Yahoo Tuning Groups Ultimate Backup tuning-math names and definitions: schismic

family name: schismic
period: octave
generator: fourth or fifth

5-limit

name: schismic
comma: 32805/32768
mapping: [<1 2 1|, <0 -1 8|]
poptimal generator: 120/289
MOS: 12, 17, 29, 41, 53, 65, 118, 171

7-limit

name: schismic, 118&171
wedgie: <<1 -8 39 -15 59 113||
mapping: [<1 2 -1 19|, <0 -1 8 -39|]
7&9 limit copoptimal generator: 732/1763
TM basis: {4375/4374, 32805/32768}
MOS: 12, 17, 29, 41, 53, 65, 118, 171

name: garibaldi, 41&53
wedgie: <<1 -8 -14 -15 -25 -10||
mapping: [<1 2 -1 -3|, <0 -1 8 14|]
7&9 limit copoptimal generator: 39/94
TM basis: {3125/3087}
MOS: 12, 17, 29, 41, 53

name: schism, 12&17
wedgie: <<1 -8 -2 -15 -6 18||
mapping: [<1 2 -1 2|, <0 -1 8 2|]
7 limit poptimal generator: 27/65
9 limit poptimal generator: 22/53
TM basis: {64/63, 360/343}
MOS: 12, 17, 29, 41, 53

name: grackle, 65&77
wedgie: <<1 -8 -26 -15 -44 -38||
mapping: [<1 2 -1 -8|, <0 -1 8 26|]
7 limit poptimal generator: 170/409
9 limit poptimal generator: 133/320
TM basis: {126/125, 32805/32768}
MOS: 12, 17, 29, 41, 53, 65, 77, 89

11 limit

name: garibaldi, 41&53
wedgie: <<1 -8 -14 23 -15 -25 33 -10 81 113||
mapping: [<1 2 -1 -3 13|, <0 -1 8 14 -23|]
poptimal generator: 95/229
TM basis: {225/224, 385/384, 2200/2187}
MOS: 12, 17, 29, 41, 53, 94

name: garybald, 29&41
wedgie: <<1 -8 -14 -18 -15 -25 -32 -10 -14 -2||
mapping: [<1 2 -1 -3 -4|, <0 -1 8 14 18|]
poptimal generator: 63/152
TM basis: {100/99, 225/224, 245/242}
MOS: 12, 17, 29, 41, 70

🔗Gene Ward Smith <gwsmith@svpal.org>

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

Gene Ward Smith wrote:

> name: schismic, 118&171
> wedgie: <<1 -8 39 -15 59 113||
> mapping: [<1 2 -1 19|, <0 -1 8 -39|]
> 7&9 limit copoptimal generator: 732/1763
> TM basis: {4375/4374, 32805/32768}
> MOS: 12, 17, 29, 41, 53, 65, 118, 171

What, *this* is the default septimal schismic? But it's the absurdly complex one! If it has to have a name, make it "microschismic".

> name: garibaldi, 41&53
> wedgie: <<1 -8 -14 -15 -25 -10||
> mapping: [<1 2 -1 -3|, <0 -1 8 14|]
> 7&9 limit copoptimal generator: 39/94
> TM basis: {225/224, 3125/3087}
> MOS: 12, 17, 29, 41, 53

This should be "schismic". It's consistent with all those ETs except 17 (which isn't 7-limit consistent in itself).

> name: schism, 12&17
> wedgie: <<1 -8 -2 -15 -6 18||
> mapping: [<1 2 -1 2|, <0 -1 8 2|]
> 7 limit poptimal generator: 27/65
> 9 limit poptimal generator: 22/53
> TM basis: {64/63, 360/343}
> MOS: 12, 17, 29, 41, 53

That name's confusingly similar to "schismic". It also looks like a white elephant. And certainly don't call it "12&17" because this is ambiguous with the 7/9-limit optimal mapping of 17-equal, which happens to give an all-round better temperament in this case.

> 11 limit
> > name: garibaldi, 41&53
> wedgie: <<1 -8 -14 23 -15 -25 33 -10 81 113||
> mapping: [<1 2 -1 -3 13|, <0 -1 8 14 -23|]
> poptimal generator: 95/229
> TM basis: {225/224, 385/384, 2200/2187}
> MOS: 12, 17, 29, 41, 53, 94
> > name: garybald, 29&41
> wedgie: <<1 -8 -14 -18 -15 -25 -32 -10 -14 -2||
> mapping: [<1 2 -1 -3 -4|, <0 -1 8 14 18|]
> poptimal generator: 63/152
> TM basis: {100/99, 225/224, 245/242}
> MOS: 12, 17, 29, 41, 70

More confusingly similar names.

Graham

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > name: schismic, 118&171
> > wedgie: <<1 -8 39 -15 59 113||
> > mapping: [<1 2 -1 19|, <0 -1 8 -39|]
> > 7&9 limit copoptimal generator: 732/1763
> > TM basis: {4375/4374, 32805/32768}
> > MOS: 12, 17, 29, 41, 53, 65, 118, 171
>
> What, *this* is the default septimal schismic?

Check the TOP tuning before blowing your top. This temperament is a
straightforward extension of 5-limit schismic; the TOP tunings are
very close. Garibaldi, which Paul wants to name something other than
schismic anyway, has a different tuning--instead of slightly flat,
like 171 or 118, the fifth of 94 is slightly sharp.

But it's the absurdly
> complex one! If it has to have a name, make it "microschismic".

I propose to keep a little consistency and order in the naming of
these things, by making the tunings in the various limits correspond.
Dave likes that because he likes things being systematic, and Paul
does not seem eager to give the same name to different limits at all,
so presumably at least having the tunings correspond would be
important to him.

> > name: garibaldi, 41&53
> > wedgie: <<1 -8 -14 -15 -25 -10||
> > mapping: [<1 2 -1 -3|, <0 -1 8 14|]
> > 7&9 limit copoptimal generator: 39/94
> > TM basis: {225/224, 3125/3087}
> > MOS: 12, 17, 29, 41, 53
>
> This should be "schismic". It's consistent with all those ETs
except 17
> (which isn't 7-limit consistent in itself).

I did name it that at first, and then changed it because the tunings
were different; morever Paul wants to name it garibaldi anyway. I
think it makes sense not to claim this is really schismic, given that
the fifth is actually sharp and schismic wants it to be very slightly
flat, by 2/9 of a cent.

The name is supposed to tell you it is like schismic, but clearly is
not--which it does. Confusing schism with schismic seems prettyn
confused, but recalling they are related should be easy.

It also looks like a
> white elephant.

Well, it's been sitting on my 7-limit list, down there at Number 89,
and people keep finding uses for temperaments far down on that list;
lemba is Number 82 and Herman loves it, along with superpelog (Number
107.) Anyway I don't think it is any more of a white elephant than
grackle.

And certainly don't call it "12&17" because this is
> ambiguous with the 7/9-limit optimal mapping of 17-equal, which happens
> to give an all-round better temperament in this case.

So how is the optimal 17-equal temperament defined? I called it 12&17
because the "standard" 17 using closest approximations to primes gives
us this.

> > name: garibaldi, 41&53
> > wedgie: <<1 -8 -14 23 -15 -25 33 -10 81 113||
> > mapping: [<1 2 -1 -3 13|, <0 -1 8 14 -23|]
> > poptimal generator: 95/229
> > TM basis: {225/224, 385/384, 2200/2187}
> > MOS: 12, 17, 29, 41, 53, 94
> >
> > name: garybald, 29&41
> > wedgie: <<1 -8 -14 -18 -15 -25 -32 -10 -14 -2||
> > mapping: [<1 2 -1 -3 -4|, <0 -1 8 14 18|]
> > poptimal generator: 63/152
> > TM basis: {100/99, 225/224, 245/242}
> > MOS: 12, 17, 29, 41, 70
>
> More confusingly similar names.

For temperaments which happen to be the same in the 7-limit.

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

Gene Ward Smith wrote:

> Check the TOP tuning before blowing your top. This temperament is a
> straightforward extension of 5-limit schismic; the TOP tunings are
> very close. Garibaldi, which Paul wants to name something other than
> schismic anyway, has a different tuning--instead of slightly flat,
> like 171 or 118, the fifth of 94 is slightly sharp.

I can guess the TOP tuning, but why? I don't give a shit about TOP. I never tune to the theoretical optima. I can't even get the 7-limit tetrads on my keyboard.

> I propose to keep a little consistency and order in the naming of
> these things, by making the tunings in the various limits correspond.
> Dave likes that because he likes things being systematic, and Paul
> does not seem eager to give the same name to different limits at all,
> so presumably at least having the tunings correspond would be
> important to him.

I thought Dave called a stop to the jargon explosion. Paul may decide to give redundant names to everything, but let's not suddenly change the meaning of "schismic". It should be perfectly simple to construct a consistent naming scheme that calls "schismic" "schismic".

