, and 5:13 S-diesis entries, which are each grouped with 11-
limit "commas" of approximately the same size (for which the notation
is the same).

These "schisminas" vanish at the medium and high-precision levels of
Sagittal JI, but there is also an extreme-precision JI level (not
discussed in the article) at which 13-limit consonances have separate
symbols -- in case anyone really needs to distinguish certain pairs
of pitches differing less than a cent.

Anyway, the bottom line is that it looks as if the term "sagittal
bridge" would not be a good one for the category of small rational
intervals in the form 2^a 3^b p or 2^a 3^b / p.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> Anyway, the bottom line is that it looks as if the term "sagittal
> bridge" would not be a good one for the category of small rational
> intervals in the form 2^a 3^b p or 2^a 3^b / p.

Most of your notation symbols are what I wanted to call a sagittal
bridge, so what would you suggest as an alternative?

hi Paul,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Paul,
> >
> >
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > > hi Carl,
> > > >
> > > >
> > > > --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > > >
> > > > > I guess I don't understand xenharmonic bridges.
> > > > > For a comma p/q, how would I know if it's a bridge?
> > > >
> > > >
> > > >
> > > > according to what i mean by the term, every ratio which
> > > > contains at least two prime-factors > 3 is a
xenharmonic-bridge.
> > > >
> > > > ... i guess that should be in my definition.
> > > >
> > > >
> > > >
> > > > -monz
> > >
> > > Hi Monz,
> > >
> > > I thought you just finished agreeing with me that a xenharmonic
> > > bridge has to have an exponent of 1 or -1 on its highest prime.
> > >
> > > What happened?
> > >
> > > -Paul
> >
> >
> > yes, actually, i did agree with you, and you are right.
> >
> > i was thinking along these lines: any ratio which has
> > at least two prime-factors >= 3 will bridge from one
> > prime-dimension to another. in most cases, the bridge
> > will not connect to the higher prime at the +/-1 exponent,
> > but of course all replicants of that interval are identical,
> > so that the one which *does* connect to the higher prime
> > at the +/-1 exponent is the one i would use to define the
> > bridge.
>
> Hmmm? I don't understand this. Either the exponent at the final
> position the vector is +/-1, or it isn't. If it isn't, the vector
> can't be a xenharmonic bridge. Didn't you agree with this?

yes, this is correct. i guess i got confused.

boy, Yahoo's search engine really sucks ... it took me
about an hour of active searching (meaning that i couldn't
run the search in the background, but instead had to keep
clicking "Next") to find some old posts i sent to this
list concerning xenharmonic-bridges in ancient Greek and
Roman tunings.

/tuning/topicId_33762.html#33763
/tuning/topicId_26618.html#26618
/tuning/topicId_7159.html#7159

to summarize:

Eratosthenes 3==19 bridge:
2,3,5,19-monzo [6 -5, -1 1> = ratio 1216:1215 = ~1.424297941 cents

Boethius 3==19 bridge:
2,3,19-monzo [-9 3, 1> = ratio 513:512 = ~3.378018728 cents

-monz

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

> to summarize:
>
> Eratosthenes 3==19 bridge:
> 2,3,5,19-monzo [6 -5, -1 1> = ratio 1216:1215 = ~1.424297941 cents
>
> Boethius 3==19 bridge:
> 2,3,19-monzo [-9 3, 1> = ratio 513:512 = ~3.378018728 cents

Why is the first one a 3==19 bridge if it includes 5 as one of its
factors?

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

> boy, Yahoo's search engine really sucks ... it took me
> about an hour of active searching (meaning that i couldn't
> run the search in the background, but instead had to keep
> clicking "Next") to find some old posts i sent to this
> list concerning xenharmonic-bridges in ancient Greek and
> Roman tunings.

I suggested on tuning-math that it would be very helpful if someone
could make a file just of the subject lines. Can this be done?

> Boethius 3==19 bridge:
> 2,3,19-monzo [-9 3, 1> = ratio 513:512 = ~3.378018728 cents

Hmmm--a "boethius bridge" rather than a "sagittal bridge"?

hi Gene,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > Boethius 3==19 bridge:
> > 2,3,19-monzo [-9 3, 1> = ratio 513:512 = ~3.378018728 cents
>
> Hmmm--a "boethius bridge" rather than a "sagittal bridge"?

i think it would be right and proper to honor Boethius
by naming something after him!

i'm still trying to understand why George objects to
"sagittal bridge" ... but if that name really won't
stick, i'm all for "boethius bridge" ... and of course,
i'll hyphenate it. :)

-monz

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > to summarize:
> >
> > Eratosthenes 3==19 bridge:
> > 2,3,5,19-monzo [6 -5, -1 1> = ratio 1216:1215 = ~1.424297941 cents
> >
> > Boethius 3==19 bridge:
> > 2,3,19-monzo [-9 3, 1> = ratio 513:512 = ~3.378018728 cents
>
> Why is the first one a 3==19 bridge if it includes 5 as one of its
> factors?

hmmm ... that *is* a good question!

the oldest post i linked to was this one:
/tuning/topicId_7159.html#7159

which contains a tabulation of all instances of this
particular xenharmonic-bridge in Eratosthenes's complete
system. this one in particular was the first one i
noticed:

64/81 [= ~-407.8 cents] diatonic parhypate meson
15/19 [= ~-409.2 cents] chr. parhypate meson / enh. lichaonos mes.

what struck me back then was that he was using a 19-limit
ratio to "impersonate" our old friend, the Pythagorean
ditone 64/81 ... or in other words, a bridging from 3 to 19.

but it sure does look to me like i should be calling
this a 5==19 bridge ... feedback appreciated.

-monz

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

/tuning/topicId_53869.html#53929

> I don't agree with that. I don't think they had any periodicity
> blocks in mind. To use the analogy provided by the block, the
> meantone (1480-1780) composers were working within a
> periodicity "strip" (which can be bent around into a cylinder),
while
> the 12-tone composers (1780-1980) were working within a
> periodicity "sheet" (which can be bent into a torus).
>

***This is really fascinating, Paul, but could you please elaborate a
little bit in such a way that I can understand it?? Does it have to
do with the number of commas that are eliminated??

Thanks!

Joseph

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > to summarize:
> > >
> > > Eratosthenes 3==19 bridge:
> > > 2,3,5,19-monzo [6 -5, -1 1> = ratio 1216:1215 = ~1.424297941
cents
> > >
> > > Boethius 3==19 bridge:
> > > 2,3,19-monzo [-9 3, 1> = ratio 513:512 = ~3.378018728 cents
> >
> > Why is the first one a 3==19 bridge if it includes 5 as one of
its
> > factors?
>
>
> hmmm ... that *is* a good question!
>
> the oldest post i linked to was this one:
> /tuning/topicId_7159.html#7159
>
> which contains a tabulation of all instances of this
> particular xenharmonic-bridge in Eratosthenes's complete
> system. this one in particular was the first one i
> noticed:
>
> 64/81 [= ~-407.8 cents] diatonic parhypate meson
> 15/19 [= ~-409.2 cents] chr. parhypate meson / enh. lichaonos mes.
>
>
> what struck me back then was that he was using a 19-limit
> ratio to "impersonate" our old friend, the Pythagorean
> ditone 64/81 ... or in other words, a bridging from 3 to 19.
>
>
> but it sure does look to me like i should be calling
> this a 5==19 bridge ... feedback appreciated.
>
>
> -monz

Ah, now I see why you called it a 3==19 bridge. Makes sense, but I
guess this "=" terminology is almost asking for confusion. Maybe it's
best just to give the "monzo", Monzo :)

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_53869.html#53929
>
>
> > I don't agree with that. I don't think they had any periodicity
> > blocks in mind. To use the analogy provided by the block, the
> > meantone (1480-1780) composers were working within a
> > periodicity "strip" (which can be bent around into a cylinder),
> while
> > the 12-tone composers (1780-1980) were working within a
> > periodicity "sheet" (which can be bent into a torus).
> >
>
> ***This is really fascinating, Paul, but could you please elaborate
a
> little bit in such a way that I can understand it?? Does it have
to
> do with the number of commas that are eliminated??

Exactly. Every time you eliminate a comma, you effectively slice the
lattice in two places (a comma apart), and connect the two ends
together. I've made a more detailed post on this in the past, and
Monz, with his new Tonalsoft software, can make it much more explicit
with actual pictures of this cylinder and torus.

I think I had you do this at one point -- you started with a big,
flat 5-limit lattice. Then you cut a random line through the lattice,
and cut a second line, parallel to the first, a syntonic comma away.
Then you took what remained between the two cuts, and bent it into a
cylinder, with the two cuts meeting each other, so that the
connections between notes matched up correctly across the cuts. Do
you remember this?

Now if you had done this with a flexible rubber sheet, you'd be able
to repeat the process, only now making two parallel cuts that were
separated by an *enharmonic equivalence* -- that is, if one cut went
through Ab at a certain angle, the other would go through G# at the
same angle. If you then took the portion of the cylinder between
these two cuts, and connected the ends so that the two cuts met each
other, you'd have a torus. And if you did it so that the connections
between notes matched up correctly across the cuts, you'd have a
perfect geometrical model of 12-equal.

Make sense?

Now should I still try to elaborate my original post?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
>
> > Anyway, the bottom line is that it looks as if the term "sagittal
> > bridge" would not be a good one for the category of small rational
> > intervals in the form 2^a 3^b p or 2^a 3^b / p.
>
> Most of your notation symbols are what I wanted to call a sagittal
> bridge, so what would you suggest as an alternative?

Notational bridges?

Surely they existed and were used as such with other symbols before the sagittal symbols
were invented for them.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Notational bridges?
>
> Surely they existed and were used as such with other symbols before
the sagittal symbols
> were invented for them.

As eg Boethius. Boethius bridge it is, I guess, but it's clear these
are very important to Sagittal.

hi Paul and Joseph,

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

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > /tuning/topicId_53869.html#53929
> >
> >
> > > I don't agree with that. I don't think they had any
> > > periodicity blocks in mind. To use the analogy provided
> > > by the block, the meantone (1480-1780) composers were
> > > working within a periodicity "strip" (which can be bent
> > > around into a cylinder), while the 12-tone composers
> > > (1780-1980) were working within a periodicity "sheet"
> > > (which can be bent into a torus).
> > >
> >
> > ***This is really fascinating, Paul, but could you please
> > elaborate a little bit in such a way that I can understand
> > it?? Does it have to do with the number of commas that
> > are eliminated??
>
> Exactly. Every time you eliminate a comma, you effectively
> slice the lattice in two places (a comma apart), and connect
> the two ends together. I've made a more detailed post on this
> in the past, and Monz, with his new Tonalsoft software, can
> make it much more explicit with actual pictures of this
> cylinder and torus.
>
> <snip, even tho it was relevant>

i'll elaborate a bit more, in hope of helping both of you
understand me better ...

of course i understand that a composer who is working
with a tuning system which reduces the dimensions of the
lattice, will be doing things harmonically which are
impossible on a regular JI lattice.

but, i do believe that the JI lattice lies at the basis
of all the harmonic moves that composers imagines.
perhaps this is simply a biased personal opinion.

anyway, the Tonalsoft software will indeed draw toroidal
lattices for equal-temperaments and helical lattices for
meantones and other linear-temperaments. however, the
user always has the option to "flatten" those lattices
into either rectangular or triangular regular JI lattices.

i have to put "flatten" in quotes, because if the JI
lattice is >2D (i.e., the prime-space uses more than
two prime-factors), it's already not flat. but the point
is that the software can convert the closed geometry of
the temperament into the open-ended geometry of JI.
the user can even have both views (or even all three)
open simultaneously.

-monz

A similar but different question on the same terms JP quoted:

> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> > /tuning/topicId_53869.html#53929
> > > >>I don't agree with that. I don't think they had any periodicity >>blocks in mind. To use the analogy provided by the block, the >>meantone (1480-1780) composers were working within a >>periodicity "strip" (which can be bent around into a cylinder), > > while > >>the 12-tone composers (1780-1980) were working within a >>periodicity "sheet" (which can be bent into a torus).
>>

While I can see the derivation of these geometrical temperament models, I completely failed to see why "block" shouldn't be used for any of these. Certainly, if you apply that kind of stringency to all aspects of the tone space, a 5-limit scale is not a block at all, but just a face?

