Some {2,3,17} 12 note Fokker blocks

With an eye to increasing this group's content in the JI department
I'm posting this here, instead of on tuning-math (where yes, a lot of
stuff about JI gets posted.)

I did a search for the Fokker blocks with commas 153/152, 289/288,
513/512 and found four of them; however the 153/152 and 513/512 were
only involved as a pair, canceling out the 19-limit stuff. The result
were four 17-limit blocks with 289/288 and 4131/4096 as commas. Having
done that, it seemed reasonable to check for 289/228 and 2187/2176 as
well, which led to five more blocks.

While these are all honest 17-limit JI scales, they also can very
logically serve as well-temperaments. This is particularly true of
the five temperaments using 289/288 and 2187/2176 to define the block,
which are good enough well temperaments that Scala calls them that. I
doubt playing the third Brandenburg in one of these tunings and
claiming the result to be just intonation would impress anyone,
however. It seems it is not just what a tuning is which makes it JI,
but what you do with it, because it would certainly *also* be possible
to treat one of these as a pure JI scale with just harmonies.

! sc3_17_1.scl
{153/152, 289/288, 513/512} Fokker block #1
12
!
18/17
9/8
153/128
64/51
4/3
24/17
3/2
27/17
27/16
16/9
32/17
2

! sc3_17_2.scl
{153/152, 289/288, 513/512} Fokker block #2
12
!
17/16
9/8
153/128
64/51
4/3
17/12
3/2
51/32
27/16
16/9
17/9
2

! sc3_17_3.scl
{153/152, 289/288, 513/512} Fokker block #3
12
!
18/17
9/8
153/128
64/51
4/3
24/17
3/2
51/32
27/16
16/9
32/17
2

! sc3_17_4.scl
{153/152, 289/288, 513/512} Fokker block #4
12
!
17/16
9/8
153/128
64/51
4/3
17/12
3/2
51/32
27/16
16/9
32/17
2

! sk3_17_1.scl
{289/288, 2187/2176} block #1
12
!
17/16
9/8
81/68
34/27
4/3
17/12
3/2
51/32
27/16
16/9
17/9
2

! sk3_17_2.scl
{289/288, 2187/2176} block #2
12
!
18/17
9/8
81/68
34/27
4/3
17/12
3/2
27/17
27/16
16/9
17/9
2

! sc3_17_3.scl
{153/152, 289/288, 513/512} Fokker block #3
12
!
18/17
9/8
153/128
64/51
4/3
24/17
3/2
51/32
27/16
16/9
32/17
2

! sk3_17_4.scl
{289/288, 2187/2176} block #4
12
!
17/16
9/8
81/68
34/27
4/3
17/12
3/2
27/17
27/16
16/9
17/9
2

! sk3_17_5.scl
{289/288, 2187/2176} block #5
12
!
18/17
9/8
81/68
34/27
4/3
24/17
3/2
27/17
27/16
16/9
17/9
2

Great post, Gene. I'll check these scales out
straight away.

What's the occasion for ressurecting ideaofgod?!

-Carl

hi Gene,

--- In tuning@yahoogroups.com, "ideaofgod" <gwsmith@s...> wrote:

> With an eye to increasing this group's content in the
> JI department I'm posting this here, instead of on
> tuning-math (where yes, a lot of stuff about JI gets posted.)
>
> I did a search for the Fokker blocks with commas 153/152,
> 289/288, 513/512 and found four of them; however the 153/152
> and 513/512 were only involved as a pair, canceling out the
> 19-limit stuff. The result were four 17-limit blocks with
> 289/288 and 4131/4096 as commas. Having done that, it seemed
> reasonable to check for 289/228 and 2187/2176 as
> well, which led to five more blocks.
>
> While these are all honest 17-limit JI scales, they also
> can very logically serve as well-temperaments. This is
> particularly true of the five temperaments using 289/288
> and 2187/2176 to define the block, which are good enough
> well temperaments that Scala calls them that. I doubt
> playing the third Brandenburg in one of these tunings
> and claiming the result to be just intonation would impress
> anyone, however. It seems it is not just what a tuning is
> which makes it JI, but what you do with it, because it
> would certainly *also* be possible to treat one of these
> as a pure JI scale with just harmonies.
>
> <snip Scala files>

have you ever read this? ...

http://tonalsoft.com/monzo/ganassi/ganassi.htm

Ganassi developed what i think is a very clever rational
12-tone well-temperament, basing his tuning on the
arithmetic string-length proportions 20:19:18:17:16:15
to divide the tetrachord.

thus, it's a 19-limit system which features ratios
of 17 and 19 quite heavily.