See here:

http://web.archive.org/web/19991010023317/http://www.cix.co.uk/~gbreed/schismic.htm

I've been calling this "schismic" for the past 5 years, on a site that has been indexed and archived. The spelling's in flux, but it's too late to change the meaning now.

> I did name it that at first, and then changed it because the tunings
> were different; morever Paul wants to name it garibaldi anyway. I
> think it makes sense not to claim this is really schismic, given that
> the fifth is actually sharp and schismic wants it to be very slightly
> flat, by 2/9 of a cent.

Schismic doesn't "want" anything. I always preferred sharp fifths anyway, even in the 5-limit. The Pythagorean commas get too small otherwise. Schismic is quite often described with Pythagorean tuning. It's rarely given enough notes for this microtempered mapping to make sense.

> The name is supposed to tell you it is like schismic, but clearly is
> not--which it does. Confusing schism with schismic seems prettyn
> confused, but recalling they are related should be easy.

The more closely things are related, the more chance there is of confusion if they have similar names. If somebody talked about "schism" temperament I'd assume they meant schismic. Especially as that has a variety of spellings already. I certainly wouldn't pick up on the missing "ic" denoting an obscure 7-limit mapping.

> Well, it's been sitting on my 7-limit list, down there at Number 89,
> and people keep finding uses for temperaments far down on that list;
> lemba is Number 82 and Herman loves it, along with superpelog (Number
> 107.) Anyway I don't think it is any more of a white elephant than
> grackle. Number 89??? And how many of these are you planning to name? I see you've already broken your resolution about reminding us what the names are supposed to refer to.

> So how is the optimal 17-equal temperament defined? I called it 12&17
> because the "standard" 17 using closest approximations to primes gives
> us this.

The one with the lowest 7- or 9-limit minimax (they agree in this case).

>>More confusingly similar names.
> > For temperaments which happen to be the same in the 7-limit.

Yes, so who's going to remember which one the trailing "i" is supposed to refer to?

Graham

🔗monz <monz@attglobal.net>

hi Gene (and Graham and everyone else too),

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

> family name: schismic
> period: octave
> generator: fourth or fifth
>
> 5-limit
>
> name: schismic
> comma: 32805/32768
> mapping: [<1 2 1|, <0 -1 8|]
> poptimal generator: 120/289
> MOS: 12, 17, 29, 41, 53, 65, 118, 171
>
> 7-limit
>
> <etc. -- snip>

thanks much for that. i was going to copy it into
my webpage right away, as i did with your meantone defitions,
but i see that there's already been a lot of discussion
about what you posted here.

what do you guys think? should i put Gene's data into
my "schismic" Encyclpaedia webpage, and correct as we
go along? or should i wait until there's been more of
a consensus here?

while we're at it, i'd like to do something about the
schismic/schismatic business. i have webpages under
both names, with different content. i'm appealing to
all of you to read both of them and give me advice on
how to combine them or separate them more clearly,
whichever is advisable.

-monz

🔗Herman Miller <hmiller@IO.COM>

Graham Breed wrote:

>>name: garibaldi, 41&53
>>wedgie: <<1 -8 -14 -15 -25 -10||
>>mapping: [<1 2 -1 -3|, <0 -1 8 14|]
>>7&9 limit copoptimal generator: 39/94
>>TM basis: {225/224, 3125/3087}
>>MOS: 12, 17, 29, 41, 53
> > > This should be "schismic". It's consistent with all those ETs except 17 > (which isn't 7-limit consistent in itself).

I agree; "schismic" is essentially a 5-limit name (based on tempering out the [-15 8 1> "schisma"), so the 7-limit extension (if any) shouldn't be more complex than it needs to be.

>>name: schism, 12&17
>>wedgie: <<1 -8 -2 -15 -6 18||
>>mapping: [<1 2 -1 2|, <0 -1 8 2|]
>>7 limit poptimal generator: 27/65
>>9 limit poptimal generator: 22/53
>>TM basis: {64/63, 360/343}
>>MOS: 12, 17, 29, 41, 53
> > > That name's confusingly similar to "schismic". It also looks like a > white elephant. And certainly don't call it "12&17" because this is > ambiguous with the 7/9-limit optimal mapping of 17-equal, which happens > to give an all-round better temperament in this case.

This looks like a strange hybrid of schismic and dominant; something you might use if you don't have enough notes for schismic, but you're willing to substitute a 16/9 for a 7/4. It would be nice to see some error values for comparison purposes.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > Check the TOP tuning before blowing your top. This temperament is a
> > straightforward extension of 5-limit schismic; the TOP tunings are
> > very close. Garibaldi, which Paul wants to name something other than
> > schismic anyway, has a different tuning--instead of slightly flat,
> > like 171 or 118, the fifth of 94 is slightly sharp.
>
> I can guess the TOP tuning, but why? I don't give a shit about TOP. I
> never tune to the theoretical optima. I can't even get the 7-limit
> tetrads on my keyboard.

Someone who is tuning schismic to get highly accurate 5-limit
intonation will be tuning using sligtly flatted fifths--in the range
1/8 to 1/9-schisma schismic. This, historically, is what it has been
used for. Someone wanting that degree of accuracy, and who has tuned
up a 53 note MOS, will find that 39 of these slightly flattened fifths
will give a better 7 than will 14 of the slightly sharpened fourths;
those will be a decent enough 6.8 to 7 cents sharp, and certainly
useable, but the other 7s will be between 1 and 1.5 cents flat, which
is much better. Optimizing for the 7s will then do little damage to
5-limit harmony, since it is a matter of moving from 1/9 to 1/10
schisma flat. The fifth of 94-et is 1/11 schisma *sharp*, and not at
all optimal for 5-limit harmony; 94 would never be considered for
5-limit schisma.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> > This should be "schismic". It's consistent with all those ETs
except 17
> > (which isn't 7-limit consistent in itself).
>
> I agree; "schismic" is essentially a 5-limit name (based on tempering
> out the [-15 8 1> "schisma"), so the 7-limit extension (if any)
> shouldn't be more complex than it needs to be.

This reasoning is backwards, and leads to bizarre conclusions. If
schismic is essentially a 5-limit temperament, then you'd better keep
close to the 5-limit tuning seems like the way to reason from your
premise. Moreover, it leads to the conclusion that dominant should get
the name "meantone" in the 7-limit, and that is a conclusion no one
seems to buy. We'd also end up renaming pajara to diaschismic, I suppose.

> >>name: schism, 12&17
> >>wedgie: <<1 -8 -2 -15 -6 18||
> >>mapping: [<1 2 -1 2|, <0 -1 8 2|]
> >>7 limit poptimal generator: 27/65
> >>9 limit poptimal generator: 22/53
> >>TM basis: {64/63, 360/343}
> >>MOS: 12, 17, 29, 41, 53

> This looks like a strange hybrid of schismic and dominant; something
you
> might use if you don't have enough notes for schismic, but you're
> willing to substitute a 16/9 for a 7/4. It would be nice to see some
> error values for comparison purposes.

It would indeed be what you'd get if you were in Schismic[12] and
wanted something to serve as a 7; people would be happy with it if for
no other reason than because they are used to noisy dominant seventh
chords resolving; this would resolve something in the manner of
meantone, where a brash V7 moves to a smooth I. With 17 notes of
course, it would be like Meantone[12], with both choices available for
certain chords.

🔗Herman Miller <hmiller@IO.COM>

Gene Ward Smith wrote:

> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > >>>This should be "schismic". It's consistent with all those ETs
> > except 17 > >>>(which isn't 7-limit consistent in itself).
>>
>>I agree; "schismic" is essentially a 5-limit name (based on tempering >>out the [-15 8 1> "schisma"), so the 7-limit extension (if any) >>shouldn't be more complex than it needs to be.
> > > This reasoning is backwards, and leads to bizarre conclusions. If
> schismic is essentially a 5-limit temperament, then you'd better keep
> close to the 5-limit tuning seems like the way to reason from your
> premise. Moreover, it leads to the conclusion that dominant should get
> the name "meantone" in the 7-limit, and that is a conclusion no one
> seems to buy. We'd also end up renaming pajara to diaschismic, I suppose.

12&19 meantone isn't excessively complex, and it's not a good idea to change established names in any case. If you have to go all the way to 118&171 for schismic, it's way too complex to be of much use to anyone; very few people will bother with it at all, so why give it the familiar name when there are better options? Even the 13-limit Cassandra 1 is only 41&94.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> 12&19 meantone isn't excessively complex...

"Excessively complex" is a value judgment which only makes sense in
the context of some projected use; hence it is really meaningless for
us, since the possible uses are varied. So long as the *commas* are
not complex it certainly might find a use in electronic music (and
this *is* the 21st century; this aspect is likely to assume
ever-increasing importance) if nowhere else. And from a comma point of
view, this so-called "excessively complex temperament" is not that
complex! It has commas of 4375/4374 (Hahn size 7) and 32805/32768
(Hahn size 9.) You are simply failing to think in terms of the 7-limit
lattice when you call it "excessively complex".