(And I don't mention this because I think another term is needed. On the contrary, less names and more qualifications/explanations/overt criteria for differentiation would suit me fine.)

klaus

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

/tuning/topicId_53869.html#54263

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > /tuning/topicId_53869.html#53929
> >
> >
> > > I don't agree with that. I don't think they had any
periodicity
> > > blocks in mind. To use the analogy provided by the block, the
> > > meantone (1480-1780) composers were working within a
> > > periodicity "strip" (which can be bent around into a
cylinder),
> > while
> > > the 12-tone composers (1780-1980) were working within a
> > > periodicity "sheet" (which can be bent into a torus).
> > >
> >
> > ***This is really fascinating, Paul, but could you please
elaborate
> a
> > little bit in such a way that I can understand it?? Does it
have
> to
> > do with the number of commas that are eliminated??
>
> Exactly. Every time you eliminate a comma, you effectively slice
the
> lattice in two places (a comma apart), and connect the two ends
> together. I've made a more detailed post on this in the past, and
> Monz, with his new Tonalsoft software, can make it much more
explicit
> with actual pictures of this cylinder and torus.
>
> I think I had you do this at one point -- you started with a big,
> flat 5-limit lattice. Then you cut a random line through the
lattice,
> and cut a second line, parallel to the first, a syntonic comma
away.
> Then you took what remained between the two cuts, and bent it into
a
> cylinder, with the two cuts meeting each other, so that the
> connections between notes matched up correctly across the cuts. Do
> you remember this?
>

***Hi Paul,

I vaguely remember making some kind of torus or cylinder at one
time...

> Now if you had done this with a flexible rubber sheet, you'd be
able
> to repeat the process, only now making two parallel cuts that were
> separated by an *enharmonic equivalence* -- that is, if one cut
went
> through Ab at a certain angle, the other would go through G# at
the
> same angle. If you then took the portion of the cylinder between
> these two cuts, and connected the ends so that the two cuts met
each
> other, you'd have a torus. And if you did it so that the
connections
> between notes matched up correctly across the cuts, you'd have a
> perfect geometrical model of 12-equal.
>
> Make sense?
>

***Well, that's very interesting, and I'm going to have to try to do
this again when I get a chance. It's a great tuning "learning by
doing" project... Tuning workshop, anybody?? :)

> Now should I still try to elaborate my original post?

***Sure... Gee, this list has suddenly gotten very active again
lately... :)

J. Pehrson

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
> A similar but different question on the same terms JP quoted:
>
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> >
> > /tuning/topicId_53869.html#53929
> >
> >
> >
> >>I don't agree with that. I don't think they had any periodicity
> >>blocks in mind. To use the analogy provided by the block, the
> >>meantone (1480-1780) composers were working within a
> >>periodicity "strip" (which can be bent around into a cylinder),
> >
> > while
> >
> >>the 12-tone composers (1780-1980) were working within a
> >>periodicity "sheet" (which can be bent into a torus).
> >>
>
> While I can see the derivation of these geometrical
> temperament models,

The Robert Kelly paper Monz cited may help others see it too.

> I completely failed to see why "block"
> shouldn't be used for any of these.

Because a "block" has confining borders which these don't.

> Certainly, if you apply
> that kind of stringency to all aspects of the tone space, a
> 5-limit scale is not a block at all, but just a face?

Can you please elaborate on what you have in mind here? Sounds very
interesting, but I don't know quite what you mean. I'd like to be
able to answer your question, though.

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> > Now should I still try to elaborate my original post?
>
> ***Sure...

Would you be so kind as to read the Robert Kelly article Monz just
dug up? Much of it concerns this exact topic, and it's from "music
academia", so it might be closer to "your language" . . .

hi Paul and klaus,

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

> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:

> > Certainly, if you apply that kind of stringency to
> > all aspects of the tone space, a 5-limit scale is not
> > a block at all, but just a face?
>
> Can you please elaborate on what you have in mind here?
> Sounds very interesting, but I don't know quite what you
> mean. I'd like to be able to answer your question, though.

i think klaus simply means that the "8ve"-equivalent
(i.e., ignore prime-factor 2) 5-limit lattice is a 2D
structure, thus, it cannot contain a "block", but only
a "face" (or plane).

a block can only occur in a >2D lattice ...
or pehaps *only* in a 3D lattice? the higher dimensions
can have equivalent periodicities, but i don't think
they'll be shaped like "blocks".

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> the higher dimensions

> can have equivalent periodicities, but i don't think
> they'll be shaped like "blocks".

Well, they're "hyper-blocks", certainly. They've always been referred
to simply as "blocks", since for example just saying "11-limit"
already implies that you're beyond 3 dimensions, so there's no need
to be redundant about that point . . .

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > Notational bridges?
> >
> > Surely they existed and were used as such with other symbols
before
> the sagittal symbols
> > were invented for them.
>
> As eg Boethius. Boethius bridge it is, I guess, but it's clear
these
> are very important to Sagittal.

Yes. But Sagittal also has symbols whose primary comma role is of
the form
2^n.3^m.p^2 or
2^n.3^m.p.q or
2^n.3^m.p/q

where n,m are integers and p,q are primes greater than 3.

wallyesterpaulrus schrieb:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >>the higher dimensions
> > >>can have equivalent periodicities, but i don't think
>>they'll be shaped like "blocks".
> > > Well, they're "hyper-blocks", certainly. They've always been referred > to simply as "blocks", since for example just saying "11-limit" > already implies that you're beyond 3 dimensions, so there's no need > to be redundant about that point . . .

Monz understood. If it's o.k. to call the flat side of a potentially multidimensional (array? forgive my choice of words, if not technically correct) a block, it seems just as o.k. to me refer to cylinders and torusses as blocks, where the warping does introduce a third dimension. You can chisel a torus from a block, so it "is" a block.

klaus

hi Paul and klaus,

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:

> Monz understood. If it's o.k. to call the flat side of a
> potentially multidimensional (array? forgive my choice of
> words, if not technically correct) a block, it seems just as
> o.k. to me refer to cylinders and torusses as blocks, where
> the warping does introduce a third dimension. You can chisel
> a torus from a block, so it "is" a block.
>
> klaus

in our Tonalsoft software, when the user creates a temperament,
the resulting lattice can be shown in either (or all) of
three geometries: rectangular-JI, triangular-JI, and "closed".

this last choice is the one which warps the JI-block
into a shape more appropriate for that temperament:
a helix for a linear-temperament, a torus for EDO, etc.

if the user creates a temperament, the program's default
lattice view is the closed geometry, but the user is
always free to flatten it out into one of the JI views,
or to open those windows in addition to the closed view
to see multiple perspectives.

should we continue to call these "blocks"? i propose
that we create a better general term, which can be
modified maybe with a pseudo-Greek prefix to indicate
the dimensionality ... similar to the way we did "enamu,
doamu, triamu ..." etc for the MIDI tuning units.

or better yet, just use the number itself as a prefix.

i recall that we've been thru this before, and i
suggested "cell". "nucleus" would be even better,
because it allows us to omit the long word "periodicity"
without losing that aspect of the meaning. so:

1-nucleus, for linear systems like Pythagorean
2-nucleus, for planar systems like 5-limit JI
3-nucleus, for 3D (i.e., 7-limit)
4-nucleus, for 4D (i.e., 11-limit)

etc.

and if speaking generally about periodicity-blocks
without having to specify dimensionality, "nucleus"
by itself is excellent.

-monz

monz schrieb:

> hi Paul and klaus,

...

> should we continue to call these "blocks"? i propose
> that we create a better general term, which can be > modified maybe with a pseudo-Greek prefix to indicate
> the dimensionality ... similar to the way we did "enamu,
> doamu, triamu ..." etc for the MIDI tuning units.
> > or better yet, just use the number itself as a prefix.
>
> i recall that we've been thru this before, and i
> suggested "cell". "nucleus" would be even better,
> because it allows us to omit the long word "periodicity"
> without losing that aspect of the meaning. so:
>
>
> 1-nucleus, for linear systems like Pythagorean
> 2-nucleus, for planar systems like 5-limit JI
> 3-nucleus, for 3D (i.e., 7-limit)
> 4-nucleus, for 4D (i.e., 11-limit)

or, still better, say "n-dimensional" ... stencil, to introduce yet another gratuitious possibility.

i have nothing against "block" if its meaning is understood loosely and allowed to be used loosely - assuming that it's fockers original term. i (as a non-mathematician) believe that the different ways of warping can already be expressed in terms of set theory, and many laypeople will be able to follow there if the resident eggheads talked about "invariance or closure in this or that respect" instead of calling something an Abelian group. so that there are plain flat blocks (tiles?), blocks where x comes out y (cylinders or helixes), blocks where x and y come out z (torus) and so on into hyperspace.

more friendly explicitness, less jargon. i'm sure i'm speaking for the silent majority here.

klaus

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

/tuning/topicId_53869.html#54275

> i have to put "flatten" in quotes, because if the JI
> lattice is >2D (i.e., the prime-space uses more than
> two prime-factors), it's already not flat. but the point
> is that the software can convert the closed geometry of
> the temperament into the open-ended geometry of JI.
> the user can even have both views (or even all three)
> open simultaneously.
>

***Well... that's pretty amazing. That should be fun to do, and
probably worth the price of admission alone...

J. Pehrson

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

/tuning/topicId_53869.html#54311

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > > Now should I still try to elaborate my original post?
> >
> > ***Sure...
>
> Would you be so kind as to read the Robert Kelly article Monz just
> dug up? Much of it concerns this exact topic, and it's from "music
> academia", so it might be closer to "your language" . . .

***Yes, I looked through this and found this dissertation very
interesting! Anybody who tries to analyze Tristan and Isolde
according to Just Intonation principles gets *my* eyebrows raised a
bit right away.

I also found the discussion of the reductions and diatonicism as
related to the geometric shapes (cylinders) of interest...

At this very moment I don't have time to really fully digest this,
but I was thinking how much fun it would be to take a course (for
*credit* of course!) with this very paper as the text matter. I
think it really could take up a full semester...

Thanks for pointing this out!

Joseph

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
> more friendly explicitness, less jargon. i'm sure i'm
> speaking for the silent majority here.

Thanks Klaus. I believe so! I've been trying to convince the other
egg-heads of this for some time now, to no avail. Please, some more
of you occasional posters, speak out on this.

More reference by description and less of the cryptic new names, I
say. I know I've been guilty myself on occasion, but it's just going
too far. We don't need to start making up short names for things
until they have been regularly discussed for _years_, and even then
they should preferably be abbreviated forms of the description
rather than deliberately cryptic. And for a long time after
agreement on a short term, the description should be given in
parenthesis.

The recent explosion of jargon sickens me. I'd much rather we called
something "the simple 7-limit minor thirds temperament" instead
of "keenan", and "prime exponent vector" rather than "monzo",
and "prime generator mapping" rather than "val". At least then folks
have a chance of figuring out what we're talking about, instead of
wondering where to find the database to look up this jargon. And
even if they had it, it would be every second word, so you'd just
give up.

Sure it's a few more key strokes, but hey that's only an effort for
one person, the author, not the hundreds(?) of readers. It's a
privilege to post to this list. An author owes it to his readers to
be as transparent as possible.

Can someone find some software for these guys so they can type their
short cryptic terms and it looks up the database at _their_ end, so
that what gets posted is something meaningful to the rest of us? :-)

-- Dave Keenan

Another vote for cleaning up the specialized language.
All it does is to make an elite.

Dave Keenan wrote:

> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
> > more friendly explicitness, less jargon. i'm sure i'm
> > speaking for the silent majority here.
>
> Thanks Klaus. I believe so! I've been trying to convince the other
> egg-heads of this for some time now, to no avail. Please, some more
> of you occasional posters, speak out on this.
>
> More reference by description and less of the cryptic new names, I
> say. I know I've been guilty myself on occasion, but it's just going
> too far.

>
>
> The recent explosion of jargon sickens me. I'd much rather we called
> something "the simple 7-limit minor thirds temperament" instead
> of "keenan", and "prime exponent vector" rather than "monzo",
> and "prime generator mapping" rather than "val". At least then folks
> have a chance of figuring out what we're talking about, instead of
> wondering where to find the database to look up this jargon. s? :-)
>
> -- Dave Keenan

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Dave Keenan wrote:

>--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
> >
>>more friendly explicitness, less jargon. i'm sure i'm >>speaking for the silent majority here.
>> >>
>
>Thanks Klaus. I believe so! I've been trying to convince the other >egg-heads of this for some time now, to no avail. Please, some more >of you occasional posters, speak out on this. >
>More reference by description and less of the cryptic new names, I >say. I know I've been guilty myself on occasion, but it's just going >too far. We don't need to start making up short names for things >until they have been regularly discussed for _years_, and even then >they should preferably be abbreviated forms of the description >rather than deliberately cryptic. And for a long time after >agreement on a short term, the description should be given in >parenthesis.
>
>The recent explosion of jargon sickens me.
>

Oh oh....I think hell just froze over...