Gene, if you would do some mathematical analysis of
this tuning, i'll add it to my webpage. (but please post
it to tuning-math instead of here.)

time to make some lattices of this one, too ...

PS -- shortly after i put this webpage up and he saw it,
Rick McGowan wrote a nice piece using Ganassi's tuning,
which had a run off-Broadway in New York.

PPS -- i don't think i've put this on the webpage yet,
but Barbour cautions that Ganassi's arithmetic was
somewhat careless, so that in practice his tuning is
not actually as precise as what i've presented on my
webpage. what i have there is his theoretical ideal.
or at least that's how i remember it ...

-monz

--- In tuning@yahoogroups.com, "ideaofgod" <gwsmith@s...> wrote:

> While these are all honest 17-limit JI scales, they also can very
> logically serve as well-temperaments. This is particularly true of
> the five temperaments using 289/288 and 2187/2176 to define the block,
> which are good enough well temperaments that Scala calls them that. I
> doubt playing the third Brandenburg in one of these tunings and
> claiming the result to be just intonation would impress anyone,
> however.

Anyone who wants to give it a shot can try the following midi file of
the first Brandenberg, first movement, tuned to sk3_17_2.scl.

/tuning/files/17limit/brand1_1.mid

http://tinyurl.com/6lgfm

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Great post, Gene. I'll check these scales out
> straight away.
>
> What's the occasion for ressurecting ideaofgod?!

Yahoo is screwed up, as usual.

>I did a search for the Fokker blocks with commas 153/152, 289/288,
>513/512 and found four of them;

...restricted to 12 notes, right?

>! sk3_17_2.scl
>{289/288, 2187/2176} block #2
//
>! sc3_17_3.scl
>{153/152, 289/288, 513/512} Fokker block #3

Any significance to the fact that some of the scales in
this second group are marked "block" rather than
"Fokker block"?

-Carl

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

> >I did a search for the Fokker blocks with commas 153/152, 289/288,
> >513/512 and found four of them;
>
> ...restricted to 12 notes, right?

Right.

> Any significance to the fact that some of the scales in
> this second group are marked "block" rather than
> "Fokker block"?

Obviously, you are wrong in thinking I should drink less coffee.

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

> have you ever read this? ...
>
> http://tonalsoft.com/monzo/ganassi/ganassi.htm
>
>
> Ganassi developed what i think is a very clever rational
> 12-tone well-temperament, basing his tuning on the
> arithmetic string-length proportions 20:19:18:17:16:15
> to divide the tetrachord.
>
> thus, it's a 19-limit system which features ratios
> of 17 and 19 quite heavily.
>
> Gene, if you would do some mathematical analysis of
> this tuning, i'll add it to my webpage. (but please post
> it to tuning-math instead of here.)
>
> time to make some lattices of this one, too ...
>
>
>
> PS -- shortly after i put this webpage up and he saw it,
> Rick McGowan wrote a nice piece using Ganassi's tuning,
> which had a run off-Broadway in New York.
>
> PPS -- i don't think i've put this on the webpage yet,
> but Barbour cautions that Ganassi's arithmetic was
> somewhat careless, so that in practice his tuning is
> not actually as precise as what i've presented on my
> webpage. what i have there is his theoretical ideal.
> or at least that's how i remember it ...

Hi, Monz!

What a beautiful, beautiful tuning!

It is the most equal 12-tone superparticular scale and it is
relatively low-numbered as a fragment of harmonic series too. And
most importantly it sounds good!