The prejudice against complex temperaments is obvious on this group,
but it all comes from people who never actually *use* them. Until
you've used them, it really resembles the people who kvetch to you
that some of your favorite temperaments are excessively poorly tuned.

and it's not a good idea to
> change established names in any case.

We've *been* changing established names; I don't think 7-limit
schismic was ever as established as some Paul wants to deep-six.
Moreover, sticking to a consistent scheme means we have a better idea
what the name means; in this case, that the name in the higher limit
has a tuning in accord with the lower limit.

If you have to go all the way to
> 118&171 for schismic, it's way too complex to be of much use to
anyone;

Wrong! I should probably write something in it; would that convince
anyone of anything or is everyone hypotized by the idea that the only
way to make music is to tune up a guitar?

> very few people will bother with it at all, so why give it the familiar
> name when there are better options?

This is the ***21st century***. I would *not* assume people in the
future are going to ignore complex tunings.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Herman Miller <hmiller@IO.COM>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Herman Miller <hmiller@IO.COM>

Gene Ward Smith wrote:

> We've *been* changing established names; I don't think 7-limit
> schismic was ever as established as some Paul wants to deep-six.
> Moreover, sticking to a consistent scheme means we have a better idea
> what the name means; in this case, that the name in the higher limit
> has a tuning in accord with the lower limit.

Looking over this reply, I'm wondering if my meaning wasn't clear enough. I'm not referring to Paul's substitution of new names for the 7- and 5-limit versions of temperaments; those substitutions don't change the meanings of the existing names. If for consistency we wanted to give [<1, 2, 4, 2|, <0, -1, -4, 2|] the same name as [<1, 2, 4|, <0, -1, -4|], it would be confusing to use the name "meantone", since that name is already associated with [<1, 2, 4, 7|, <0, -1, -4, -10|]. So we'd have to come up with a new name, like "syntonic".

But of course there could be occasional cases where there's a good reason to change a higher-limit name. I've argued for #56 [<1, 1, 2, 4|, <0, 2, 1, -4|] to get the name "dicot" in place of #23 [<1, 1, 2, 1|, <0, 2, 1, 6|], which would be "pseudo-dicot". It sounds like there are good arguments for changing "schismic" as well. I'm just not convinced that 118&171 is the best candidate for the name, and I'm dubious about applying the same criteria to other temperaments.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> At some point a temperament's errors are so low and its complexity
> so high, that you might as well use strict rational intervals (or
> the best approximation of them you can get on your
> instrument/computer). If that's the case, why would anyone care
> about the temperament?

Try and and you'll see that the Tenney height of your rational
intervals quickly grows. You can, to be sure, use rational intervals
and rejoice in the fact that if you land 2401/2400 away from where you
started, no one is likely to notice. But why do that? It's harder!

My 171-et piece Kotekant constantly made use of the "too complex"
commas 4375/4374 and 32805/32768, so I'm really not impressed with the
idea they are too complex to be made use of. There is something to be
said for the theory that if you are going to use ennealimmal or this
version of schismic or various other 171-et supported 7-limit
temperaments such as sesquiquartififths you may as well simply park
yourself in 171-equal, but even if you maintain that, understanding an
equal temperament is helped by understanding the linear temperaments
it supports. The notion that the only good version of 7-limit schismic
is 41&53 is really all wet.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Looking over this reply, I'm wondering if my meaning wasn't clear
> enough. I'm not referring to Paul's substitution of new names for
the 7-
> and 5-limit versions of temperaments; those substitutions don't change
> the meanings of the existing names. If for consistency we wanted to
give
> [<1, 2, 4, 2|, <0, -1, -4, 2|] the same name as [<1, 2, 4|, <0, -1,
> -4|], it would be confusing to use the name "meantone", since that name
> is already associated with [<1, 2, 4, 7|, <0, -1, -4, -10|]. So we'd
> have to come up with a new name, like "syntonic".

Out there in the wide, wide world people may be thinking the right way
to get a meantone 7 is as an approximate 16/9, so I don't know how
established this really is.

> But of course there could be occasional cases where there's a good
> reason to change a higher-limit name. I've argued for #56 [<1, 1, 2,
4|,
> <0, 2, 1, -4|] to get the name "dicot" in place of #23 [<1, 1, 2, 1|,
> <0, 2, 1, 6|], which would be "pseudo-dicot".

You have? When was that?

It sounds like there are
> good arguments for changing "schismic" as well. I'm just not convinced
> that 118&171 is the best candidate for the name, and I'm dubious about
> applying the same criteria to other temperaments.

It's the only temperament with a decent badness figure and an optimal
tuning close to 5-limit schismic; that seems to give it a strong claim.

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>
> > 12&19 meantone isn't excessively complex...
>
> "Excessively complex" is a value judgment which only
> makes sense in the context of some projected use; hence
> it is really meaningless for us, since the possible uses
> are varied. So long as the *commas* are not complex it
> certainly might find a use in electronic music (and
> this *is* the 21st century; this aspect is likely to
> assume ever-increasing importance) if nowhere else. And
> from a comma point of view, this so-called "excessively
> complex temperament" is not that complex! It has commas
> of 4375/4374 (Hahn size 7) and 32805/32768 (Hahn size 9.)
> You are simply failing to think in terms of the 7-limit
> lattice when you call it "excessively complex".

please give me enough data (or point to it if it's already
posted somewhere) to *make* that 7-limit lattice for my
webpage.

this is *precisely* the kind of thing i want to enhance
the Encyclopaedia with, to help as many people as possible
understand the amazing rapid developments that have
occured in tuning theory within the last few years.

> > and it's not a good idea to change established names
> > in any case.
>
> We've *been* changing established names; I don't think
> 7-limit schismic was ever as established as some Paul
> wants to deep-six. Moreover, sticking to a consistent
> scheme means we have a better idea what the name means;
> in this case, that the name in the higher limit has a
> tuning in accord with the lower limit.

i've been thinking a lot about the naming thread
lately, and about an hour before i read this post,
i was thinking that "orwell" really provides a clue here.

orwell is already the *family* name for these temperaments,
and it looks like Gene made "george" a member of the
family. so now we can have "emily", "mike", etc.

maybe there will be a "bundy" family, with members
like "al", "peggy", "kelly", and "bud".

how about a "simpson" family, with "homer", "marge",
"bart", "lisa", and "maggie"?

or the small "petri" family, consisting only of
"rob", "laura", and "ritchie" ... but with friends
and co-workers like "buddy", "sally", "mel",
"jerry & millie" (a pair of married temperaments),
and occasional appearances by "alan" and "stacey".

... for the younger people on the list who might not
get that last reference, try this:

http://www.museum.tv/archives/etv/D/htmlD/dickvandyke/dickvandyke.htm

all kidding aside ... i can sympathize with those
who are alarmed by the "jargon explosion", but really,
naming tunings is a good thing, because it makes it
much easier to talk about them and remember stuff
about them. putting the names together in families
like this is a great way to keep them organized.

i've created a webpage for "family" in the Encyclopaedia

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

it contains a sample table of some 5-limit families
and an incomplete list of EDOs which belong to them.
(i got the data from my own bingo-card webpage)

others are welcome to help fill out that table with
other family names, and to construct whole new tables
for other prime-limits.

> This is the ***21st century***. I would *not* assume people
> in the future are going to ignore complex tunings.

you go, Gene!

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> please give me enough data (or point to it if it's already
> posted somewhere) to *make* that 7-limit lattice for my
> webpage.

The TM basis is {4375/4374, 32805/32768}, and the Hahn reduced basis
is {4375/4374, 65625/65536}; these are the two most interesting block
commas to use; of course {32805/32768, 65625/65536} is another
possibility, and there are other commas out there, such as
95703125/95551488.

Incidentally, if you are putting up the web page I don't know what you
are going to use as a name, but whatever it is may need a "?" after
it. However, instead of calling it 118&171 we should call it 53&118,
since 53 and 118 are really the extreme ranges, and 171 is a good
choice for tuning, and those are the rules for deciding the question.
I just did it wrong.

> i've been thinking a lot about the naming thread
> lately, and about an hour before i read this post,
> i was thinking that "orwell" really provides a clue here.
>
> orwell is already the *family* name for these temperaments,
> and it looks like Gene made "george" a member of the
> family. so now we can have "emily", "mike", etc.

Fortunately I don't think too many more family members are likely to
show up.

> maybe there will be a "bundy" family, with members
> like "al", "peggy", "kelly", and "bud".

Don't forget Ted!

> all kidding aside ... i can sympathize with those
> who are alarmed by the "jargon explosion", but really,
> naming tunings is a good thing, because it makes it
> much easier to talk about them and remember stuff
> about them. putting the names together in families
> like this is a great way to keep them organized.

As I say, naming tunings is not such a big deal for me, but I strongly
resist the idea that concepts like "val" and "monzo" do not deeserve
names. They are fundamental concepts.

> > This is the ***21st century***. I would *not* assume people
> > in the future are going to ignore complex tunings.

> you go, Gene!