--
* David Beardsley
* microtonal guitar
* http://biink.com/db

Thanks Klaus. I agree with you completely. I also think "nucleus"
*does* lose the meaning -- in what context do we see "nuclei" forming
a regular pattern with no gaps or overlaps, Monz?

If you think invariance and closure are words that will help
communicate the "egghead" ideas better, Klaus, I encourage you to
help inject them into the dialogue. The more "translators" helping
make the stuff comprehensible to everyone, the better.

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
> monz schrieb:
>
> > hi Paul and klaus,
>
> ...
>
>
> > should we continue to call these "blocks"? i propose
> > that we create a better general term, which can be
> > modified maybe with a pseudo-Greek prefix to indicate
> > the dimensionality ... similar to the way we did "enamu,
> > doamu, triamu ..." etc for the MIDI tuning units.
> >
> > or better yet, just use the number itself as a prefix.
> >
> > i recall that we've been thru this before, and i
> > suggested "cell". "nucleus" would be even better,
> > because it allows us to omit the long word "periodicity"
> > without losing that aspect of the meaning. so:
> >
> >
> > 1-nucleus, for linear systems like Pythagorean
> > 2-nucleus, for planar systems like 5-limit JI
> > 3-nucleus, for 3D (i.e., 7-limit)
> > 4-nucleus, for 4D (i.e., 11-limit)
>
> or, still better, say "n-dimensional" ... stencil, to
> introduce yet another gratuitious possibility.
>
> i have nothing against "block" if its meaning is understood
> loosely and allowed to be used loosely - assuming that it's
> fockers original term. i (as a non-mathematician) believe
> that the different ways of warping can already be expressed
> in terms of set theory, and many laypeople will be able to
> follow there if the resident eggheads talked about
> "invariance or closure in this or that respect" instead of
> calling something an Abelian group. so that there are plain
> flat blocks (tiles?), blocks where x comes out y (cylinders
> or helixes), blocks where x and y come out z (torus) and so
> on into hyperspace.
>
> more friendly explicitness, less jargon. i'm sure i'm
> speaking for the silent majority here.
>
> klaus

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> The recent explosion of jargon sickens me.

This method is standard in math, because it works very well.
Moreover, saying you'd rather something was called "closed" or
"invariant" rather than an "abelian group" is simply asking to be
confused about things you don't need to be confused about. If you
don't know what an abelian group is, you can google for the answer. If
you don't know what someone means by "an invariant set of notes", you
are going to remain lost and clueless.

I'd much rather we called
> something "the simple 7-limit minor thirds temperament" instead
> of "keenan", and "prime exponent vector" rather than "monzo",
> and "prime generator mapping" rather than "val".

"Simple 7-limit minor thirds temperament" is a mouthful, and "prime
exponent vector" suggests wen should be talking about vectors and
vector spaces. "Prime generator mapping" is vague and confusing, and
certainly does not tell us that *every* interval in some p-limit is
being mapped, nor that the mappings do not necessarily have to do with
generators.

At least then folks
> have a chance of figuring out what we're talking about, instead of
> wondering where to find the database to look up this jargon.

If they go find an explanation, they have a better chance of figuring
out what it means than if they try to cook one up and get it wrong,
which is what's likely to happen using your system. I really prefer
not to sow unneeded confusion.

> Can someone find some software for these guys so they can type their
> short cryptic terms and it looks up the database at _their_ end, so
> that what gets posted is something meaningful to the rest of us? :-)

Try web sites.

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
> wallyesterpaulrus schrieb:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> >>the higher dimensions
> >
> >
> >>can have equivalent periodicities, but i don't think
> >>they'll be shaped like "blocks".
> >
> >
> > Well, they're "hyper-blocks", certainly. They've always been
referred
> > to simply as "blocks", since for example just saying "11-limit"
> > already implies that you're beyond 3 dimensions, so there's no
need
> > to be redundant about that point . . .
>
> Monz understood. If it's o.k. to call the flat side of a
> potentially multidimensional (array? forgive my choice of
> words, if not technically correct) a block,

Flat side? I don't get it.

> it seems just as
> o.k. to me refer to cylinders and torusses as blocks, where
> the warping does introduce a third dimension.

I don't see it this way. Blocks have borders, edges. The "warping"
can be seen in a number of ways, including maintaining the
dimensionality and eliminating some borders. For example, look at my
paper _The Forms Of Tonality_:

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

The bottom of figure 4 shows two diatonic blocks. Now look at figure
5 -- the portion of the lattice representing the diatonic scale under
81:80 equivalence. Is this structure a block? I prefer to stay closer
to the visual, so instead of "periodicity block", I refer to this
structure as a "periodicity strip". It's not a higher-dimensional
structure -- even the surface of a cylinder or torus still has only
two dimensions.

hi Joe,

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> /tuning/topicId_53869.html#54275
>
> > i have to put "flatten" in quotes, because if the JI
> > lattice is >2D (i.e., the prime-space uses more than
> > two prime-factors), it's already not flat. but the point
> > is that the software can convert the closed geometry of
> > the temperament into the open-ended geometry of JI.
> > the user can even have both views (or even all three)
> > open simultaneously.
> >
>
> ***Well... that's pretty amazing. That should be fun to do,
> and probably worth the price of admission alone...
>
> J. Pehrson

yes, it *is* a lot of fun! not to mention how nice
it is, after all these years of laboriously drawing
lattices by hand, to set up a tuning system and then
immediately see not only one but *three* different views
of it!

... and don't forget, you can then use the lattice to
select notes to pop into the score of your musical piece!
(but of course, we don't really care about *that* here
on the tuning list!)

;-)

-monz

hi Joe,

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> >
> > Would you be so kind as to read the Robert Kelly article
> > Monz just dug up? Much of it concerns this exact topic,
> > and it's from "music academia", so it might be closer to
> > "your language" . . .
>
>
> ***Yes, I looked through this and found this dissertation very
> interesting! Anybody who tries to analyze Tristan and Isolde
> according to Just Intonation principles gets *my* eyebrows
> raised a bit right away.

well, if you thought *that* was fun, then you should find
some of Martin Vogel's numerous books (all in German except one)
where he has a *field day* transforming the opening of
_Tristan und Isolde_ into 7-limit JI.

-monz

Hi Monz,

The part of your message I was objecting to before is:

> and if speaking generally about periodicity-blocks
> without having to specify dimensionality, "nucleus"
> by itself is excellent.

As I said, I don't think "nucleus" should replace "block" in any
instance.

However, you brought up something else entirely here:

> i recall that we've been thru this before, and i
> suggested "cell". "nucleus" would be even better,
> because it allows us to omit the long word "periodicity"
> without losing that aspect of the meaning. so:
>
>
> 1-nucleus, for linear systems like Pythagorean
> 2-nucleus, for planar systems like 5-limit JI
> 3-nucleus, for 3D (i.e., 7-limit)
> 4-nucleus, for 4D (i.e., 11-limit)

None of these systems were ever called "blocks" to begin with --
blocks are *finite*, while these systems are *infinite*.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Can someone find some software for these guys so they
> can type their short cryptic terms and it looks up the
> database at _their_ end, so that what gets posted is
> something meaningful to the rest of us? :-)
>
> -- Dave Keenan

hmmm ... i wish someone could write to Gene-Ward-Smith-ese
-to-English translation program!

but seriously ... i know from the emails i get that
people *do* often turn to my Encyclopaedia of Tuning
for help on the jargon. but for a few reasons, it's
been lacking lately:

1) doing so much work on the software and especially
the redesign of the website lately, i simply haven't
had time to satisfy my addiction to reading the tuning lists,
and concommitantly to keep adding new terms to the Encyclopaedia;

2) partly because of this lack of attention, i've gotten
serious behind on following the theory developments here
for over a year now.

but the good news is that i *am* updating the Encyclopaedia,
and will be hunting for new terms out of which to make
webpages.

eventually, all the graphics in the Encyclopaedia
will be converted to applets which will run under the
free lattice-viewer we (at Tonalsoft) are also building.
readers will be able to play around with the diagrams
interactively, which should help to teach the concepts.

-monz

hi Paul,

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

> Hi Monz,
>
> The part of your message I was objecting to before is:
>
> > and if speaking generally about periodicity-blocks
> > without having to specify dimensionality, "nucleus"
> > by itself is excellent.
>
> As I said, I don't think "nucleus" should replace "block"
> in any instance.

yeah, upon further reflection, i think "nucleus" is
only synonymous with "kernel". so i still like "cell".

> However, you brought up something else entirely here:
>
> > i recall that we've been thru this before, and i
> > suggested "cell". "nucleus" would be even better,
> > because it allows us to omit the long word "periodicity"
> > without losing that aspect of the meaning. so:
> >
> >
> > 1-nucleus, for linear systems like Pythagorean
> > 2-nucleus, for planar systems like 5-limit JI
> > 3-nucleus, for 3D (i.e., 7-limit)
> > 4-nucleus, for 4D (i.e., 11-limit)
>
> None of these systems were ever called "blocks" to begin with --
> blocks are *finite*, while these systems are *infinite*.

huh? of course i know that blocks are finite.
but a 4D infinite JI system is still going to have
4D blocks (or cells).

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> but seriously ... i know from the emails i get that
> people *do* often turn to my Encyclopaedia of Tuning
> for help on the jargon. but for a few reasons, it's
> been lacking lately:

Monz,

I have no complaints about your encyclopedia. I don't know what we'd
do without it. You can't be expected to keep up the the current
deluge of hundreds of new names for temperaments and commas, as well
as the mathematical terminology being used with no concession to
what it actually means when interpreted for tuning. It's bad enough
on tuning-math, but on _this_ list, it's beyond the pale.

>Monz,
>
>I have no complaints about your encyclopedia. I don't know what we'd
>do without it. You can't be expected to keep up the the current
>deluge of hundreds of new names for temperaments and commas, as well
>as the mathematical terminology being used with no concession to
>what it actually means when interpreted for tuning. It's bad enough
>on tuning-math, but on _this_ list, it's beyond the pale.

Here, here. Often, I sit and think, "Damn, the worst part about the
list is that there is no FAQ." Then, I remember: the Encyclopedia!
It isn't quite a FAQ, but it's the work of one person! Sure, you've
had a lot of help Joe, but wow, I sometimes think, "If Joe hadn't
started collecting these things, THEN where would we be?"

So, I'm bound to say:

Congratulations! You're the first ever recipient of the all-new
Weekly Tuning Community Honor Awards!!!!!!!!!!!!!!!!!!!!!!!!!!!

. .. .. .. .. .. .. .. .. .. ..
Weekly Tuning Community Honor Awards
.. last Sunday of every week! ..

This week's winner:

Joe Monzo, for the Tonalsoft Encyclopedia

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .

Stay Tuned for next week's installment, and Thanks again, Joe!

-Carl

The Weekly Tuning Community Honor Awards were founded on the
principle that, since nobody else was giving awards, somebody
could just start giving awards.

Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> You can't be expected to keep up the the current
> deluge of hundreds of new names for temperaments and commas, as well
> as the mathematical terminology being used with no concession to
> what it actually means when interpreted for tuning. It's bad enough
> on tuning-math, but on _this_ list, it's beyond the pale.

I'm certainly thankful someone of your background (on the list and
elsewhere) has seen fit to put the above in the public record. There
will always be need for technical language for various endeavors, but
I can not believe (say that as John McEnroe would...) that every
single blessed nuanced temperment or comma or whatever *needs* a name.
After all, how many people are discussing or interested in them?

In time, it may become clear how much of this will be ultimately of
continued and frequent use, and one should have name. But the
continual rush to name things seems a bit bizarre, almost as if the
naming is the most important aspect.

The difficulty of the 'technology' (math) behind much of this, the
special 'language' in use, it all harkens back to the early days of
the mainframe computers, where only the 'priests' were allowed in to
make things go.

I'm the last person to know exactly when something needs an easy
identifier, but it looks like things have gotten carried away. And,
yes, I believe a lot of the silent majority have glazed over...

Cheers,
Jon (who gets a lot of mileage out of Type Pilot, a Windows macro
utility that makes long phrases just 2 or 3 keystrokes away...)

wow! thanks Carl.
it means a lot to me to be the first recipient.