Kalle

hi Kalle,

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > http://tonalsoft.com/monzo/ganassi/ganassi.htm
>
> Hi, Monz!
>
> What a beautiful, beautiful tuning!
>
> It is the most equal 12-tone superparticular scale and
> it is relatively low-numbered as a fragment of harmonic
> series too. And most importantly it sounds good!
>
> Kalle

and don't forget ... it's another excellent example
of "rational tuning" which is not JI.

i haven't personally used Ganassi's tuning in any
of my pieces yet, so i only have Rick's piece and
the chord examples on my webpage to go on.

but i recognized an aesthetic appeal to it right
away, when i first wrote about it in my book c.1995.

Rick's piece was incidental music for Moliere's play
_The Miser_:

http://tonalsoft.com/monzo/marchetto/marchetto.htm

Marchetto of Padua may have been the first theorist
to recognize 17 as a viable prime-factor, by approximating
the two Pythagorean semitones as 16:17:18, in his
_Lucidarium_ of 1318.

Ganassi may have been the first theorist to actually
advocate ratios of 17, with this scale, in his
_Regula Rubertina_ of 1543.

17 became firmly recognized as a useful prime-factor in
the late 1500s, with Vincenzo Galilei's recommendation
of 18:17 as the fretting for pseduo-12-ET semitones on
the lute. AFAIK, this became fairly standard ... lute
experts are urged to correct me if i'm wrong.

19 had been a part of rational tunings as long ago
as ancient Greece (Eratosthenes, 200s BC), probably
because the 3==19 xenharmonic bridges had been
discovered early on, actually possibly long before
Greece (Sumer?).

here's an old post where i wrote more about some
prominent 17 and 19-limit tunings:

/tuning/topicId_7159.html#7159

and my page on Marchetto:

http://tonalsoft.com/monzo/marchetto/marchetto.htm

(where i speculatively find ratios with primes that go
far beyond anything that Marchetto actually quantified
himself.)

-monz

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

> > > http://tonalsoft.com/monzo/ganassi/ganassi.htm
>
> Rick's piece was incidental music for Moliere's play
> _The Miser_:
>

oops! ... my bad. copied the wrong URL.
here's Rick McGowan's music to _The Miser_:

http://rm-and-jo.laughingsquid.org/Music/

-monz

From: "ideaofgod" <gwsmith@s>
> While these are all honest 17-limit JI scales, they also can very
> logically serve as well-temperaments. This is particularly true of
> the five temperaments using 289/288 and 2187/2176 to define the block,
> which are good enough well temperaments that Scala calls them that. I
> doubt playing the third Brandenburg in one of these tunings and
> claiming the result to be just intonation would impress anyone,
> however. It seems it is not just what a tuning is which makes it JI,
> but what you do with it, because it would certainly *also* be possible
> to treat one of these as a pure JI scale with just harmonies.

It's really interesting how many of my ideas are similar to your ones (and
also some views on various topics). Don't know if you have the newest
edition of Manuel's scale archive, but there are also three scales that I
made as 19-limit well-temperaments so maybe you'll be interested. You'll
find them in the list of filenames begining with "parizek".
It may also be interesting for me to examine how much of our "similar views"
appears in you music. Is there any way to hear it?
Thanks.
Petr

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> Hi, Monz!
>
> What a beautiful, beautiful tuning!
>
> It is the most equal 12-tone superparticular scale and it is
> relatively low-numbered as a fragment of harmonic series too. And
> most importantly it sounds good!

The super_12 scale in the Scala archive is actually inverse of the
Ganassi tuning. Super_12 is the one that is low in the harmonic
series. It can be expressed as

45:48:51:54:57:60:64:68:72:76:80:85:90

And thanks Monz for the links!