Not everyone agrees; but I will inflict another piece of music on the
world fairly soon now, and theory, despite what many think, does
actually translate to practice.

🔗monz <monz@attglobal.net>

hi Dave,

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

> At some point a temperament's errors are so low and
> its complexity so high, that you might as well use
> strict rational intervals (or the best approximation
> of them you can get on your instrument/computer).
> If that's the case, why would anyone care about
> the temperament?

it's been a basis of my theories for a long time
that humans make use of two fundametally different
pitch-perception mechanisms, both of which recognize
some equivalence-interval: one which is based on
the harmonicity of the conglomerate sounds (ratios),
and the other which is based on approximately equal
logarithmic divisions of that equivalence-interval (ETs).

from this perspective, it's *very* valuable to have
ways of comparing and relating similar tunings by means
of calculations based on either of the two mechanisms.

-monz

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > please give me enough data (or point to it if it's already
> > posted somewhere) to *make* that 7-limit lattice for my
> > webpage.
>
> The TM basis is {4375/4374, 32805/32768}, and the
> Hahn reduced basis is {4375/4374, 65625/65536}; these are
> the two most interesting block commas to use; of course
> {32805/32768, 65625/65536} is another possibility, and there
> are other commas out there, such as 95703125/95551488.

i totally missed out on "Hahn reduced" ... can you please
explain that in a way similar to what's on my "TM-reduced" page?

i've made a graphic of the TM-basis version, here:

http://tonalsoft.com/enc/tm-reduced-lattice-53-118-schismic.gif

but without being able to rotate (which i can do here with
my software), the static graphic of this tuning is too
complex to really convey much information. so i didn't
want to put it onto the webpage yet.

hope that helps a little, Herman!

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> i totally missed out on "Hahn reduced" ... can you please
> explain that in a way similar to what's on my "TM-reduced" page?

First I need to explain the Hahn metric. If <O a b c| is a 7-limit
monzo, then the octave class it represents does not depend on O, and a
metric on octave classes will ignore it. The Hahn metric is

|| <O a b c| ||_Hahn = max(|a|,|b|,|c|,|a+b|,|b+c|,|c+a|,|a+b+c|)

It counts the number of 7-limit consonaces needed to get to the octave
class. If we have a set of 7-limit commas, they are Hahn reduced by
finding the comma with smallest Hahn distance, with ties broken by
Tenney distance. Then we find the comma with second smallest Hahn
distance independent from the first, and so forth.

In the case of the "pontiac" commas, the Hahn distance of 4375/4374
and 65625/65536 is 7, with 4375/4374 having the smaller Tenney
distance. These define the Hahn reduced comma set for pontiac, since
any other commas for it, such as 32805/32768 with Hahn distance 9,
would under this measure be regarded as more complex.

This gives a rather different perspective on complexity for
temperaments than figuring out how many generator steps it takes to
reach a consonace; it figures out how many consonances it takes to
reach the commas. Pontiac would, using the measure of the largest Hahn
distance needed in a reduced set, have a complexity of 7. In
comparison, meantone would have a complexity of 4, but orwell or
miracle would have exactly the same complexity. Hemiwuerscmidt would
have a complexity of 5, as would garibaldi. Shrutar would have a
complexity of 6, but so would the <0 0 34 0 54 79| temperament, with
mapping [<34 54 79 95|, <0 0 0 1|], which comes out with too high a
badness to bother with because of its high complexity according to the
usual ways to measure complexity. I've been pointing out that for some
purposes, this isn't the relevant measure, which would in fact serve
to rehabilitate temperaments such as this 34&68 temperament. I don't
think I've managed to get my point across to anyone as yet.

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

Gene Ward Smith wrote:

> Someone who is tuning schismic to get highly accurate 5-limit
> intonation will be tuning using sligtly flatted fifths--in the range
> 1/8 to 1/9-schisma schismic. This, historically, is what it has been
> used for. Someone wanting that degree of accuracy, and who has tuned
> up a 53 note MOS, will find that 39 of these slightly flattened fifths
> will give a better 7 than will 14 of the slightly sharpened fourths;
> those will be a decent enough 6.8 to 7 cents sharp, and certainly
> useable, but the other 7s will be between 1 and 1.5 cents flat, which
> is much better. Optimizing for the 7s will then do little damage to
> 5-limit harmony, since it is a matter of moving from 1/9 to 1/10
> schisma flat. The fifth of 94-et is 1/11 schisma *sharp*, and not at
> all optimal for 5-limit harmony; 94 would never be considered for
> 5-limit schisma.

There are so many "ifs" here. Where are you getting your history from? I don't know of anybody doing what you said. What I remember is:

The Arab/Persian systematists who used a 17 note scale that may have been Pythagorean or 17-equal at various times. Ratios can be deduced involving 7 that are consistent with two different mappings, the 12&29 one and a simpler one from 17-equal.

European composers who used schismic chords in 12 note Pythagorean music.

Helmholtz, who described 1/8 and 1/9 schisma tuning theoretically.

Bosanquet, who built a 53 note organ. I'm not sure of the tuning, most assume 53-equal.

Groven, who built a piano, I think with 36 notes to the octave. I don't know the precise tuning.

Wilson, who used a 41 note, 11-limit template.

Margo Schulter, who uses sharp-fifth tunings, but often without the 5 approximation.

Of these, only Helmholtz is notably interested in flattened fifths, and may find this micro (or nano?) temperament useful.

The 53&118 mapping is useless in a 53 note MOS. You only get 6 complete chords! They cover around 20 notes. If you want those 20 notes, why not tune up 20 notes instead of a 53 note MOS? Furthermore, how do you expect these highly accurate temperaments to be playable on a schismic keyboard?

Historically, temperament in general is certainly advocated for physical instruments.

When did you get to be an authority on what is or isn't going to be considered? 94-equal is a perfectly good schismic tuning. The schismic range goes from 12-equal on one side to beyond 29-equal on the other. It's something of a straw man anyway, because the much more common Pythagorean tuning is also outside of the 53&118 range. I note you haven't answered my criticisms of flattened fifth schismics used melodically.

Graham

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

Gene Ward Smith wrote:

> This reasoning is backwards, and leads to bizarre conclusions. If
> schismic is essentially a 5-limit temperament, then you'd better keep
> close to the 5-limit tuning seems like the way to reason from your
> premise. Moreover, it leads to the conclusion that dominant should get
> the name "meantone" in the 7-limit, and that is a conclusion no one
> seems to buy. We'd also end up renaming pajara to diaschismic, I suppose.

Actually, another thing I object to is this idea that things like "schismic" and "meantone" are temperaments. They are not. 1/4-comma meantone and 1/8-schisma schismic are temperaments. Schismic and meantone are what I used to properly (and I may have slipped up) call "temperament families". That term seems to be used for larger groups now, but still, schismic is not a temperament. It does not lead to "the tuning". It is valid for a wide range of tunings, 5-limit or otherwise. In the widest sense, any tuning that tempers out the schisma is schismic.

I take it "dominant" is the meantone extension with 8:7 equal to a tone? That could be called 5&12, as 5-equal is 7-limit consistent. The problem is that it optimizes with a fifth sharp of 12-equal, which is an uncharacteristic tuning for meantone. 12 is the pivot ET[1] between meantone and schismic. Anything with a fifth sharp of 12-equal looks like schismic, not meantone.

The "diaschismic" issue is more debatable. There isn't really a single, obvious 7-limit diaschismic. The argument agains pajara is that it optimizes so close to 22-equal that it doesn't gain you much over using 22-equal.

Graham

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

Gene Ward Smith wrote:

> "Excessively complex" is a value judgment which only makes sense in
> the context of some projected use; hence it is really meaningless for
> us, since the possible uses are varied. So long as the *commas* are
> not complex it certainly might find a use in electronic music (and
> this *is* the 21st century; this aspect is likely to assume
> ever-increasing importance) if nowhere else. And from a comma point of
> view, this so-called "excessively complex temperament" is not that
> complex! It has commas of 4375/4374 (Hahn size 7) and 32805/32768
> (Hahn size 9.) You are simply failing to think in terms of the 7-limit
> lattice when you call it "excessively complex".

Those figures should really be based on the 9-limit, because schismic (of either variety) is more compelling in the 9-limit than the 7-limit. As such, the sizes get reduced quite a bit. 4375:4374 is size 4 and the schisma is size 5.

What get me, though, is why you need fixed temperaments for this kind of work. Can't you get the computer to temper out one comma at a time as you need it? The whole approach seems to depend on having a computer available to realize it.

> The prejudice against complex temperaments is obvious on this group,
> but it all comes from people who never actually *use* them. Until
> you've used them, it really resembles the people who kvetch to you
> that some of your favorite temperaments are excessively poorly tuned. Of course you only hear from people who don't use such temperament. You're the only person in the entire world who does, or has ever used them. And yet you're trying to change existing terminology for your own convenience.

Is Paul proposing to change other names? The name "schismic" is established on my web page, which has had thousands of hits since I put it up. It must be about as visible as a Xenharmonikon article. It also originated on URL I no longer have control over, so anybody chasing that will depend on the archived copies.