-monz

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > Monz,
> >
> > I have no complaints about your encyclopedia. I don't
> > know what we'd do without it. You can't be expected to
> > keep up the the current deluge of hundreds of new names
> > for temperaments and commas, as well as the mathematical
> > terminology being used with no concession to what it
> > actually means when interpreted for tuning. It's bad enough
> > on tuning-math, but on _this_ list, it's beyond the pale.
>
> Here, here. Often, I sit and think, "Damn, the worst part
> about the list is that there is no FAQ." Then, I remember:
> the Encyclopedia! It isn't quite a FAQ, but it's the work
> of one person! Sure, you've had a lot of help Joe, but wow,
> I sometimes think, "If Joe hadn't started collecting these
> things, THEN where would we be?"
>
> So, I'm bound to say:
>
> Congratulations! You're the first ever recipient of the all-new
> Weekly Tuning Community Honor Awards!!!!!!!!!!!!!!!!!!!!!!!!!!!
>
> . .. .. .. .. .. .. .. .. .. ..
> Weekly Tuning Community Honor Awards
> .. last Sunday of every week! ..
>
> This week's winner:
>
> Joe Monzo, for the Tonalsoft Encyclopedia
>
> .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
>
> Stay Tuned for next week's installment, and Thanks again, Joe!
>
> -Carl
>
>
> The Weekly Tuning Community Honor Awards were founded on the
> principle that, since nobody else was giving awards, somebody
> could just start giving awards.

hi Dave and Jon,

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:

> Dave,
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > You can't be expected to keep up the the current
> > deluge of hundreds of new names for temperaments
> > and commas, as well as the mathematical terminology
> > being used with no concession to what it actually means
> > when interpreted for tuning. It's bad enough on
> > tuning-math, but on _this_ list, it's beyond the pale.
>
> I'm certainly thankful someone of your background (on the
> list and elsewhere) has seen fit to put the above in the
> public record. There will always be need for technical
> language for various endeavors, but I can not believe
> (say that as John McEnroe would...) that every single
> blessed nuanced temperment or comma or whatever *needs*
> a name.
> After all, how many people are discussing or interested
> in them?
>
> In time, it may become clear how much of this will be
> ultimately of continued and frequent use, and one should
> have name. But the continual rush to name things seems
> a bit bizarre, almost as if the naming is the most
> important aspect.
>
> The difficulty of the 'technology' (math) behind much of
> this, the special 'language' in use, it all harkens back
> to the early days of the mainframe computers, where only
> the 'priests' were allowed in to make things go.
>
> I'm the last person to know exactly when something needs
> an easy identifier, but it looks like things have gotten
> carried away. And, yes, I believe a lot of the silent
> majority have glazed over...
>
> Cheers,
> Jon (who gets a lot of mileage out of Type Pilot, a Windows macro
> utility that makes long phrases just 2 or 3 keystrokes away...)

i'm really sorry to disagree with both of you ...
especially since so often i think the three of us
are on the same wavelength.

the outburst of naming is simply an expression of the
exhilaration of discovering so many tunings, and so
many ways in which they relate to each other.

most of the names are not of single tunings, but of
families of temperaments. those names provide easy
handles to grab onto when discussing such a plethora
of new discovery.

what we need is more diagrams, musical staff-notation,
and most of all, audio examples, to illustrate the
concepts that lie behind the differences between
those names. i promise to do what i can in that
regard ... but you all know i'm busy with other jobs ...

but i do agree that Gene and others of his mathematical
league should really make an effort to post the more arcane
mathematical jargon on tuning-math instead of here.
a single post like that tends to promote a big response,
which just clutters this list's already huge archive.

i don't see any reason why complex mathematical jargon
would be "bad enough" on the tuning-math list -- that's
*precisely* why that list was created.

-monz

Dave:
>>The recent explosion of jargon sickens me.

Gene:
> This method is standard in math, because it works very well.
> Moreover, saying you'd rather something was called "closed" or
> "invariant" rather than an "abelian group" is simply asking to be
> confused about things you don't need to be confused about. If you
> don't know what an abelian group is, you can google for the answer. If
> you don't know what someone means by "an invariant set of notes", you
> are going to remain lost and clueless.

I can't say I'm sickened, but I'm certainly deeply troubled by this attitude. From my POV, many of the conventions of mathematical discourse are a considerable barrier to understanding. The worst cases are those books or papers which begin by introducing a set of terms or even, *shudder*, notation before saying what it's going to be used for. They then proceed to state a load of theorems using that notation and, if you're lucky, explain them using the new terminology. Which means you have to keep flicking back to see what it's supposed to mean. Quite often you also have to know some obscure terminology from a field that is given a vague name such as "linear algebra" that you thought you were already familiar with. So you need to read a few more books to find out what things are called today. At the end of all this, you might find a useful result that can be expressed in a few lines of C code.

Now, perhaps among professional mathematicians this kind of thing really does work. When you've spent enough years learning the field, and you absolutely need to understand something in all its complexity, and don't worry about things outside your field, the effort may be worth it for the gain of precision and conciseness. But for outsiders, it's really a disaster. That you don't seem to appreciate this goes some way to explaining why you have such difficulty communicating.

I'm also tempted to say that anybody who thinks that 29 is a multiple of 12 has no right to start pontificating about precision of language. There you go, I've said it.

As for your example, yes, you can google for "abelian group". You'll get a few different definitions, all of which refer you to the definition of "group". From "group", you immediately get hit with "set". Frankly, if you haven't already studied group theory your best bet is to get a book on it. Schaum's Outlines is good, and only assumes a high school background for really smart readers. Abelian Groups are introduced on page 177.

If the reader has studied group theory, they should already know what an abelian group is, of course. So they won't be Googling.

The worst case is that the reader will think they already know what a group or set is, as these words have different meanings in different contexts. They will then get deeply confused both as to what it is you're talking about and why they don't understand it. This is the very problem you persist in having with the word "lattice".

Certainly as far as this list is concerned, whenever you feel the temptation to talk about abelian groups you should really stop to think of a better way of explaining whatever point you were trying to get across.

Dave:
> I'd much rather we called >>something "the simple 7-limit minor thirds temperament" instead >>of "keenan", and "prime exponent vector" rather than "monzo", >>and "prime generator mapping" rather than "val". Gene:
> "Simple 7-limit minor thirds temperament" is a mouthful, and "prime
> exponent vector" suggests wen should be talking about vectors and
> vector spaces. "Prime generator mapping" is vague and confusing, and
> certainly does not tell us that *every* interval in some p-limit is
> being mapped, nor that the mappings do not necessarily have to do with
> generators.

Yes, "simple 7-limit minor thirds temperament" is a bit of a mouthful, but at least it gives people some idea what you're talking about. Presumably, a linear temperament with a generator of a minor third that approximated 7-limit JI. And, if you can think of more than one such example, the simplest one. "Keenan", on the other hand, has to be looked up. And where to do that? My list of linear temperaments is incomplete in the face of the barrage of new, and sometimes contradictory names being proposed and immediately adopted. I think Paul has a database but it's only 5-limit. Both require you to know that you want a linear temperament. Monz's dictionary may one day include them, with the result that the index will get bloated down and make it harder for people to find more useful terms. However much trouble you find it to type, you should really make that effor for your readers, because it makes it so much easier for them.

The only value I can see in this naming processes is if somebody can fix a definitive list of names, and everybody else sticks to it. The only thing else than calling a temperament by an obscure name is calling it by lots of different obscure names.

Hey, how about we find cute names for all the equal temperaments up to 100?

There are a whole host of contexts where the term "vector" gets used with no reference to vector spaces. Mathematicians who are so out of touch with the rest of the world that they make this assumption could always try a Google search (as you expect mere mortals to do upon being confronted with an abelian group). The first result I get is in Japanese. The second is to the Java API documentation. A vector in this case is a list of things with no pre-defined size. Nothing to do with vector spaces, and close to our "prime exponent vector". Next up, something about DNA. Probably nothing whatsoever to do with our "vectors".

Then, Scalable Vector Graphics. Vectors here are roughly what they are in Physics -- something with a magnitude and a direction. You could call this a special case of a vector space, like you could call addition a special case of an abelian group. But if you think addition suggests abelian groups, you're crazy.

The term "val" doesn't suggest anything to most people, as it's utterly meaningless. To understand your own definition, they'll have to understand free abelian groups, torsion elements and non-canonically isomorphic dual groups. They're really far more likely to guess what "prime generator mapping" means. If every interval in the p-limit must be mapped for whatever you're explaning, you should say so regardless. Even people who understood your "val" definition may have forgotten this detail. Most people, and this may come as a shock to you, don't worry about terms being vague. They work out the intended meaning from the context.

And heaven help you if you try to search for "val" on Google.

> If they go find an explanation, they have a better chance of figuring
> out what it means than if they try to cook one up and get it wrong,
> which is what's likely to happen using your system. I really prefer
> not to sow unneeded confusion.

If you think most people here have a hope of understanding the explanations they'll find on your site you're sadly deluded. You certainly aren't sowing confusion there -- you fend off confusion with an initial barrier of total incomprehensibility.

Graham

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

/tuning/topicId_53869.html#54465
>
> Hey, how about we find cute names for all the equal temperaments up
to 100?
>

***Please put me down in favor of "mouthfulls" rather than cutsie
terminology...

JP

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

> If you think most people here have a hope of understanding the
> explanations they'll find on your site you're sadly deluded. You
> certainly aren't sowing confusion there -- you fend off confusion with
> an initial barrier of total incomprehensibility.

Someone, somewhere should give precise definitions, and this stuff is
vastly easier than the math which gets applied to many other fields.
Of course in most areas the experts learn the math, because it is
their job. Music has humanities people trying to do math, and thinking
1 is a prime number, and techno types trying to do music, when like me
they've probably only had a few undergraduate courses. The academic
experts on using math for music theory don't have to be very good at it.

> Jon (who gets a lot of mileage out of Type Pilot, a Windows macro
> utility that makes long phrases just 2 or 3 keystrokes away...)

http://www.colorpilot.com/

Hey, this is a interesting firm.

Some of our math people might be interested in these...

http://www.colorpilot.com/mathsoft.html

-Carl

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

> Yes, "simple 7-limit minor thirds temperament" is a bit of a mouthful,
> but at least it gives people some idea what you're talking about.

This is partly because you are lucky (6/5, a well-known consonance, is
a generator) and partly because it really is a name (the word "simple"
is in there, with no precise meaning attached to it.) Pajara would be
the "7-limit septimal tritone and fifths temperament", which is
getting ugly. Negri could be the "7-limit simple major or maybe minor
diatonic semitone temperament", and miracle the "7-limit complex major
or maybe minor diatonic semitone temperament". 35/32 is also a
generator for negri, but porcupine would grab that one, becoming the
"7-limit simple septimal neutral seconds temperament", whereas
hemikleismic would be the "7-limit complex septimal neutral seconds
temperament".

Still, calculating the simplest p-limit generators for a given
temperament seems like useful code to write, so I think I will.

"Keenan", on the other hand, has to be
> looked up.

So does "7-limit simple minor thirds temperament."

>> This method is standard in math, because it works very well.
>> Moreover, saying you'd rather something was called "closed" or
>> "invariant" rather than an "abelian group" is simply asking to be
>> confused about things you don't need to be confused about. If you
>> don't know what an abelian group is, you can google for the answer.
>> If you don't know what someone means by "an invariant set of
>> notes", you are going to remain lost and clueless.
>
>I can't say I'm sickened, but I'm certainly deeply troubled by this
>attitude. From my POV, many of the conventions of mathematical
>discourse are a considerable barrier to understanding. The worst cases
>are those books or papers which begin by introducing a set of terms or
>even, *shudder*, notation before saying what it's going to be used for.
> They then proceed to state a load of theorems using that notation and,
>if you're lucky, explain them using the new terminology. Which means
>you have to keep flicking back to see what it's supposed to mean. Quite
>often you also have to know some obscure terminology from a field that
>is given a vague name such as "linear algebra" that you thought you were
>already familiar with. So you need to read a few more books to find out
>what things are called today. At the end of all this, you might find a
>useful result that can be expressed in a few lines of C code.

You go, Graham!

>Now, perhaps among professional mathematicians this kind of thing really
>does work. When you've spent enough years learning the field, and you
>absolutely need to understand something in all its complexity, and don't
>worry about things outside your field, the effort may be worth it for
>the gain of precision and conciseness. But for outsiders, it's really a
>disaster. That you don't seem to appreciate this goes some way to
>explaining why you have such difficulty communicating.

This needed to be said. In Gene's defense, he seems to be trying.
I've always just viewed it as perhaps the 'cost', at least in his
case, of the thinking he used to get to where he is.

>I'm also tempted to say that anybody who thinks that 29 is a multiple
>of 12 has no right to start pontificating about precision of language.
>There you go, I've said it.

That's a different kind of thing. But again, the oblivious professor
is perhaps romanticized for a reason. By this I mean that if you're
oblivious to some things, maybe you're paying that much more attention
to others.