Kalle

From: "monz" <monz@a>
> have you ever read this? ...
>
> http://tonalsoft.com/monzo/ganassi/ganassi.htm

Ahhh, just ... How to say that? You're getting me in doubt about the purpose
of well-temperaments. I always thought that their main goal was to allow
playing in all the keys with only 12 tones per octave (that means all the 12
fifths to be more or less usable). But I'm not so sure this one also works
like that. It was just the first time I saw the scale when I found an
evidently unusable fifth of G-D (this needs no detailed analysis) which is a
syntonic comma smaller than the pure 3/2. If a scale like this is considered
a well-temperament, then another such "well-temperament" may be a scale that
has 10 pure fifths, one fifth a schisma smaller, and one fifth a syntonic
comma smaller. Indeed, Scala calls such a scale a "well-temperament".
Someone may argue that the syntonic comma can be used as a unison vector. Of
course, but that means to distribute it in more than one fifth. I would
consider a fifth mistuned even if it was only 1/2 syntonic comma smaller. I
don't know the exact definition of "well-temperament". But if such a scale
is allowed to be called like this, then there must be something inaccurate
or wrong about it. Perhaps others know better?
Petr

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > Hi, Monz!
> >
> > What a beautiful, beautiful tuning!
> >
> > It is the most equal 12-tone superparticular scale and it is
> > relatively low-numbered as a fragment of harmonic series too. And
> > most importantly it sounds good!
>
> The super_12 scale in the Scala archive is actually inverse of the
> Ganassi tuning.

There are also versions of it in there attributed to Malcolm and to
Aaron Johnson, so it has a history of its own.

Scala tells us it is epimorphic for {2,3,5,17,19}, and gives the
mapping <12 19 28 * * * 49 51| for it. This is what you'd get if you
obtained a mapping starting from slightly flattened octaves--for
instance, the octaves 5-limit TOP tuning gives. Scala does not
consider it close enough to 12-equal to call it a well-temperament,
and in fact one of the fifths is a comma flat, so it has an
out-and-out wolf as well as exotic fifths sharp by 136/135 and
256/255, and flat by 153/152 and 171/170. It's also got a 51/40 major
third, which is a 51/50 above 5/4 and perhaps counts more as a 14/11,
and other exotic beasts which may or may not be wolves but certainly
have teeth.

hi Petr,

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@w...> wrote:
> From: "monz" <monz@a>
> > have you ever read this? ...
> >
> > http://tonalsoft.com/monzo/ganassi/ganassi.htm
>
> Ahhh, just ... How to say that? You're getting me in doubt
> about the purpose of well-temperaments. I always thought that
> their main goal was to allow playing in all the keys with
> only 12 tones per octave (that means all the 12 fifths
> to be more or less usable). But I'm not so sure this one
> also works like that. It was just the first time I saw the
> scale when I found an evidently unusable fifth of G-D (this
> needs no detailed analysis) which is a syntonic comma smaller
> than the pure 3/2. If a scale like this is considered
> a well-temperament, then another such "well-temperament"
> may be a scale that has 10 pure fifths, one fifth a schisma
> smaller, and one fifth a syntonic comma smaller. Indeed,
> Scala calls such a scale a "well-temperament".

hmm ... that might be a distincition worth making in the
name. perhaps "well-temperament" *should* be restricted to
a tuning in which all "5ths" are "usable".

i've always thought of well-temperaments in pretty much
the same way as Carl: "perturbations from 12-ET".

> Someone may argue that the syntonic comma can be used
> as a unison vector. Of course, but that means to distribute
> it in more than one fifth.

only if it's a *vanishing* unison-vector!

the unison-vector need not vanish by being distributed
... it can also be a sudden disruption in the JI lattice,
which is exactly what happens when the syntonic comma
is not distributed.

-monz

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

> Scala tells us it is epimorphic for {2,3,5,17,19}, and gives
> the mapping <12 19 28 * * * 49 51| for it. This is what you'd
> get if you obtained a mapping starting from slightly flattened
> octaves--for instance, the octaves 5-limit TOP tuning gives.
> Scala does not consider it close enough to 12-equal to call
> it a well-temperament, and in fact one of the fifths is a
> comma flat, so it has an out-and-out wolf as well as exotic
> fifths sharp by 136/135 and 256/255, and flat by 153/152
> and 171/170. It's also got a 51/40 major third, which is a
> 51/50 above 5/4 and perhaps counts more as a 14/11, and
> other exotic beasts which may or may not be wolves but
> certainly have teeth.

hmmm ... evidence seems to be mounting that i erred in
calling Ganassi's tuning a "well-temperament".