How can the higher limit tuning ever be invalid in the lower limit?

If you want a consistent scheme, how about taking the mapping covering the largest number of simple, consistent equal temperaments?

Graham

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > Someone who is tuning schismic to get highly accurate 5-limit
> > intonation will be tuning using sligtly flatted fifths--in the range
> > 1/8 to 1/9-schisma schismic. This, historically, is what it has been
> > used for.

> There are so many "ifs" here. Where are you getting your history from?
> I don't know of anybody doing what you said.

Read Helmholtz on how he tuned his organ and what he used it for, and
check out the stuff on Groven in Joe's encyclopedia.

What I remember is:
>
> The Arab/Persian systematists who used a 17 note scale that may have
> been Pythagorean or 17-equal at various times. Ratios can be deduced
> involving 7 that are consistent with two different mappings, the 12&29
> one and a simpler one from 17-equal.
>
> European composers who used schismic chords in 12 note Pythagorean
music.
>
> Helmholtz, who described 1/8 and 1/9 schisma tuning theoretically.

Helmholtz, who did a hell of a lot more than that.

> Bosanquet, who built a 53 note organ. I'm not sure of the tuning, most
> assume 53-equal.
>
> Groven, who built a piano, I think with 36 notes to the octave. I
don't
> know the precise tuning.

I think 1/8-schisma.

> Of these, only Helmholtz is notably interested in flattened fifths, and
> may find this micro (or nano?) temperament useful.

Not true.

> Historically, temperament in general is certainly advocated for
physical
> instruments.

Historically, that was the only kind there was; this comment makes no
sense.

> When did you get to be an authority on what is or isn't going to be
> considered? 94-equal is a perfectly good schismic tuning.

It's excellent for the 7-limit, it's pointless if all you want is the
5-limit, which is why Helmholtz and Groven wouldn't touch it with a
barge pole.

The schismic
> range goes from 12-equal on one side to beyond 29-equal on the other.

Using one definition of what this means, but you are assuming your
conclusion if this is supposed to be an argument.

> It's something of a straw man anyway, because the much more common
> Pythagorean tuning is also outside of the 53&118 range. I note you
> haven't answered my criticisms of flattened fifth schismics used
> melodically.

I regarded it as transparently bogus, frankly.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Actually, another thing I object to is this idea that things like
> "schismic" and "meantone" are temperaments. They are not. 1/4-comma
> meantone and 1/8-schisma schismic are temperaments. Schismic and
> meantone are what I used to properly (and I may have slipped up) call
> "temperament families".

Not in my lexicon. To me and I think to most of us, you are confusing
a temperament with a tuning by saying this; they are not the same thing.

> In the widest sense, any tuning that tempers out the schisma is
schismic.

That's a pretty wide sense. Is the 7-limit temperament which tempers
out 135/128 and 10/9 actually schismic?

> I take it "dominant" is the meantone extension with 8:7 equal to a
tone?
> That could be called 5&12, as 5-equal is 7-limit consistent. The
> problem is that it optimizes with a fifth sharp of 12-equal, which
is an
> uncharacteristic tuning for meantone.

Which is why I don't propose to call it meantone, the name it would be
given if we followed the naming plan being proposed of going with the
least complex temperament.

12 is the pivot ET[1] between
> meantone and schismic. Anything with a fifth sharp of 12-equal looks
> like schismic, not meantone.

"Looks like" is pretty meaningless unless you know how it is being
used. A diatonic scale in dominant, tuned to 41-equal, "looks like"
schismic except that it doesn't get to have any thirds.

> The "diaschismic" issue is more debatable. There isn't really a
single,
> obvious 7-limit diaschismic. The argument agains pajara is that it
> optimizes so close to 22-equal that it doesn't gain you much over using
> 22-equal.

Shrutar (46&68) is what you get if you want to keep the poptimal
generator of 5-limit diaschismic, but you can't call it diashismic
since the generator has been cut in half; ignoring that small
difficuty the tunings are close, and it seems to me on your theory of
what temperaments actually are, namely tunings, it probably *should*
be called diaschismic. But of course no one wants to call it that!

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > "Excessively complex" is a value judgment which only makes sense in
> > the context of some projected use; hence it is really meaningless for
> > us, since the possible uses are varied. So long as the *commas* are
> > not complex it certainly might find a use in electronic music (and
> > this *is* the 21st century; this aspect is likely to assume
> > ever-increasing importance) if nowhere else. And from a comma point of
> > view, this so-called "excessively complex temperament" is not that
> > complex! It has commas of 4375/4374 (Hahn size 7) and 32805/32768
> > (Hahn size 9.) You are simply failing to think in terms of the 7-limit
> > lattice when you call it "excessively complex".
>
> Those figures should really be based on the 9-limit, because schismic
> (of either variety) is more compelling in the 9-limit than the 7-limit.
> As such, the sizes get reduced quite a bit. 4375:4374 is size 4 and
> the schisma is size 5.

You can use 4375/4374 and 65625/65526, for a 7-limit Hahn complexity of 7.

> What get me, though, is why you need fixed temperaments for this
kind of
> work.

It's easier, for starters. Actually equal temperaments are what are
really easy, and so if we were using "pontiac" it would make sense,
for instance, to keep track of everything in 171-et, and then use
pontiac as a way of generating ideas for harmonic connections. If you
take a certain number of fifths in a row (which need not amount to a
MOS) and the associated 7-limit harmonies, one has a ready-made system
for making use of 171-equal which is not the same as the system you'd
get from ennealimmal, for example. Of course if I *only* planned on
using pontiac, I'd probably tune it to 1763-equal; using Scala seq
files for my scores, this keeps track of things nicely and really
causes no more problems than

Can't you get the computer to temper out one comma at a time as
> you need it?

That's adaptive tempering, and is a hell of a lot harder to use, and
for that matter to think about in a compositional setting.

The whole approach seems to depend on having a computer
> available to realize it.

Welcome to the 21st century.

> > The prejudice against complex temperaments is obvious on this group,
> > but it all comes from people who never actually *use* them. Until
> > you've used them, it really resembles the people who kvetch to you
> > that some of your favorite temperaments are excessively poorly tuned.
>
> Of course you only hear from people who don't use such temperament.
> You're the only person in the entire world who does, or has ever used
> them.

Are you certain of this?

And yet you're trying to change existing terminology for your own
> convenience.

There really isn't much *existing* terminology. The Yahoo gang is
creating it.

> > We've *been* changing established names; I don't think 7-limit
> > schismic was ever as established as some Paul wants to deep-six.
> > Moreover, sticking to a consistent scheme means we have a better idea
> > what the name means; in this case, that the name in the higher limit
> > has a tuning in accord with the lower limit.
>
> Is Paul proposing to change other names? The name "schismic" is
> established on my web page, which has had thousands of hits since I put
> it up.

A little contradiction here and there will keep our juices flowing.
I've concluded your use of the same for the 7-limit isn't that good an
idea, so I won't use it, but there's no reason you should not. Pontiac
probably makes sense as a name if someone (me, for example) wanted to
write something in it and put the piece up on a web page, along with
a description of the tuning.

It must be about as visible as a Xenharmonikon article. It also
> originated on URL I no longer have control over, so anybody chasing
that
> will depend on the archived copies.
>
> How can the higher limit tuning ever be invalid in the lower limit?

Eh? What does this mean, exactly?

> If you want a consistent scheme, how about taking the mapping covering
> the largest number of simple, consistent equal temperaments?

How do I go about defining this, and do I have to worry about 2-equal?

🔗monz <monz@attglobal.net>

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

> Actually, another thing I object to is this idea that
> things like "schismic" and "meantone" are temperaments.
> They are not. 1/4-comma meantone and 1/8-schisma schismic
> are temperaments. Schismic and meantone are what I used
> to properly (and I may have slipped up) call "temperament
> families". That term seems to be used for larger groups
> now, but still, schismic is not a temperament. It does
> not lead to "the tuning". It is valid for a wide range
> of tunings, 5-limit or otherwise.

i just earlier today made a webpage Encyclopaedia entry
for "family" at

http://tonalsoft.com/enc

(anyone who goes to the Tonalsoft site these days should
be clicking "refresh/reload" a lot, because i keep changing
things, including the left-side index.)

i plan to be adding a lot more to this page.

i'm interested in the history of describing temperament
families. did it originate with you, Graham? i think
i recall suggesting it myself in a tuning-list post
around 1999. or is the idea much older than that?

-monz

🔗Herman Miller <hmiller@IO.COM>

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >>Gene Ward Smith wrote:
>>
>>
>>>and it's not a good idea to >>>
>>>
>>>>change established names in any case. >>>
>>>
>>>We've *been* changing established names; I don't think 7-limit
>>>schismic was ever as established as some Paul wants to deep-six.
>>
>>I was referring to "meantone".
> > > Who changed that?

This is getting tedious.