>As for your example, yes, you can google for "abelian group". You'll
>get a few different definitions, all of which refer you to the
>definition of "group". From "group", you immediately get hit with
>"set". Frankly, if you haven't already studied group theory your best
>bet is to get a book on it. Schaum's Outlines is good, and only
>assumes a high school background for really smart readers. Abelian
>Groups are introduced on page 177.

As someone who's struggled with Gene's stuff for years now, I will
say that as much as I hope for better explanations from him, I'm
also thankful for the improvement in my understanding of mathematics
our conversations have brought.

>The only value I can see in this naming processes is if somebody can
>fix a definitive list of names, and everybody else sticks to it.

Yes, that would be nice, wouldn't it. Changing kleismic to hanson
and pelogic to mavilla rather peeved me. I admire Paul's desire to
give credit, though.

>Hey, how about we find cute names for all the equal temperaments
>up to 100?

So you want to call linear and planar temperaments by their
wedgies?

>There are a whole host of contexts where the term "vector" gets used
>with no reference to vector spaces. Mathematicians who are so out of
>touch with the rest of the world that they make this assumption could
>always try a Google search (as you expect mere mortals to do upon
>being confronted with an abelian group). The first result I get is
>in Japanese. The second is to the Java API documentation. A vector
>in this case is a list of things with no pre-defined size. Nothing
>to do with vector spaces, and close to our "prime exponent vector".
>Next up, something about DNA. Probably nothing whatsoever to do with
>our "vectors".

I couldn't have said it better myself.

>And heaven help you if you try to search for "val" on Google.

!

However, in light of the inequality you gave about *changing* obscure
names, I think we should keep "val".

>If you think most people here have a hope of understanding the
>explanations they'll find on your site you're sadly deluded. You
>certainly aren't sowing confusion there -- you fend off confusion
>with an initial barrier of total incomprehensibility.

:):)

Again in Gene's defense, though, I'm glad there *is* a mathematically
precise statement of this stuff somewhere, so that mathematicians
can read it and get interested in it. The problem is a lack of other
explanations, not the existence of any particular one.

So I guess Gene might say, "what are you waiting for?". Unfortunately,
since he's the *only* person that understands some of this stuff, we
can't very well offer other versions. But we can hope that he'll
continue to get better at offering the other versions himself, so that
we can get onboard.

It's a real pleasure to know people as gifted as the ones on this
list. Seriously, I've never met a bunch of such incredibly talented
folks.

-Carl

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

> So I guess Gene might say, "what are you waiting for?". Unfortunately,
> since he's the *only* person that understands some of this stuff, we
> can't very well offer other versions.

Which things are the ones only I understand?

Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> Which things are the ones only I understand?

When you think about it, that is a pretty funny question. There are a
lot of easy (and inaccurate) responses, but no one, true answer.

Cheers,
Jon

>> So I guess Gene might say, "what are you waiting for?".
>> Unfortunately, since he's the *only* person that understands
>> some of this stuff, we can't very well offer other versions.
>
>Which things are the ones only I understand?

Have Graham, Herman and the two Pauls duplicated all your methods
between them?

-Carl

Carl Lumma wrote:

> As someone who's struggled with Gene's stuff for years now, I will
> say that as much as I hope for better explanations from him, I'm
> also thankful for the improvement in my understanding of mathematics
> our conversations have brought.

That's what tuning-math's for. People who want to understand mathematics can always go over there.

> So you want to call linear and planar temperaments by their
> wedgies?

Certainly not! Linear temperaments can be defined either as a pair of equal temperaments, or a set of commas, or an octave-equivalent mapping with period and generator sizes. Planar temperaments it's usually simplest in terms of commas. If you talk about the same temperaments a lot, you can use their names. It's still helpful go give clues about which ones you mean.

We can assume people know "meantone" is 7&12, the same as they know "chromatic" is 12.

> However, in light of the inequality you gave about *changing* obscure
> names, I think we should keep "val".

It's fine for tuning-math, but I don't think there's any need for it here.

> Again in Gene's defense, though, I'm glad there *is* a mathematically
> precise statement of this stuff somewhere, so that mathematicians
> can read it and get interested in it. The problem is a lack of other
> explanations, not the existence of any particular one.

Yes, but while that situation persists the terms shouldn't be used with a non specialist audience, such as this one.

> So I guess Gene might say, "what are you waiting for?". Unfortunately,
> since he's the *only* person that understands some of this stuff, we
> can't very well offer other versions. But we can hope that he'll
> continue to get better at offering the other versions himself, so that
> we can get onboard.

I'm still not sure what "val" means -- in that I don't know exactly which borderline cases are covered. I didn't know before Gene said recently that the primes have to be consecutive, so presumably you can't have a val for 2.3.7 space. Actually, this does follow from the definition on his website, but not that I can see the one in Monzo's dictionary (now encyclopedia).

> It's a real pleasure to know people as gifted as the ones on this
> list. Seriously, I've never met a bunch of such incredibly talented
> folks.

I'll get round to my American holidays one day. Of course, it'll have to be one for each coast.

Graham

>> So you want to call linear and planar temperaments by their
>> wedgies?
>
>Certainly not! Linear temperaments can be defined either as a pair of
>equal temperaments, or a set of commas, or an octave-equivalent mapping
>with period and generator sizes. Planar temperaments it's usually
>simplest in terms of commas. If you talk about the same temperaments
>a lot, you can use their names. It's still helpful go give clues about
>which ones you mean.

I think a standard repository would be a good idea. I'm sure Paul's
and/or monz will put up an updated catalog sooner or later.

>> However, in light of the inequality you gave about *changing* obscure
>> names, I think we should keep "val".
>
>It's fine for tuning-math, but I don't think there's any need for
>it here.

Agreed.

-Carl

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

> I'm still not sure what "val" means -- in that I don't know exactly
> which borderline cases are covered. I didn't know before Gene said
> recently that the primes have to be consecutive, so presumably you
can't
> have a val for 2.3.7 space. Actually, this does follow from the
> definition on his website, but not that I can see the one in Monzo's
> dictionary (now encyclopedia).

A {2,3,7}-val would be fine, so long as you called it that; otherwise
you would have no way of knowing <118 187 331| means the mapping
applies to 2, 3 and 7 except by guessing. If "val", without
qualification, is what you say then you know what the prime limit is
by counting.

Graham Breed wrote:

> The only value I can see in this naming processes is if somebody can fix > a definitive list of names, and everybody else sticks to it. The only > thing else than calling a temperament by an obscure name is calling it > by lots of different obscure names.
> > Hey, how about we find cute names for all the equal temperaments up to 100?

If linear temperaments were as simple as equal temperaments there wouldn't be any point in naming them. I still think there's value in giving names to the more useful temperaments. But there's an alternative to arbitrary names. Take for example the temperaments identified as "catakleismic" (or now alternatively "hanson") and "parakleismic". Both are minor-third temperaments, but they have different maps:

[<1, 0, 1, -3|, <0, 6, 5, 22|] catakleismic
[<1, 5, 6, 12|, <0, -13, -14, -35|] parakleismic

The first one has a TOP generator of 316.9 cents and the second one 315.1 cents. But it's really the mapping that distinguishes them, more than the sizes of the generators.

Fortunately, many (but not all) of these temperaments can be found with your temperament finder (http://x31eq.com/temper/twoet.html). So one way of identifying the [<1, 0, 1, -3|, <0, 6, 5, 22|] temperament is "the 7-limit 19&53 linear temperament", or "7-limit 19&53-LT" for short. The other one, [<1, 5, 6, 12|, <0, -13, -14, -35|], is similarly the "7-limit 19&80-LT". Anyone who's familiar with 19-ET can probably guess why there's a bunch of LT's with approximate minor third generators in that general area. You can also compare the numbers "19&53" with "19&80" and get a rough idea of the general complexity of the temperament.

For those few temperaments that can't be found this way, it might be best just to give the map. An example:

[<1, 2, -1, 2|, <0, -1, 8, 2|]

You can tell this is a 7-limit version of schismic (5-limit 12&29-LT to use the suggested convention), which has the map

[<1, 2, -1|, <0, -1, 8|]

But it's not one of the usual varieties of schismic, and doesn't have a name. The 7-limit version of 12&29-LT has a different map:

[<1, 2, -1, -3|, <0, -1, 8, 14|]

Fortunately, practically all of the useful 7-limit temperaments can be named this way, and this can be extended to higher limits without the awkwardness of wedgies. "Shrutar" is simply "11-limit 22&46", while "Cassandra 1" becomes "13-limit 41&94". (Anyone care to guess what "13-limit 29&58" is?)

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> > >>Yes, "simple 7-limit minor thirds temperament" is a bit of a mouthful, >>but at least it gives people some idea what you're talking about. > > > This is partly because you are lucky (6/5, a well-known consonance, is
> a generator) and partly because it really is a name (the word "simple"
> is in there, with no precise meaning attached to it.) Pajara would be
> the "7-limit septimal tritone and fifths temperament", which is
> getting ugly. Negri could be the "7-limit simple major or maybe minor
> diatonic semitone temperament", and miracle the "7-limit complex major
> or maybe minor diatonic semitone temperament". 35/32 is also a
> generator for negri, but porcupine would grab that one, becoming the > "7-limit simple septimal neutral seconds temperament", whereas
> hemikleismic would be the "7-limit complex septimal neutral seconds
> temperament". .. whereas I'd call them:

pajara "7-limit 10&12"
negri "7-limit 9&10"
miracle "7-limit 10&31"
porcupine "7-limit 15&22"
hemikleismic "7-limit 15&53"

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> If linear temperaments were as simple as equal temperaments there
> wouldn't be any point in naming them. I still think there's value in
> giving names to the more useful temperaments. But there's an
alternative
> to arbitrary names.

There's more than one alternative. Naming N&M is good if we have two
standard vals giving the temperament, but it is not usually unique.
Someone (you?) proposed using the smallest pair which works, and that
would give a concise name.

An alternative is comma naming, using the TM basis or as I've
proposed, succesive limit commas.

Take for example the temperaments identified as
> "catakleismic" (or now alternatively "hanson") and "parakleismic". Both
> are minor-third temperaments, but they have different maps:
>
> [<1, 0, 1, -3|, <0, 6, 5, 22|] catakleismic
> [<1, 5, 6, 12|, <0, -13, -14, -35|] parakleismic
>
> The first one has a TOP generator of 316.9 cents and the second one
> 315.1 cents. But it's really the mapping that distinguishes them, more
> than the sizes of the generators.

Or commas, of course. Hanson/catakleismic would be (15625/15552,
225/224) which is not as concise as 19&53 but which gives important
information. Parakleismic would give two different commas,
(|8 14 -13 0>, |6 0 -5 2>).

> Fortunately, practically all of the useful 7-limit temperaments can be
> named this way, and this can be extended to higher limits without the
> awkwardness of wedgies. "Shrutar" is simply "11-limit 22&46", while
> "Cassandra 1" becomes "13-limit 41&94". (Anyone care to guess what
> "13-limit 29&58" is?)

It has definite advantages. Paul would either have to put up with
standard vals, or with some replacement, however.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> .. whereas I'd call them:

I'm not getting the same names:

> pajara "7-limit 10&12" I get 2&10
> negri "7-limit 9&10" I get 1&9
> miracle "7-limit 10&31" I get 10&11
> porcupine "7-limit 15&22" I get 7&8
> hemikleismic "7-limit 15&53" I get 15&23

None of the names are the same!

ARE WE SURPRISED?

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> > .. whereas I'd call them:
>
> I'm not getting the same names:
>
> > pajara "7-limit 10&12" I get 2&10
> > negri "7-limit 9&10" I get 1&9
> > miracle "7-limit 10&31" I get 10&11
> > porcupine "7-limit 15&22" I get 7&8
> > hemikleismic "7-limit 15&53" I get 15&23
>
> None of the names are the same!
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > >>.. whereas I'd call them:
> > > I'm not getting the same names:
> > >>pajara "7-limit 10&12" I get 2&10
>>negri "7-limit 9&10" I get 1&9
>>miracle "7-limit 10&31" I get 10&11
>>porcupine "7-limit 15&22" I get 7&8
>>hemikleismic "7-limit 15&53" I get 15&23
> > > None of the names are the same!

I'm using the convention of picking the smallest 7-limit-consistent ETs to avoid ambiguity. 1-ET, 2-ET, 7-ET, 8-ET, 11-ET, and 23-ET aren't consistent in the 7-limit.