-monz

Gene wrote 26-7:
> Scala does not
>consider it close enough to 12-equal to call it a well-temperament,
>and in fact one of the fifths is a comma flat, so it has an
>out-and-out wolf as well as exotic fifths sharp by 136/135 and
>256/255, and flat by 153/152 and 171/170.

The criterion is that fifths should not be sharper than 3/2
and major thirds not flatter than 5/4 because this leads to
harmonic waste (more beating than necessary) so it's not a
well-temperament.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul" <manuel.op.de.
coul@e...> wrote:

>
> Gene wrote 26-7:
> > Scala does not
> >consider it close enough to 12-equal to call it a well-temperament,
> >and in fact one of the fifths is a comma flat, so it has an
> >out-and-out wolf as well as exotic fifths sharp by 136/135 and
> >256/255, and flat by 153/152 and 171/170.
>
> The criterion is that fifths should not be sharper than 3/2
> and major thirds not flatter than 5/4 because this leads to
> harmonic waste (more beating than necessary) so it's not a
> well-temperament.
>
> Manuel

ok, then i have to revise my definition of well-temperament
to make it more specific.

anyone care to help?

and what *would* be a good label for Ganassi's tuning,
if it's not a well-temperament? does anyone see a
category that it fits into?

-monz

>> >Scala does not consider it close enough to 12-equal to
>> >call it a well-temperament, and in fact one of the
>> >fifths is a comma flat, so it has an out-and-out wolf
>> >as well as exotic fifths sharp by 136/135 and 256/255,
>> >and flat by 153/152 and 171/170.
>>
>> The criterion is that fifths should not be sharper than 3/2
>> and major thirds not flatter than 5/4 because this leads to
>> harmonic waste (more beating than necessary) so it's not a
>> well-temperament.
>
>ok, then i have to revise my definition of well-temperament
>to make it more specific.
>
>anyone care to help?
>
>and what *would* be a good label for Ganassi's tuning,
>if it's not a well-temperament? does anyone see a
>category that it fits into?

IIRC Jorgenson coined the term "well temperament". What does
he say about it?

-Carl

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> The criterion is that fifths should not be sharper than 3/2
> and major thirds not flatter than 5/4 because this leads to
> harmonic waste (more beating than necessary) so it's not a
> well-temperament.

It might be worthwhile to also set a criterion for circulation; for
instance that no third or fifth was off by more than 16 cents. This
would make 12-equal circulating, but just barely.

on 8/10/04 12:29 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> It might be worthwhile to also set a criterion for circulation; for
> instance that no third or fifth was off by more than 16 cents. This
> would make 12-equal circulating, but just barely.

You mean like a point is a trivial case of a circle?

In that case I've tempered out 0/1 in all my tunings. ;)

-Kurt

From: "Gene Ward Smith" <gwsmith@s>

> It might be worthwhile to also set a criterion for circulation; for
> instance that no third or fifth was off by more than 16 cents. This
> would make 12-equal circulating, but just barely.

A few days ago, someone successfully confused myself so I wasn't sure how it
is with well-temperaments. Now I'm beginning to realize that we should make
a difference between "well-temperaments" and "circular temperaments". As far
as I can understand it now, the tunings you spoke about are circular
temperaments, while those Manuel spoke about are well-temperaments. Or if
it's not like this, why? And how is it then?
Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@t...> wrote:
> From: "Gene Ward Smith" <gwsmith@s>

> A few days ago, someone successfully confused myself so I wasn't
sure how it
> is with well-temperaments. Now I'm beginning to realize that we
should make
> a difference between "well-temperaments" and "circular
temperaments". As far
> as I can understand it now, the tunings you spoke about are circular
> temperaments, while those Manuel spoke about are well-temperaments.
Or if
> it's not like this, why? And how is it then?

As far as I can tell how it is is a mess you could take to
tuning-jargon and argue over for months. There are, as I pointed out,
temperaments ordinaire, with sharp fifths. There are my temperaments,
which have some thirds so sharp they become 14/11 or even 9/7
intervals, yet which still circulate in some manner because all of the
fifths are usable as fifths. There are, therefore, more kinds of
circulating temperaments/scales than there is jargon to classify them.