You said:
"This reasoning is backwards, and leads to bizarre conclusions. If
schismic is essentially a 5-limit temperament, then you'd better keep
close to the 5-limit tuning seems like the way to reason from your
premise. Moreover, it leads to the conclusion that dominant should get
the name "meantone" in the 7-limit, and that is a conclusion no one
seems to buy. We'd also end up renaming pajara to diaschismic, I suppose."

I replied:
"12&19 meantone isn't excessively complex, and it's not a good idea to
change established names in any case."

It should have been perfectly clear from the context that I was talking about meantone. To be precise, in case this still isn't clear, I was trying to explain why we don't need to change the meaning of "meantone" to refer to what we've been calling "dominant". But since you were talking about changing the definition of "schismic", I can see how this could be misinterpreted. Now can we please stop arguing pointlessly and start talking about music again?

🔗Herman Miller <hmiller@IO.COM>

Gene Ward Smith wrote:

> Out there in the wide, wide world people may be thinking the right way
> to get a meantone 7 is as an approximate 16/9, so I don't know how
> established this really is.

This of course is a perfectly valid thing to do, if the version of meantone you're using is the one that's been in common use for the last century or so (12-ET). Illustrations of the harmonic series typically use Bb for the 7th harmonic. But if I'm not mistaken, originally the word "meantone" was applied to what we now call "quarter-comma meantone", and this is a much better fit with 12&19.

>>But of course there could be occasional cases where there's a good >>reason to change a higher-limit name. I've argued for #56 [<1, 1, 2,
> > 4|, > >><0, 2, 1, -4|] to get the name "dicot" in place of #23 [<1, 1, 2, 1|, >><0, 2, 1, 6|], which would be "pseudo-dicot".
> > > You have? When was that?

Okay, it looks like what I actually did was suggest that #94 [<1, 2, 1, 5|, <0, -1, 3, -5|] should get the name "pelogic" instead of #19 [<1, 2, 1, 1|, <0, -1, 3, 4|], and then the next day posted that "dicot" has the same problem. So my memory isn't perfect. But you get the idea.

/tuning-math/message/9670
/tuning-math/message/9745

🔗Carl Lumma <ekin@lumma.org>

>> i totally missed out on "Hahn reduced" ... can you please
>> explain that in a way similar to what's on my "TM-reduced" page?
>
>First I need to explain the Hahn metric. If <O a b c| is a 7-limit
>monzo, then the octave class it represents does not depend on O, and a
>metric on octave classes will ignore it. The Hahn metric is
>
>|| <O a b c| ||_Hahn = max(|a|,|b|,|c|,|a+b|,|b+c|,|c+a|,|a+b+c|)
>
>It counts the number of 7-limit consonaces needed to get to the octave
>class. If we have a set of 7-limit commas, they are Hahn reduced by
>finding the comma with smallest Hahn distance, with ties broken by
>Tenney distance. Then we find the comma with second smallest Hahn
>distance independent from the first, and so forth.

Great explanation Gene. Thanks!

>In the case of the "pontiac" commas, the Hahn distance of 4375/4374
>and 65625/65536 is 7, with 4375/4374 having the smaller Tenney
>distance. These define the Hahn reduced comma set for pontiac, since
>any other commas for it, such as 32805/32768 with Hahn distance 9,
>would under this measure be regarded as more complex.
>
>This gives a rather different perspective on complexity for
>temperaments than figuring out how many generator steps it takes to
>reach a consonace; it figures out how many consonances it takes to
>reach the commas. Pontiac would, using the measure of the largest Hahn
>distance needed in a reduced set,

Instead of max Hahn distance and max Tenney distance, what about the
number of notes in the Fokker blocks delimited by the Hahn-reduced
basis vs. the number in the TM-reduced basis?

>In comparison, meantone would have a complexity of 4, but orwell or
>miracle would have exactly the same complexity. Hemiwuerscmidt would
>have a complexity of 5, as would garibaldi. Shrutar would have a
>complexity of 6, but so would the <0 0 34 0 54 79| temperament, with
>mapping [<34 54 79 95|, <0 0 0 1|], which comes out with too high a
>badness to bother with because of its high complexity according to the
>usual ways to measure complexity. I've been pointing out that for some
>purposes, this isn't the relevant measure, which would in fact serve
>to rehabilitate temperaments such as this 34&68 temperament. I don't
>think I've managed to get my point across to anyone as yet.

Can you list some of these paradoxes, reached by using Hahn
complexity instead of Tenney complexity? It does seem that
meantone should be simpler than orwell in the 7-limit, but can
you give some other examples?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Instead of max Hahn distance and max Tenney distance, what about the
> number of notes in the Fokker blocks delimited by the Hahn-reduced
> basis vs. the number in the TM-reduced basis?

The trouble with that is that you need another comma to define a
block; and then the number of notes is simply where this comma is
being mapped in terms of generators by the temperament. In other
words, I could get a Fokker block for pontiac using any two commas
which generate the kernel by adding 225/224; which is mapped to -22
octaves and 53 fourths via pontiac, meaning the Fokker block would
have 53 notes to the octave. This tells us the complexity of 225/224
in pontiac is 53, but it isn't clear why we would care more about that
than the complexity of the 7-limit consonances, which are what usually
concern us.

> Can you list some of these paradoxes, reached by using Hahn
> complexity instead of Tenney complexity? It does seem that
> meantone should be simpler than orwell in the 7-limit, but can
> you give some other examples?

I suppose the thing to do would be to find Hahn reduced bases, or
minimax reduced bases in the sense that the maximum Hahn distance is
minimum, and then simply take a look at what we have. A badness
measure in terms of Hahn complexity would sort 7-limit temperaments in
quite a different way, giving a much higher goodness to miracle than
to meantone. It would certainly make for an interesting alternative
picture.

🔗Carl Lumma <ekin@lumma.org>

>> Instead of max Hahn distance and max Tenney distance, what about the
>> number of notes in the Fokker blocks delimited by the Hahn-reduced
>> basis vs. the number in the TM-reduced basis?
>
>The trouble with that is that you need another comma to define a
>block;

D'oh! As Paul keeps pointing out to me. LTs are strips. Still,
there might be some sort of measure we can apply here.

Otherwise, avg. distance seems better than max, since the "limit"
abstraction is loaded in the first place -- biased towards people
who want saturated chords.

>and then the number of notes is simply where this comma is
>being mapped in terms of generators by the temperament. In other
>words, I could get a Fokker block for pontiac using any two commas
>which generate the kernel by adding 225/224; which is mapped to -22
>octaves and 53 fourths via pontiac, meaning the Fokker block would
>have 53 notes to the octave. This tells us the complexity of 225/224
>in pontiac is 53, but it isn't clear why we would care more about
>that than the complexity of the 7-limit consonances, which are what
>usually concern us.

Interesting... I agree that we shouldn't be adding commas to
close the blocks (of course!). The sort of 'dual' (I'm probably
using this incorrectly) space of generators, where the 'basis'
is the chord you want does seem better. Which is Graham
complexity, I suppose...

>> Can you list some of these paradoxes, reached by using Hahn
>> complexity instead of Tenney complexity? It does seem that
>> meantone should be simpler than orwell in the 7-limit, but can
>> you give some other examples?
>
>I suppose the thing to do would be to find Hahn reduced bases, or
>minimax reduced bases in the sense that the maximum Hahn distance is
>minimum, and then simply take a look at what we have. A badness
>measure in terms of Hahn complexity would sort 7-limit temperaments in
>quite a different way, giving a much higher goodness to miracle than
>to meantone. It would certainly make for an interesting alternative
>picture.

Well if you ever find the time, yes, it does sound interesting.

-Carl

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

Gene Ward Smith wrote:

> It's easier, for starters. Actually equal temperaments are what are
> really easy, and so if we were using "pontiac" it would make sense,
> for instance, to keep track of everything in 171-et, and then use
> pontiac as a way of generating ideas for harmonic connections. If you
> take a certain number of fifths in a row (which need not amount to a
> MOS) and the associated 7-limit harmonies, one has a ready-made system
> for making use of 171-equal which is not the same as the system you'd
> get from ennealimmal, for example. Of course if I *only* planned on
> using pontiac, I'd probably tune it to 1763-equal; using Scala seq
> files for my scores, this keeps track of things nicely and really
> causes no more problems than Yes, but the "easy" argument seems to depend on the complexity of the mapping. Or, as you say, leads to equal temperaments.

> That's adaptive tempering, and is a hell of a lot harder to use, and
> for that matter to think about in a compositional setting.

Ah, but if you're already thinking in terms of lattices, and the 21st Century provides improvements in software wouldn't this be the most direct way of working?

The 21st Century is also likely to provide software for dynamically tuning simpler linear temperaments. So the 12&29 mapping would end up much closer to JI.

>>Of course you only hear from people who don't use such temperament. >>You're the only person in the entire world who does, or has ever used >>them. > > Are you certain of this? It's difficult to prove a negative. I don't know of any other examples.