But inconsistent ET's could be allowed if you can notate the prime mapping concisely:

7a = take the best 7-ET approximation of each of the primes
<7, 11, 16, 20|
7b = substitute the second best 3:1 approximation
<7, 12, 16, 20|
7c = substitute the second best 5:1 approximation
<7, 11, 17, 20|
7d = substitute the second best 7:1 approximation
<7, 11, 16, 19|

In the case of 7-ET, it probably only makes sense to substitute the 7:1, but you might want all three alternatives for 8-ET.

So then porcupine could be 7a&8a. If you use the alternative mappings of 7d <7, 11, 16, 19| and 8d <8, 13, 19, 23|, you end up with <<3, 5, 9, 1, 6, 7|| (map [<1, 2, 3, 4|, <0, -3, -5, -9|]), which was "Number 59" on the big list from tuning-math (generator 158.147c, period 1193.416c).

Naming linear temperaments by giving a pair of ETs isn't too bad.
But I always want to know the approximate size of the generator and
how many chains to the octave. Is there a simple pen & paper
algorithm that will give me those from the two ETs.

The other thing is, I'd expect the two ETs to be ones that
meaningfully contain the linear temperament. For example 1 step of
10-ET seems just too far from any kind of optimum for the "Miracle"
generator. It makes more sense to me to think of Miracle as the
31&41-LT, and then I can add those together and get 72-ET which
supports it even better than those two which essentially give the
extremes. Similarly meantone seems like the 12&19-LT to me.

But of course I'm not proposing to stop using the names Miracle or
meantone for these, as these are among the few really good linear
temperaments that get talked about a lot. I was just using them for
familiar examples.

Dave Keenan wrote:
> Naming linear temperaments by giving a pair of ETs isn't too bad. > But I always want to know the approximate size of the generator and > how many chains to the octave. Is there a simple pen & paper > algorithm that will give me those from the two ETs.

Oh, simple. Pen. Paper. Yes, well, the periods per octave (is that what you mean by "chains"?) is the greatest common divisor, which Euclid's algorithm will give you if you can't guess it. For the generator, you need to solve an inverse modulo problem, which can be done using the extended Euclidian algorithm. Number theory text books may well have that.

The easiest way with pen and paper is to build up the relevant scale tree and find where the two ETs occur adjacent to one another. It helps if they're simple, or you have some idea what the generator is supposed to be. But you can still work backwards from the ETs, and then forwards to fill in the generators.

If we take meantone for the example, say you're trying to decipher it as 12&19. Well, you do 19-12=7. That gives you part of the scale tree

7 12
19

Then you do 12-7=5

5 7
12
19

then 7-5=2

5 2
7
12
19

Now you can work forwards to get 5 and 2

1 2
3
5

and build the whole tree with generators

0/1 1/2
1/3
2/5
3/7
5/12
8/19

If you aren't restricted to pen and paper, there's a script here:

http://x31eq.com/temper/twoet.html

> The other thing is, I'd expect the two ETs to be ones that > meaningfully contain the linear temperament. For example 1 step of > 10-ET seems just too far from any kind of optimum for the "Miracle" > generator. It makes more sense to me to think of Miracle as the > 31&41-LT, and then I can add those together and get 72-ET which > supports it even better than those two which essentially give the > extremes. Similarly meantone seems like the 12&19-LT to me. But it's also useful for the smaller number to refer to a "white note" scale. That gives you some idea of the melodic properties. You can also guess where the intervals are likely to fall in a small scale, and it's nice to know when it's consistent.

Graham

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Dave Keenan wrote:
> > Naming linear temperaments by giving a pair of ETs isn't too
bad.
> > But I always want to know the approximate size of the generator
and
> > how many chains to the octave. Is there a simple pen & paper
> > algorithm that will give me those from the two ETs.
>
> Oh, simple. Pen. Paper. Yes, well, the periods per octave (is
that
> what you mean by "chains"?)

Yes.

> is the greatest common divisor, which
> Euclid's algorithm will give you if you can't guess it.

OK. Well that's simple enough. Then once we have the period from
that, we know we can find a generator smaller than half the period.

> For the
> generator, you need to solve an inverse modulo problem, which can
be
> done using the extended Euclidian algorithm. Number theory text
books
> may well have that.
>
> The easiest way with pen and paper is to build up the relevant
scale
> tree and find where the two ETs occur adjacent to one another. It
helps
> if they're simple, or you have some idea what the generator is
supposed
> to be. But you can still work backwards from the ETs, and then
forwards
> to fill in the generators.
... <example elided>

Thanks for that, Not real simple when the numbers get bigger, eh?

Can we take some shortcuts and not have to know what a scale tree is?

Will this work in general:

Do successive subtractions, stopping just before the first result
that is bigger than the previous one. e.g.
19-12=7, 12-7=5, 7-5=2, (but not 5-2=3).

Take that last result (2) and put a 1 over it, 1/2. Then divide the
second-last result (5) by the last result (2) and round to nearest
integer (if equidistant, round down, but it doesn't really matter,
you'll get a generator either way). So we get Round(5/2) = 2. Put
this over the second-last result, 2/5.

So now we have th efirst two approximations to the generator (as
fractions of an octave) 1/2 and 2/5. We then add the two numerators
to get the next numerator, and the two denominators to get the next
denominator, contiinuing Fibinacci-like until we get back to our
original two numbers as denominators, or one past them. e.g.

1/2, 2/5, 3/7, 5/12, 8/19, 13/31

We can then multiply the last by 1200 if we want the approximate
generator in cents.

1200 * 13/31 ~= 503 c <OK I cheated and used a calculator here>

----------------------------------
Try Miracle as 31&41

GCD = 1 so period is whole octave

41, 31, 10, (21)

1/10

Round(31/10) = 3

1/10, 3/31, 4/41, 7/72

1200*7/72 ~= 117 c

---------------------------------
Try 15&22

GCD = 1 so period is whole octave

22, 15, 7, (8)

Round(15/7) = 2

1/7, 2/15, 3/22, 5/37

------------------------------------

> > The other thing is, I'd expect the two ETs to be ones that
> > meaningfully contain the linear temperament. For example 1 step
of
> > 10-ET seems just too far from any kind of optimum for
the "Miracle"
> > generator. It makes more sense to me to think of Miracle as the
> > 31&41-LT, and then I can add those together and get 72-ET which
> > supports it even better than those two which essentially give
the
> > extremes. Similarly meantone seems like the 12&19-LT to me.
>
> But it's also useful for the smaller number to refer to a "white
note"
> scale. That gives you some idea of the melodic properties. You
can
> also guess where the intervals are likely to fall in a small scale,

Right. But you can get this by successive subtraction can't you?

> and it's nice to know when it's consistent.

A fairly minor nicety surely. You can find this elsewhere. Wouldn't
you rather know when it's accurate enough to be worth the complexity
of its mapping?

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Naming linear temperaments by giving a pair of ETs isn't too bad.
> But I always want to know the approximate size of the generator and
> how many chains to the octave. Is there a simple pen & paper
> algorithm that will give me those from the two ETs.

If the two ets are reasonable ones, which you suggest below, then my
white keys/black keys continued fraction algorithm will give you the
period and generator using pen & paper for the tuning which is the sum
of the pair.

> The other thing is, I'd expect the two ETs to be ones that
> meaningfully contain the linear temperament. For example 1 step of
> 10-ET seems just too far from any kind of optimum for the "Miracle"
> generator. It makes more sense to me to think of Miracle as the
> 31&41-LT, and then I can add those together and get 72-ET which
> supports it even better than those two which essentially give the
> extremes. Similarly meantone seems like the 12&19-LT to me.

This suggests picking two ets which sum to a good (not necessarily
poptimal) generator and represent the ends of the range of reasonable
choices. This is what people (me, for instance) have mostly been doing
in practice when using this informally; the question of how to
formalize it and make it precise then arises. How do we know 12 and 19
represent the extremes of meantone, or 31 and 41 of miracle? We could
describe ennealimmal very reasonably as 171&270, but some people might
find 72&99 better. Which is it? 171 and 270 sum to 441, clearly a
"good" choice for ennealimmal, and gives the range of the true
microtempered versions, but 71 and 99 sum to 171, which for most
people is going to be plenty good already. Some rule which would
automatically tell us we pick 171&270 would be nice.

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

> Oh, simple. Pen. Paper. Yes, well, the periods per octave (is that
> what you mean by "chains"?) is the greatest common divisor, which
> Euclid's algorithm will give you if you can't guess it. For the
> generator, you need to solve an inverse modulo problem, which can be
> done using the extended Euclidian algorithm. Number theory text books
> may well have that.

I could explain how to do it easily enough, but activity such as that
is banned on this list. In fact, "Euclidean" is one of those words
which should not have been used.

> The easiest way with pen and paper is to build up the relevant scale
> tree and find where the two ETs occur adjacent to one another.

It doesn't look as easy as applying the E*&^%$#@$ alforithm to me,
judging by your example below.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> So now we have th efirst two approximations to the generator (as
> fractions of an octave) 1/2 and 2/5. We then add the two numerators
> to get the next numerator, and the two denominators to get the next
> denominator, contiinuing Fibinacci-like until we get back to our
> original two numbers as denominators, or one past them. e.g.
>
> 1/2, 2/5, 3/7, 5/12, 8/19, 13/31
>
> We can then multiply the last by 1200 if we want the approximate
> generator in cents.
>
> 1200 * 13/31 ~= 503 c <OK I cheated and used a calculator here>

It seems to me a situation where some people are allowed to discuss
math and other people are not allowed is clearly obnoxious and highly
unfair. If there was a rule saying math was not allowed on this list,
would not this post be subject to cancellation?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> This suggests picking two ets which sum to a good (not necessarily
> poptimal) generator and represent the ends of the range of
reasonable
> choices. This is what people (me, for instance) have mostly been
doing
> in practice when using this informally; the question of how to
> formalize it and make it precise then arises. How do we know 12
and 19
> represent the extremes of meantone, or 31 and 41 of miracle? We
could
> describe ennealimmal very reasonably as 171&270, but some people
might
> find 72&99 better. Which is it? 171 and 270 sum to 441, clearly a
> "good" choice for ennealimmal, and gives the range of the true
> microtempered versions, but 71 and 99 sum to 171, which for most
> people is going to be plenty good already. Some rule which would
> automatically tell us we pick 171&270 would be nice.

Agreed.

The extremes of generator size have to do with when a different
mapping (of similar complexity) becomes a better choice. Also when
the error grows significantly larger than the minimum. I know these
are still vague. Sorry.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > So now we have th efirst two approximations to the generator (as
> > fractions of an octave) 1/2 and 2/5. We then add the two
numerators
> > to get the next numerator, and the two denominators to get the
next
> > denominator, contiinuing Fibinacci-like until we get back to our
> > original two numbers as denominators, or one past them. e.g.
> >
> > 1/2, 2/5, 3/7, 5/12, 8/19, 13/31
> >
> > We can then multiply the last by 1200 if we want the approximate
> > generator in cents.
> >
> > 1200 * 13/31 ~= 503 c <OK I cheated and used a calculator here>
>
> It seems to me a situation where some people are allowed to discuss
> math and other people are not allowed is clearly obnoxious and
highly
> unfair. If there was a rule saying math was not allowed on this
list,
> would not this post be subject to cancellation?

Gene,

I think most folks appreciate the difference between the above,
which is primary school math (I think you call it grade school) and
university level math-double-major-type math.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> I think most folks appreciate the difference between the above,
> which is primary school math (I think you call it grade school) and
> university level math-double-major-type math.

The subject line gives a point I think should be possible to make on
this list, but frankly I just am not sure when and why the screaming
is going to start. Why is it that "val" an explosion in a jargon
factory, but "schismino" isn't? When can you follow up a posting with
another posting, without drawing a lot of hostile responses? Why
should people interesting in tuning even be *making* hostile responses
unless the posting is off-topic or makes no sense?

Anyway, onward to the subject line. If you choose a fifth of size
(2048/11)^(1/13) it will have a wolf, for 14 notes of meantone, of
size 11/8. This fifth is 696.05 and is closely approximated by the
fifth of 50-equal, which is 696 cents exactly. Very much along the
same lines is taking 16 notes of meantone with a fifth of size
(416)^(1/15), which nets you a wolf fifth of 16/13; the fifth here is
of size 696.04.

These nifty wolf fifths are related to the excellence of 50-et for 11
and 13 overtones, as is my "ratwolf" idea of making the wolf in 12
notes of meantone exactly 20/13. I'll post something about 50 over on
tuning-math.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> The subject line gives a point I think should be possible to make
> on this list, but frankly I just am not sure when and why the
> screaming is going to start. Why is it that "val" an explosion in
> a jargon factory, but "schismino" isn't?