From: "Gene Ward Smith" <gwsmith@s>

> As far as I can tell how it is is a mess you could take to
> tuning-jargon and argue over for months.

I don't find mess here. So far I haven't found a reason why my view is
wrong.

> There are, as I pointed out,
> temperaments ordinaire, with sharp fifths. There are my temperaments,
> which have some thirds so sharp they become 14/11 or even 9/7
> intervals, yet which still circulate in some manner because all of the
> fifths are usable as fifths.

Okay, these are all circular temperaments as they all meet the condition of
having all the fifths usable. And all well-temperaments are also circular
temperaments. But not all circular temperaments are well-temperaments.
Anyway, I'm gonna find other ways to solve things like this than subscribing
to Tuning-Jargon.
Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@t...> wrote:
> From: "Gene Ward Smith" <gwsmith@s>
>
> > As far as I can tell how it is is a mess you could take to
> > tuning-jargon and argue over for months.
>
> I don't find mess here. So far I haven't found a reason why my view is
> wrong.

It's not wrong, but why is it right? I don't think the meaning has
been pinned down so precisely by anyone but Manuel.

Hello, all,

on 8/11/04 3:53 AM, Petr Parízek <p.parizek@tiscali.cz> wrote:

> From: "Gene Ward Smith" <gwsmith@s>
>
>> As far as I can tell how it is is a mess you could take to
>> tuning-jargon and argue over for months.
>
> I don't find mess here. So far I haven't found a reason why my view is
> wrong.
>
>> There are, as I pointed out,
>> temperaments ordinaire, with sharp fifths. There are my temperaments,
>> which have some thirds so sharp they become 14/11 or even 9/7
>> intervals, yet which still circulate in some manner because all of the
>> fifths are usable as fifths.
>
> Okay, these are all circular temperaments as they all meet the condition of
> having all the fifths usable. And all well-temperaments are also circular
> temperaments. But not all circular temperaments are well-temperaments.
> Anyway, I'm gonna find other ways to solve things like this than subscribing
> to Tuning-Jargon.
> Petr

So please correct my impression here as needed...

It was always my impression that a circulating temperament had a circular
organization, e.g. around the circle of fifths usually, but any other
generator for a linear temperament would do. By circular organization I
mean it changes gradually from "good" in the center key gradually to "bad"
in a far key and then gradually back to "good", in a circle. So to me, if
it stays the same through a bunch of keys, that's not quite "circulating".
Likewise if it jumps around then that's not circulating. Thus duodene would
not be circulating (never mind whether it is a temperament).

But I think Gene and others are calling things circulating that "jump
around".

I personally would feel rather robbed of a word for what I have (perhaps
nievely) been calling circulating if it is to include jumping around. Damn
I didn't think I'd be getting into jargon arguments. But no I won't argue,
this is mainly to ask the question and state my case just once.

-Kurt

hi Kurt,

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

> It was always my impression that a circulating temperament
> had a circular organization, e.g. around the circle of fifths
> usually, but any other generator for a linear temperament
> would do. By circular organization I mean it changes
> gradually from "good" in the center key gradually to "bad"
> in a far key and then gradually back to "good", in a circle.
> So to me, if it stays the same through a bunch of keys,
> that's not quite "circulating". Likewise if it jumps around
> then that's not circulating. Thus duodene would not be
> circulating (never mind whether it is a temperament).
>
> But I think Gene and others are calling things circulating
> that "jump around".
>
> I personally would feel rather robbed of a word for what
> I have (perhaps nievely) been calling circulating if it is
> to include jumping around. Damn I didn't think I'd be
> getting into jargon arguments. But no I won't argue,
> this is mainly to ask the question and state my case
> just once.

actually that is a *very* good point which you've brought up.

in Bach's time, it was *definitely* the desiderata that
a temperament should be irregular so that the concordance
of the tuning's harmonies was greatest in the most "central"
keys (on the circle-of-5ths) and became progressively worse
as the keys got more and more remote.

but the word "circulating" in itself really only signifies
that the vanishing of the promo (or "comma" if you insist)
is distributed in an irregular fashion around the whole circle.
at least that's what i always thought.

anyone here know more about the history? damn, i wish
Paul was around ... hopefully he'll answer.

if there hasn't already been a word which meant the
orderly distribution of error as described by Kurt, then
we should have one. i welcome replies on tuning-jargon.
;-)

-monz

From: "Kurt Bigler" <kkb@b>

> So please correct my impression here as needed...