> A little contradiction here and there will keep our juices flowing.
> I've concluded your use of the same for the 7-limit isn't that good an
> idea, so I won't use it, but there's no reason you should not. Pontiac
> probably makes sense as a name if someone (me, for example) wanted to
> write something in it and put the piece up on a web page, along with
> a description of the tuning.

I think there was hope for a consensus on the names in Paul's article. I have to withdraw from that consensus if "schismic" defaults to a 7-limit temperament other than 12&29 (synonymous with 53&41). If new terms get added to disambiguate it, that's fine. I can mention this on my web page.

The 53&118 mapping could be called "the 7/9-limit schismic microtemperament". I see the cutoff for microtemperaments is defined as 2.8 cents, and 12&29 can't reach this. In a piece about 7 or 9-limit microtemperaments it can be called "schismic" without ambiguity.

So is there similar name for the 12&29 mapping? That is, do we have a word for temperaments with an error *larger* than 2.8 cents, but not much larger? I think we could do with one.

>>How can the higher limit tuning ever be invalid in the lower limit?
> > Eh? What does this mean, exactly?

Lower limits are subsets of higher limits. So a temperament (and tuning) that works in a higher limit must work in the lower one.

>>If you want a consistent scheme, how about taking the mapping covering >>the largest number of simple, consistent equal temperaments?
> > How do I go about defining this, and do I have to worry about 2-equal?

You have to define "largest number" and "simple". I suggest "largest number" should be an absolute majority of whatever ETs we're considering. Then temperaments like diaschismic that have no obvious 7-limit mapping won't have any of them as the default in the 7-limit.

"Simple" is harder because you have to make a subjective decision. A quick calculation shows that most schismic ETs are not of the 12&29 family in the 9-limit.

All 9-limit consistent schismic ETs (including contorted ones):

[12, 29, 41, 53, 77, 89, 94, 118, 130, 142, 159, 171, 183, 200, 212, 224, 248, 253, 265, 289, 301, 313, 330, 354, 383, 395, 419, 431, 436, 460, 472, 484, 501, 525, 554, 566, 590, 607, 631, 643, 655, 696, 725, 737, 761, 778, 802, 814, 826, 867, 896, 908, 932, 985, 997, 1038, 1067, 1079, 1156, 1168, 1209, 1250, 1303, 1327, 1380, 1421]

The family also tempering out 4375:4374

[53, 118, 171, 224, 289, 395, 460, 566, 631, 737, 802, 908, 1079, 1250, 1303, 1421]

The family tempering out 225:224

[12, 29, 41, 53, 94]

My preference would win out for an absolute majority of ETs smaller than 100 (which doesn't scale to more complex temperaments) or the first 9 consistent ETs (which is arbitrary). It's also the most popular mapping for the first 10 consistent ETs.

2-equal won't be a problem because it isn't 7-limit consistent. The smallest such are 4, 5, 6, 7, 8 and 10. I suggest leaving them in and seeing if they cause trouble. The smallest 9-limit consistent ET is 5. Next is 12.

Graham

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

Me:
>>Actually, another thing I object to is this idea that things like >>"schismic" and "meantone" are temperaments. They are not. 1/4-comma >>meantone and 1/8-schisma schismic are temperaments. Schismic and >>meantone are what I used to properly (and I may have slipped up) call >>"temperament families". Gene:
> Not in my lexicon. To me and I think to most of us, you are confusing
> a temperament with a tuning by saying this; they are not the same thing. Hmm, well, I checked my encyclopedia, and it's ambiguous. But the point is that things like "schismic" and "meantone" should be allowed a wide range of tunings.

Me:
>> In the widest sense, any tuning that tempers out the schisma is
> > schismic.

Gene:
> That's a pretty wide sense. Is the 7-limit temperament which tempers
> out 135/128 and 10/9 actually schismic?

It must be, if it fits the definition. It doesn't look like much of a temperament.

Gene:
> Shrutar (46&68) is what you get if you want to keep the poptimal
> generator of 5-limit diaschismic, but you can't call it diashismic
> since the generator has been cut in half; ignoring that small
> difficuty the tunings are close, and it seems to me on your theory of
> what temperaments actually are, namely tunings, it probably *should*
> be called diaschismic. But of course no one wants to call it that!

I wouldn't want default names to go to contorted temperaments. You could call this "a contorted schismic". In itself it isn't contorted, so perhaps you should say something slightly different. But you get the idea.

It is different as a tuning.

Graham

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> monz wrote:
>
> > i'm interested in the history of describing temperament
> > families. did it originate with you, Graham? i think
> > i recall suggesting it myself in a tuning-list post
> > around 1999. or is the idea much older than that?
>
> AFAIK, but the new terminology seems to conflict. Although
> we could say that my families are of equal temperaments and
> the new ones are families of linear temperaments.
>
> I still don't know what "temperament" means ;)

very good point, and i'm glad you said it because
i had planned to and then forgot.

yes, let's please nail down once and for all exactly
what we mean by "tuning", "tuning system", "temperament",
and "scale", in that there seems to be a perception that
all four terms can mean the same thing.

in case it means anything ...

the Tonalsoft Musica software uses "tuning system" to mean
the collection of pitches which the user either derives from
a prime-space manually or sets up quickly with a wizard
(e.g., the fast equal-temperament wizard method).

if one is still working within the prime-space, the
prime-space panel can be used to create either a JI
tuning system or a temperament. for our purposes, the
latter is synonymous with "tempered tuning system".

tuning systems are then saved as .tuning files.

so essentially, the way we're doing it, "tuning" and
"tuning system" are synonymous, and "temperament" is
a subset under that which refers to a whole category
of tuning systems.

a "scale" is simply a source-set of pitches whose
cardinality is =/< that of the tuning system to
which it belongs ... depending on the size of the
tuning system, the scale is generally much smaller than
the whole tuning system. it also carries the specific
meaning of a Scala .scl file.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> > That's adaptive tempering, and is a hell of a lot harder to use, and
> > for that matter to think about in a compositional setting.
>
> Ah, but if you're already thinking in terms of lattices, and the 21st
> Century provides improvements in software wouldn't this be the most
> direct way of working?

Give me some software and I'll try it; but I note that using this
system, you are depending on the computer to sort things out for you,
whereas if you stick with a single tuning, you know what is going on.

> The 21st Century is also likely to provide software for dynamically
> tuning simpler linear temperaments. So the 12&29 mapping would end up
> much closer to JI.

Probably you are right, but we are not there yet; band-in-a-box is
already here, and will just get better.

> I think there was hope for a consensus on the names in Paul's article.
> I have to withdraw from that consensus if "schismic" defaults to a
> 7-limit temperament other than 12&29 (synonymous with 53&41).

I think the most likely compromise is not to use schismic as a name
for any temperament above the 5-limit; but note the TOP MOS I posted,
and how similar pontiac and 5-limit schismic are.

> The 53&118 mapping could be called "the 7/9-limit schismic
> microtemperament". I see the cutoff for microtemperaments is
defined as
> 2.8 cents, and 12&29 can't reach this. In a piece about 7 or 9-limit
> microtemperaments it can be called "schismic" without ambiguity.

I don't know what ukase decided it was 2.8 cents; I have been setting
it at 1 cent myself.

There's a problem with this system. Dave wants to call the 15&19
temperament, obtained by adding 126/125 to the kleisma, "simple
kleismic", and the 53&72 temperament, obtained by adding 225/224,
"complex kleismic", and you could do something similar by calling it
the "7/9-limit kleismic microtemperament". What, then, do we do with
the 53&87 temperament, obtained by adding 5120/5103 to the kleisma?
It's even better in tune, though the TOP tuning is not closer to
5-limit kleismic, so I didn't propose giving it the name "kleismic".
Moreover in terms of the *ratios* of TOP generators, defining the TOP
MOS system, neither one wins; instead the simple kleismic system turns
out to have exactly the same ratio, even though the TOP tunings
themselves are different.

5-limit kleismic

(7), (11), 15, 19, 34, 53, (87), (140), (193), ...

7-limit 15&19 <<6 5 3 -6 -12 -7||

The *ratio* of TOP generators is identical!

(7), (11), 15, 19, 34, 53, (87), (140), (193), ...

7-limit 53&72 <<6 5 22 -6 18 37||

(7), (11), 15, 19, (34), 53, 72, 125, 197, ...

7-limit 53&87 <<6 5 -31 -6 -66 -86||

(7), (11), 15, 19, 34, 53, (87), (140), 193, ...

> So is there similar name for the 12&29 mapping? That is, do we have a
> word for temperaments with an error *larger* than 2.8 cents, but not
> much larger? I think we could do with one.

I proposed a magnitude scale, with accuricies of better than one cent
being first magnitude or better; that is, 0.25 to 0.5, 0th magnitude,
0.5 to 1.0, 1st magnitude, 1.0 to 2.0, 2nd magnitude. Then the range
2.0 to 4.0 would be 3rd magnitude. Visually, we can see up to the 6th
magnitude, which are very faint stars, the corresponding range being
16-32 cents.