The term is actually "schismina" (an "a" on the end like comma,
kleisma, schisma). And maybe it _is_ unnecessary jargon. We should
at least still be explaining what it means in every new thread we
use it in. Sorry if we haven't.

But maybe the reason is that it is derived in a fairly obvious way
from an existing term, "schisma", that's been in use for (I think)
hundreds of years (with various spellings). I think many people,
assuming they know what a schisma is, could guess from the context
that a "schismina" is something like a schisma, only female.

Just joking. ... just like a schisma, only smaller. :-)

Perhaps I'm being too optimistic.

>> I think most folks appreciate the difference between the above,
>> which is primary school math (I think you call it grade school) and
>> university level math-double-major-type math.
>
>The subject line gives a point I think should be possible to make on
>this list, but frankly I just am not sure when and why the screaming
>is going to start. Why is it that "val" an explosion in a jargon
>factory, but "schismino" isn't? When can you follow up a posting with
>another posting, without drawing a lot of hostile responses? Why
>should people interesting in tuning even be *making* hostile responses
>unless the posting is off-topic or makes no sense?

Indeed Dave, the Sagittal project is hardly guitless in the jargon
department. I'm completely lost when I hear about "Olympian",
"Herculean" or "Trojan" subsets, for example. I personally think
it's fine. If I were more interested in Sagittal (or had more
time for the list) I'd learn these terms. They're cute. But they
seem to be jargon of the type you're railing against.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I think most folks appreciate the difference between the above,
> >> which is primary school math (I think you call it grade school)
and
> >> university level math-double-major-type math.
> >
> >The subject line gives a point I think should be possible to make
on
> >this list, but frankly I just am not sure when and why the
screaming
> >is going to start. Why is it that "val" an explosion in a jargon
> >factory, but "schismino" isn't? When can you follow up a posting
with
> >another posting, without drawing a lot of hostile responses? Why
> >should people interesting in tuning even be *making* hostile
responses
> >unless the posting is off-topic or makes no sense?
>
> Indeed Dave, the Sagittal project is hardly guitless in the jargon
> department. I'm completely lost when I hear about "Olympian",
> "Herculean" or "Trojan" subsets, for example. I personally think
> it's fine. If I were more interested in Sagittal (or had more
> time for the list) I'd learn these terms. They're cute. But they
> seem to be jargon of the type you're railing against.

You're right, at least until the mythology is complete, which should
weave them into one big consistent set of Sagittal memory aids on
the theme of Ancient Greek myths.

We should use the terms:
"extreme precision" instead of Olympian
"high precision" instead of Herculean
"medium precision" instead of Athenian
"low precision" instead of Spartan
and
"12-relative" (also medium precision) instead of Trojan

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> The term is actually "schismina" (an "a" on the end like comma,
> kleisma, schisma). And maybe it _is_ unnecessary jargon. We should
> at least still be explaining what it means in every new thread we
> use it in. Sorry if we haven't.

please write up a definition of skhismina which is similar
to my definition #2 of "skhisma", and i'll include it in
the Encyclopaedia.

> But maybe the reason is that it is derived in a fairly obvious way
> from an existing term, "schisma", that's been in use for (I think)
> hundreds of years (with various spellings).

yes, the term "schisma" has been in use since Philolaus,
c. 400 BC ... so it's really been *thousands* of years.

*but* ... Philolaus's use of it is not the same as what
we mean by it today. the current standard meaning of
skhisma dates only from the mid-late 1800s, Elli's
translation of Helmholtz.

and Ellis deliberately spelled it with a "k" so as to
avoid any religious connotations, referring to the
"Great Schism". i suppose the spelling with "c" is
much more common, but i always use "k".

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Dave,
...
> please write up a definition of skhismina which is similar
> to my definition #2 of "skhisma", and i'll include it in
> the Encyclopaedia.

schismina, skhismina

A generally imperceptiple interval, smaller than a schisma. The most
convenient cutoff between schismas and schisminas, for
classification of commas(2), is at about 1.8 cents (or precisely log
(sqrt(3^53/2^84))/log(2)*1200).

Incidentally Monz,

A graphic seems to be broken on your comma page. And I'd like to
refine that approx 10 to 40 cents to approx 12 to 35 cents.

You may be interested in including, somewhere in your encyclopedia,
the other comma category cutoffs that George and I found most useful
while developing Sagittal.

Square of lowerbound
2-exponent
3-exponent
Lowerbound (cents)
Size range name
-------------------------------------------------
0 schismina
-84 53 1.807522933 schisma
317 -200 4.499913461 kleisma
-19 12 11.73000519 comma
-57 36 35.19001558 small-diesis
8 -5 45.11249784 (medium-)diesis
-11 7 56.84250303 large-diesis
-30 19 68.57250822 small-semitone
35 -22 78.49499048 limma
-3 2 101.9550009 large-semitone
62 -39 111.8774831 apotome
-106 67 115.492529

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Dave,
> ...
> > please write up a definition of skhismina which is similar
> > to my definition #2 of "skhisma", and i'll include it in
> > the Encyclopaedia.
>
> schismina, skhismina
>
> A generally imperceptiple interval, smaller than a schisma. The most
> convenient cutoff between schismas and schisminas, for
> classification of commas(2), is at about 1.8 cents (or precisely log
> (sqrt(3^53/2^84))/log(2)*1200).

i'm so glad that i took one more peek at the tuning list
before turning off my PC! it's in there.

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

thanks! (and for being so quick, too)

> Incidentally Monz,
>
> A graphic seems to be broken on your comma page.

over the next few weeks readers will find a *lot*
of broken links on the Tonalsoft website. we had
to make it public before we were ready to, to get
the credit-card processing working, etc. i'm working
frantically on it, so please do keep the reports of
broken links coming!

> And I'd like to
> refine that approx 10 to 40 cents to approx 12 to 35 cents.
>
> You may be interested in including, somewhere in your encyclopedia,
> the other comma category cutoffs that George and I found most useful
> while developing Sagittal.
>
> Square of lowerbound
> 2-exponent
> 3-exponent
> Lowerbound (cents)
> Size range name
> -------------------------------------------------
> 0 schismina
> -84 53 1.807522933 schisma
> 317 -200 4.499913461 kleisma
> -19 12 11.73000519 comma
> -57 36 35.19001558 small-diesis
> 8 -5 45.11249784 (medium-)diesis
> -11 7 56.84250303 large-diesis
> -30 19 68.57250822 small-semitone
> 35 -22 78.49499048 limma
> -3 2 101.9550009 large-semitone
> 62 -39 111.8774831 apotome
> -106 67 115.492529

wow, thanks for that, Dave! it was something i
wanted but don't think i ever asked you for.
i'll get around to it ASAP.

-monz

hi Dave,

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

> hi Dave,
>
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> <snip> ... on your comma page.
>
> > And I'd like to refine that approx 10 to 40 cents
> > to approx 12 to 35 cents.
> >
> > You may be interested in including, somewhere in your
> > encyclopedia, the other comma category cutoffs that
> > George and I found most useful while developing Sagittal.
> >
> > Square of lowerbound
> > .. 2-exponent
> > ...... 3-exponent
> > ................ Lowerbound (cents)
> > ................................. Size range name
> > -------------------------------------------------
> > .. [ 0, 0 > ....... 0 ........... schismina
> > .. [-84, 53 > ..... 1.807522933 . schisma
> > .. [ 317,-200 > ... 4.499913461 . kleisma
> > .. [-19, 12 > .... 11.73000519 .. comma
> > .. [-57, 36 > .... 35.19001558 .. small-diesis
> > .. [ 8, -5 > ..... 45.11249784 .. (medium-)diesis
> > .. [-11, 7 > ..... 56.84250303 .. large-diesis
> > .. [-30, 19 > .... 68.57250822 .. small-semitone
> > .. [ 35, -22 > ... 78.49499048 .. limma
> > .. [-3, 2 > ..... 101.9550009 ... large-semitone
> > .. [ 62, -39 > .. 111.8774831 ... apotome
> > .. [-106, 67 > .. 115.492529
>
>
> wow, thanks for that, Dave! it was something i
> wanted but don't think i ever asked you for.
> i'll get around to it ASAP.

(i took the liberty of reformatting your table
so that it appears the way it should on the
Yahoo interface.)

thanks, Dave ... the virus my system got a few weeks
ago apparently zapped that graphic into the ether.

but in fact, this tabke is exactly what i need to
recreate that graphic, but to do it properly.

but i'm curious ... why are the cents-values you give
exactly half of the actual cents-values of those intervals?
it obivously has something to do with ranges for
sagittal notational symbols. how does this work?

-monz

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

> but i'm curious ... why are the cents-values you give
> exactly half of the actual cents-values of those intervals?
> it obivously has something to do with ranges for
> sagittal notational symbols. how does this work?

never mind, i get it. sheesh, i don't know what
wasn't clear the first time i read it. it's obvious
that the values are lowerbounds. duh.

-monz

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

> > > <snip> ... the other comma category cutoffs that
> > > George and I found most useful while developing Sagittal.
> > >
> > > Square of lowerbound
> > > .. 2-exponent
> > > ...... 3-exponent
> > > ................ Lowerbound (cents)
> > > ................................. Size range name
> > > -------------------------------------------------
> > > .. [ 0, 0 > ....... 0 ........... schismina
> > > .. [-84, 53 > ..... 1.807522933 . schisma
> > > .. [ 317,-200 > ... 4.499913461 . kleisma
> > > .. [-19, 12 > .... 11.73000519 .. comma
> > > .. [-57, 36 > .... 35.19001558 .. small-diesis
> > > .. [ 8, -5 > ..... 45.11249784 .. (medium-)diesis
> > > .. [-11, 7 > ..... 56.84250303 .. large-diesis
> > > .. [-30, 19 > .... 68.57250822 .. small-semitone
> > > .. [ 35, -22 > ... 78.49499048 .. limma
> > > .. [-3, 2 > ..... 101.9550009 ... large-semitone
> > > .. [ 62, -39 > .. 111.8774831 ... apotome
> > > .. [-106, 67 > .. 115.492529

oops ... i put the comma in the wrong place inside the
monzos ... it should come after the 3-exponent, not the 2.
it will be fixed on the webpage.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > Square of lowerbound
> > > .. 2-exponent
> > > ...... 3-exponent
> > > ................ Lowerbound (cents)
> > > ................................. Size range name
> > > -------------------------------------------------
> > > .. [ 0, 0 > ....... 0 ........... schismina
> > > .. [-84, 53 > ..... 1.807522933 . schisma
> > > .. [ 317,-200 > ... 4.499913461 . kleisma
> > > .. [-19, 12 > .... 11.73000519 .. comma
> > > .. [-57, 36 > .... 35.19001558 .. small-diesis
> > > .. [ 8, -5 > ..... 45.11249784 .. (medium-)diesis
> > > .. [-11, 7 > ..... 56.84250303 .. large-diesis
> > > .. [-30, 19 > .... 68.57250822 .. small-semitone
> > > .. [ 35, -22 > ... 78.49499048 .. limma
> > > .. [-3, 2 > ..... 101.9550009 ... large-semitone
> > > .. [ 62, -39 > .. 111.8774831 ... apotome
> > > .. [-106, 67 > .. 115.492529
>
> (i took the liberty of reformatting your table
> so that it appears the way it should on the
> Yahoo interface.)

Good.

> but in fact, this table is exactly what i need to
> recreate that graphic, but to do it properly.
>
>
> but i'm curious ... why are the cents-values you give
> exactly half of the actual cents-values of those intervals?
> it obviously has something to do with ranges for
> sagittal notational symbols. how does this work?

Yes the boundaries between categories are the square root (half the
cents value) of the interval described by the prime exponent vector.
e.g. 1.807... cents is log2(sqrt(2^-84 * 3^53))*1200
which is log2(2^-84 * 3^53)*1200/2.

For one thing, having the boundaries at irrational numbers (as these
square roots are) ensures that no comma will ever fall exactly on a
boundary.

It actually has nothing to do with the choices made about sagittal
symbols. I'll have to explain more later.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > > Square of lowerbound
> > > > .. 2-exponent
> > > > ...... 3-exponent
> > > > ................ Lowerbound (cents)
> > > > ................................. Size range name
> > > > -------------------------------------------------
> > > > .. [ 0, 0 > ....... 0 ........... schismina
> > > > .. [-84, 53 > ..... 1.807522933 . schisma
> > > > .. [ 317,-200 > ... 4.499913461 . kleisma
> > > > .. [-19, 12 > .... 11.73000519 .. comma
> > > > .. [-57, 36 > .... 35.19001558 .. small-diesis
> > > > .. [ 8, -5 > ..... 45.11249784 .. (medium-)diesis
> > > > .. [-11, 7 > ..... 56.84250303 .. large-diesis
> > > > .. [-30, 19 > .... 68.57250822 .. small-semitone
> > > > .. [ 35, -22 > ... 78.49499048 .. limma
> > > > .. [-3, 2 > ..... 101.9550009 ... large-semitone
> > > > .. [ 62, -39 > .. 111.8774831 ... apotome
> > > > .. [-106, 67 > .. 115.492529

> It actually has nothing to do with the choices made about sagittal
> symbols. I'll have to explain more later.