I'm beginning to feel pretty unsure once again but I'll do my best.

> It was always my impression that a circulating temperament had a circular
> organization, e.g. around the circle of fifths usually, but any other
> generator for a linear temperament would do. By circular organization I
> mean it changes gradually from "good" in the center key gradually to "bad"
> in a far key and then gradually back to "good", in a circle.

I've never wondered if there's a difference between "circulating" and
"circular" temperaments. Maybe there is. And if so, good for you.
Just as far as I can understand it, tunings like these DO belong to
so-called "circular temperaments" but this is not their primary goal.
Circular temperaments are those in which all the fifths are usable as fifths
(i.e. there are no wolves) to achieve a circle of fifths (i.e. not an open
chain). The fifths may be slightly sharper or flatter than the just 3/2 and
the tolerance of their detuning varies significantly from one musician to
another (while I consider a detuning of more than 8 cents to be too strong,
others accept a detuning as large as 12 or 15 cents). This means that the
regular 12-tet is also a circular temperament.
Then, there are so-called "well-temperaments" which also belong to circular
temperaments. These tunings meet the following conditions:
A) All fifths are usable as fifths.
B) A fifth is smaller than or equal to the just 3/2, not larger.
C) A major third is larger than or equal to the just 5/4, not smaller.
If someone complains I follow Manuel's definition, then I'm asking what I
should follow.
Petr

From: "Petr Par�zek" <p.parizek@t>

> A) All fifths are usable as fifths.
> B) A fifth is smaller than or equal to the just 3/2, not larger.
> C) A major third is larger than or equal to the just 5/4, not smaller.

Yet I should have added:
D) The triads in the C-major key are the most similar to JI.

on 8/12/04 8:27 AM, Petr Parízek <p.parizek@tiscali.cz> wrote:

> From: "Kurt Bigler" <kkb@b>
>
>> So please correct my impression here as needed...
>
> I'm beginning to feel pretty unsure once again but I'll do my best.
>
>> It was always my impression that a circulating temperament had a circular
>> organization, e.g. around the circle of fifths usually, but any other
>> generator for a linear temperament would do. By circular organization I
>> mean it changes gradually from "good" in the center key gradually to "bad"
>> in a far key and then gradually back to "good", in a circle.
>
> I've never wondered if there's a difference between "circulating" and
> "circular" temperaments. Maybe there is. And if so, good for you.

No I didn't mean that at all. I was ignoring the fact that you said
"circular" and figured you just meant "circulating".

> Just as far as I can understand it, tunings like these DO belong to
> so-called "circular temperaments" but this is not their primary goal.
> Circular temperaments are those in which all the fifths are usable as fifths
> (i.e. there are no wolves) to achieve a circle of fifths (i.e. not an open
> chain). The fifths may be slightly sharper or flatter than the just 3/2 and
> the tolerance of their detuning varies significantly from one musician to
> another (while I consider a detuning of more than 8 cents to be too strong,
> others accept a detuning as large as 12 or 15 cents). This means that the
> regular 12-tet is also a circular temperament.

Ah, I see. So you mean as opposed to a meantone that has a wolf and
therefore creates some "unusable" keys which breaks the circle. I never
understood it that way but the whole time I may have been making an
assumption. It will be interesting to sort this out.

> Then, there are so-called "well-temperaments" which also belong to circular
> temperaments. These tunings meet the following conditions:
> A) All fifths are usable as fifths.
> B) A fifth is smaller than or equal to the just 3/2, not larger.
> C) A major third is larger than or equal to the just 5/4, not smaller.
> If someone complains I follow Manuel's definition, then I'm asking what I
> should follow.
> Petr

It may be that Jorgenson said something more specific than your list of 3.
I think his list is longer, IIRC, but I don't own the book. Carl asked
about this a couple days ago. Can someone who has Jorgenson look this up?

-Kurt