It does however have to do with the fact that we were using comma
inflections to notate JI or rational pitches relative to the
nominals A to G, considered to be in a chain of just fifths, FCGDAEB.

The apotome is the "comma" that is normally notated with a sharp or
flat (2048:2187) in JI or Pythagoraen tunings. Apotome symbols are
purely 3-limit. They just let you extend the chain of fifths up or
down.

The first thing to notice about these comma category boundaries is
that they are complementary within the apotome. Or putting it
another way, they are symmetrical about the half apotome. All except
for the somewhat-arbitrary boundary between schismas and kleismas
that is. So we have the following complementary pairs of categories.

schisminas complementary to apotomes
schismas or kleismas complementary to large semitones
commas complementary to limmas
small dieses complementary to small semitones
(medium) dieses complementary to large dieses

We wanted to name commas (broad sense) according to what ratios they
allowed us to notate. Since we can use the same symbol to notate
ratios having any number of 2's or 3's in them just by changing
octaves or nominals on the chain of fifths. The important thing
about a ratio for notation purposes was the powers of the primes
greater than 3 that it contained. So we took to calling commas by
this part, such as "5-comma" for the Didymus or syntonic comma of
80:81, and "7-comma" for the Archytas or septimal comma of 63:64.

But there is more than one comma (broad sense) that will let you
notate a given ratio. For example a 5/4 (from C as 1/1) can be
notated as a syntonic comma less than the Pythagorean E or as a
schisma more than a Pythagorean Fb.

It was natural in this case to call one the 5-comma and the other
the 5-schisma. And through examining many such sets of multiple
commas like this, usable for notating the same ratio in different
ways, it eventually became clear that we could guarantee a unique
name for each, which would also indicate its approximate size, just
by having the boundaries of what were existing (but vague and
overlapping) categories at the above square-roots (half cents) of
various 3-limit commas.

on 7/12/04 9:09 PM, monz <monz@attglobal.net> wrote:

> please do keep the reports of broken links coming!

Maybe just load your site into Dreamweaver or something which will check the
entire site within seconds for broken links.

I haven't used Dreamweaver much for this and kind of stopped using the
program because it was such a hunk of junk, but chances are there are other
possibilities, maybe even freeware that others could suggest?

-Kurt

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

/tuning/topicId_53869.html#54577

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Dave,
> ...
> > please write up a definition of skhismina which is similar
> > to my definition #2 of "skhisma", and i'll include it in
> > the Encyclopaedia.
>
> schismina, skhismina
>
> A generally imperceptiple interval, smaller than a schisma.

I'm envisioning the scena for naming a schismina:
Not wanting to be meana, we're going for sublima
We wouldn't want to misma, by only doing schisma
That's now so very common,
One might think it's just a comma, man…

jp

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> on 7/12/04 9:09 PM, monz <monz@a...> wrote:
>
> > please do keep the reports of broken links coming!
>
> Maybe just load your site into Dreamweaver or something
> which will check the entire site within seconds for broken
> links.
>
> I haven't used Dreamweaver much for this and kind of
> stopped using the program because it was such a hunk of junk,
> but chances are there are other possibilities, maybe even
> freeware that others could suggest?
>
> -Kurt

thanks for those suggestions, Kurt.

let's please move any further discussion of this to
metatuning.

-monz

Hi there.

> Anyway, onward to the subject line. If you choose a fifth of size
> (2048/11)^(1/13) it will have a wolf, for 14 notes of meantone, of
> size 11/8. This fifth is 696.05 and is closely approximated by the
> fifth of 50-equal, which is 696 cents exactly. Very much along the
> same lines is taking 16 notes of meantone with a fifth of size
> (416)^(1/15), which nets you a wolf fifth of 16/13; the fifth here is
> of size 696.04.
>
> These nifty wolf fifths are related to the excellence of 50-et for 11
> and 13 overtones, as is my "ratwolf" idea of making the wolf in 12
> notes of meantone exactly 20/13. I'll post something about 50 over on
> tuning-math.

Oops!
I've come to this idea about two years ago but at that time I knew nothing
about the tuning list so I told only a few people about these things. Well,
I think reader's don't mind if I explore this in greater detail now. But
before I start, I'd like to introduce three other facts that will be
important later in my text. First of all, instead of using the word "wolf"
to mean the interval between the last and the first tone of the chain of
fifths, I'm gonna use "wolf" simply in the case of a diminished sixth (this
comes from the way regular 12-tone chains were being tuned over centuries).
Then, to be able to compare meantone tunings efficiently, I decided to
classify them roughly into two groups - the bright ones (i.e. the fifth is
only a little smaller than the just 3/2) and the dark ones (the fifth is
much smaller than 3/2). Finally, for comparing these temperaments, I'm gonna
use the size of the minor second instead of the fifth. The reason is that
the minor second is quite a small interval and therefore changes in its size
are more or less audible. Moreover, the minor second can be described as 3
octaves minus 5 fifths (or 5 fourths minus 2 octaves) and that's why a
change in the size of a fifth will reflect 5 times stronger (in the opposite
direction) in the size of a minor second. OK, let's get it.
I'll start with the meantone where the double-diminished fifth is tuned
to 11/8. This tuning (let's call it "MT11", for example) has a minor second
of ~119.738 cents. Then there's another tuning (I'll call it "MT13") with a
double-augmented fifth tuned to 13/8. In this case, the minor second has
~119.824 cents. Looking carefully at these two may raise another idea. If
11/8 is a double-diminished fifth (for example C-Gbb) and 13/8 is a
double-augmented fifth (i.e. C-G##), then 13/11 could be the difference
between these two (i.e. Gbb-G##). In fact, this is 4 times the size of a
regular chroma because shifting a note a chroma higher always adds a sharp
or removes a flat (Gbb, Gb, G, G#, G##). Of course, you can make such a
meantone (this could be called "MT13/11") where 4 chromas together make an
exact 13/11 interval. As a result, you get a minor second of ~119.784 cents.
So we could say that MT11 is the brightest of these three tunings, MT13 is
the darkest one, and MT13/11 lies somewhere in-between. Therefore, we can
get to a conclusion that the number 11 works here as some sort of a
"brightener" and the number 13 is some kind of a "darkener" here. Indeed,
this is true. And we can demonstrate it clearly by picking up two other
meantone temperaments (I'll call them "MT11/5" and "MT13/5"). In the first
of these, the double-diminished third is tuned to 11/10. In the other one,
the augmented third is tuned to 13/10. While in MT11/5 you find a minor
second of ~119.119 cents, MT13/5 has a minor second of ~120.812 cents. If
you think of the 11/10 as the interval between, for instance, E-Gbb, and
13/10 as the interval between E-G##, you find that 13/11 has, again, the
meaning of 4 chromas. So, indeed, between MT11/5 and MT13/5 you find just
the same as between MT11 and MT13 (i.e. MT13/11). Incidentally, the
augmented third is the octave inversion of the wolf (i.e. diminished sixth)
so the MT13/5 can be tuned on a regular 12-tone keyboard by simply dividing
the factor of 416/5 (i.e. augmented third plus 6 octaves) geometrically into
11 equal fifths.
Even more, lots of interesting similarities appear as we add 7-limit
intervals. It's a well-known matter that 7/4 can be applied beautifully as
an augmented sixth. That means we have not only a 5-limit major second of
16/15 but also a 7-limit one of 15/14. This minor second has a size of
~119.443 cents. A meantone with such a minor second is a bit brighter than
MT11 and darker than MT11/5. Moreover, if you consider 7/4 to be, for
example, C-A#, then 33/28 comes out as A#-Dbb. Making a meantone where this
is true results in a minor second of ~119.193 cents (quite close to MT11/5).
And that's still not all. If the number 7 represents A#, then 35 is C## and
so 36/35 should be the enharmonic diesis between C##-D. If you make a
meantone in which this is true, you get a minor second of ~120.321 cents
(i.e. a bit darker than MT13 and brighter than MT13/5). And, to prove the
point, this is extremely close to another meantone in which a major third
downwards (i.e. E-C) has the same beat rate as a major sixth downwards (i.e.
E-G). This tuning has a minor second of ~120.330 cents (or, to be more
precise, ~120.3302118992214 cents). When you take all these things into
account and make a few averages, soon you discover that all of these tunings
are in some ways swinging around 50-tone equal tuning! And that's why to me
the 50-TET is the "nicest" of all the ET-meantone tunings not exceeding 100
tones per octave. Having read so much about 31-TET, from time to time, I
feel a bit alone in my effort of promoting 50-TET. Fortunately, I got a note
from Monz that Woolhouse also liked 50-TET (thanks a lot for this, Monz) so
I know I'm not quite that alone.
And what is my conclusion? If someone says that the first 16 elements in
the harmonic series can't be converted to the standard notation, he's wrong.
If you don't mind using such a large chain as 28 fifths, you can do it like
this:

C C G C
E G A# C
D E Gbb G
G## A# B C

This may put a question of making such a tuning that behaves much like a
meantone though it's made purely of just intervals. Of course, these can be
made with no problem. I even suppose that someone found such temperaments
useful much earlier than I did. I remember having read somewhere that
7-limit intervals were being explored quite often during the Renaissance and
the Baroque periods. So I wouldn't be surprised if someone had come to this
idea as early as in the 16th century. Perhaps some of you know more about
this matter? To demonstrate this, I've sent one such tuning in the regular
Scala format. It was originally meant to have the 1/1 on D which makes a
chain of fifths from two flats to three sharps (the most common way of
tuning the chain). But it's also good to place the 1/1 on E. If you do it
this way, all black keys are sharps and there are no flats at all then. This
makes the chord of C-E-G-A# to be maximally in tune.

! parizek_7lqmtd2.scl
!June 2004 - Petr Parizek
!Use SET MIDDLE 62
!To tune the scale by ear, please choose the intervals in this order:
!D-F#, D-Bb, Bb-F, Bb-G#, G#-C#, C#-A, A-E, E-C, F#-Eb, Eb-G, G-B.
!
7-limit Quasi-meantone no. 2 (1/1 is D)
12
!
15/14
28/25
6/5
5/4
75/56
7/5
112/75
8/5
375/224
224/125
28/15
2/1

Any suggestions?
Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@w...> wrote:

To demonstrate this, I've sent one such tuning in the regular
> Scala format. It was originally meant to have the 1/1 on D which makes a
> chain of fifths from two flats to three sharps (the most common way of
> tuning the chain). But it's also good to place the 1/1 on E. If you
do it
> this way, all black keys are sharps and there are no flats at all
then. This
> makes the chord of C-E-G-A# to be maximally in tune.

It is indeed a something like a detempered meantone. If you look at it
the commas you might try tempering it with you find 225/224,
3136/3125, and even 703125/702464, ay two of which together leads to
meantone. One way to get such things would be via Fokker blocks where
the commas were chosen to be small ones, such as 225/224 and
3136/3125, so that the resulting block did not stray too far from
meantone.

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > schismina, skhismina
> >
> > A generally imperceptiple interval, smaller than a schisma.
>
>
>
> I'm envisioning the scena for naming a schismina:
> Not wanting to be meana, we're going for sublima
> We wouldn't want to misma, by only doing schisma
> That's now so very common,
> One might think it's just a comma, man…

This is great. I think we might have to get you to translate some of
Inanates poems for future episodes of the Sagittal mythology.

Legend has it that philosophers argued over how many angels could
dance on the head of a pin. Hermes and I found ourselves arguing
over how many schisminas could vanish on the head of an arrow.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

/tuning/topicId_53869.html#54775

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
> wrote:
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > schismina, skhismina
> > >
> > > A generally imperceptiple interval, smaller than a schisma.
> >
> >
> >
> > I'm envisioning the scena for naming a schismina:
> > Not wanting to be meana, we're going for sublima
> > We wouldn't want to misma, by only doing schisma
> > That's now so very common,
> > One might think it's just a comma, man…
>
> This is great. I think we might have to get you to translate some
of
> Inanates poems for future episodes of the Sagittal mythology.
>

***Thanks, Dave! I'm impressed that you recognized that poem as one
of Inanates gratefully lost works...

JP