> The family tempering out 225:224
>
> [12, 29, 41, 53, 94]
>
> My preference would win out for an absolute majority of ETs smaller
than
> 100 (which doesn't scale to more complex temperaments) or the first 9
> consistent ETs (which is arbitrary). It's also the most popular
mapping
> for the first 10 consistent ETs.

If you dumped "simple" you'd have a clear definition with no ad hoc
stuff tossed in. You'd also tend to end up naming things my way. I'll
need to write some code for a consistency test, I guess, and check out
kleismic.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>> That's adaptive tempering, and is a hell of a lot harder to use,
>> and for that matter to think about in a compositional setting.
>
>Ah, but if you're already thinking in terms of lattices, and the
>21st Century provides improvements in software wouldn't this be
>the most direct way of working?

Adaptive tempering necessarily hands over some of the compositional
control to software, which is not something every composer might
want to give up.

>The 21st Century is also likely to provide software for dynamically
>tuning simpler linear temperaments. So the 12&29 mapping would end
>up much closer to JI.

Not something which is always desirable, either.

>>>How can the higher limit tuning ever be invalid in the lower
>>>limit?
>>
>> Eh? What does this mean, exactly?
>
>Lower limits are subsets of higher limits. So a temperament (and
>tuning) that works in a higher limit must work in the lower one.

We need to normalize complexity for limit. One expects
hexads to require more notes than triads. So Miracle is a
better 11-limit temperament than it is a 5-limit one.

-Carl

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> There's a problem with this system. Dave wants to call the 15&19
> temperament, obtained by adding 126/125 to the kleisma, "simple
> kleismic", and the 53&72 temperament, obtained by adding 225/224,
> "complex kleismic", and you could do something similar by calling
it
> the "7/9-limit kleismic microtemperament". What, then, do we do
with
> the 53&87 temperament, obtained by adding 5120/5103 to the kleisma

supercomplex 7-limit kleismic? And if you need more, there's
hypercomplex, ultracomplex, 5-complex, 6-complex etc.

> If you dumped "simple" you'd have a clear definition with no ad hoc
> stuff tossed in.

Yes. The word "simple" was intended to be optional. I should have
written it in parenthesis. It only needs to be used when you need to
be absolutely clear about which one you're talking about,
like "major" in talking about thirds.

Sometimes the temperaments that differ from the "best" one (and
therefor the one deserving of the unqualified name) are not more
complex, but less accurate. In this case I'd
use "inaccurate", "superinaccurate", etc. for these others.

This scheme should be applicable to 7-limit and temperaments that
have the 5-limit part of their map being identical to meantone and
schismic, as well as kleismic. Although I understand that some
people may baulk at describing "dominant sevenths" as "inaccurate 7-
limit meantone."

🔗monz <monz@attglobal.net>

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

> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
> > ... I see the cutoff for microtemperaments is defined as
> > 2.8 cents, and 12&29 can't reach this. In a piece about
> > 7 or 9-limit microtemperaments it can be called "schismic"
> > without ambiguity.
>
> I don't know what ukase decided it was 2.8 cents; I have
> been setting it at 1 cent myself.
>
> <snip>
>
> > ... do we have a word for temperaments with an error
> > *larger* than 2.8 cents, but not much larger? I think
> > we could do with one.
>
> I proposed a magnitude scale, with accuricies of better
> than one cent being first magnitude or better; that is,
> 0.25 to 0.5, 0th magnitude, 0.5 to 1.0, 1st magnitude,
> 1.0 to 2.0, 2nd magnitude. Then the range 2.0 to 4.0
> would be 3rd magnitude. Visually, we can see up to the
> 6th magnitude, which are very faint stars, the
> corresponding range being 16-32 cents.

i remember being intrigued by this when you posted it
a few years ago. i finally made a webpage for it:

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

-monz

🔗Carl Lumma <ekin@lumma.org>

>> That's adaptive tempering, and is a hell of a lot harder to use,
>> and for that matter to think about in a compositional setting.
>
>Ah, but if you're already thinking in terms of lattices, and the 21st
>Century provides improvements in software wouldn't this be the most
>direct way of working?

Adaptive tempering necessarily hands over some of the compositional
control to software, which is not something every composer might
want to give up.

>The 21st Century is also likely to provide software for dynamically
>tuning simpler linear temperaments. So the 12&29 mapping would end
>up much closer to JI.

Not something which is always desirable, either.

>>>How can the higher limit tuning ever be invalid in the lower limit?
>>
>> Eh? What does this mean, exactly?
>
>Lower limits are subsets of higher limits. So a temperament (and
>tuning) that works in a higher limit must work in the lower one.

We need to normalize complexity for limit. One expects
hexads to require more notes than triads. So Miracle is a
better 11-limit temperament than it is a 5-limit one.

-Carl

🔗Herman Miller <hmiller@IO.COM>

Graham Breed wrote:

> I meant to add a footnote to that. A pivot ET is an equal temperament > that is consistent with two different linear temperament mappings. It > marks the point between the characteristic ranges of the two different > linear temperament families (or whatever we want to call families now).

I wonder if the idea of a pivot ET could be used to "define" the characteristic ranges of the associated temperaments? It seems to work for meantone, with 19-ET as a pivot on one end (with flattone on the other side) and 12-ET on the other (dominant, which is equivalent to schismic in 12-ET). Then the characteristic range of schismic could be defined by 41-ET, which is bound on the other side by the "number 53" temperament [<1, 2, 16, 14|, <0, -1, -33, -27|]. Both 12-ET and 29-ET are outside the typical range of schismic. Between schismic and 12-ET is the "number 114" temperament [<1, 2, -1, -8|, <0, -1, 8, 26|], which sets the upper bound of typical schismic to 27/65; depending on how far you go in the chain of generators, 27/65-LT can support dominant [<1, 2, 4, 2|, <0, -1, -4, 2|], schismic [<1, 2, -1, -3|, <0, -1, 8, 14|], or "number 114". You could also mix and match and get things like "number 89" ("schism") [<1, 2, -1, 2|, <0, -1, 8, 2|]. (Otherwise you can get "schism" from 7/17, but since 17-ET isn't 7-limit consistent that's probably not the best name for it. Well, 65-ET isn't 7-limit consistent either....)

Since all of these are consistent with 27/65-LT (if we ignore the inconsistency of 65-ET), it would be nice to be able to name all of them in relation to 27/65. How about giving the range of generators in parentheses?

27/65-LT (-4..2) = [<1, 2, 4, 2|, <0, -1, -4, 2|] dominant
27/65-LT (-1..8) = [<1, 2, -1, 2|, <0, -1, 8, 2|] schism
27/65-LT (-1..14) = [<1, 2, -1, -3|, <0, -1, 8, 14|] schismic
27/65-LT (-1..26) = [<1, 2, -1, -8|, <0, -1, 8, 26|] number 114

This method could be used to name temperaments, like "superpelog" [<1, 2, 1, 3|, <0, -2, 6, -1|], which otherwise would be difficult to name with a generator/period ratio.

5/23-LT (-3..2) = [<1, 2, 3, 3, 3|, <0, -2, -3, -1, 2|] bug / beep
5/23-LT (-2..6) = [<1, 2, 1, 3, 3|, <0, -2, 6, -1, 2|] superpelog
5/23-LT (-10..6) = [<1, 2, 1, 5, 5|, <0, -2, 6, -10, -7|]

A few pivot ET's that might be of interest:

9-ET: hexadecimal [<1, 2, 1, 5|, <0, -1, 3, -5|]
and pelogic [<1, 2, 1, 1|, <0, -1, 3, 4|]

12-ET: injera [<2, 3, 4, 5|, <0, 1, 4, 4|]
and pajara [<2, 3, 5, 6|, <0, 1, -2, -2|]

15-ET: hemikleismic [<1, 0, 1, 4|, <0, 12, 10, -9|]
and porcupine [<1, 2, 3, 2|, <0, -3, -5, 6|]

15-ET: kleismic [<1, 0, 1, 2|, <0, 6, 5, 3|]
and superkleismic [<1, 4, 5, 2|, <0, -9, -10, 3|]

19-ET: muggles [<1, 0, 2, 5|, <0, 5, 1, -7|]
and magic [<1, 0, 2, -1|, <0, 5, 1, 12|]

19-ET: parakleismic [<1, 5, 6, 12|, <0, -13, -14, -35|]
and catakleismic [<1, 0, 1, 16|, <0, 6, 5, -50|]

31-ET: hemithirds [<1, 4, 2, 2|, <0, -15, 2, 5|]
and hemiwuerschmidt [<1, -1, 2, 2|, <0, 16, 2, 5|]

Some other pivot ET's are less interesting; 10-ET is technically consistent with both miracle [<1, 1, 3, 3|, <0, 6, -7, -2|] and negri [<1, 2, 2, 3|, <0, -4, 3, -2|], but isn't a very good example of either one.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> I wonder if the idea of a pivot ET could be used to "define" the
> characteristic ranges of the associated temperaments? It seems to work
> for meantone, with 19-ET as a pivot on one end (with flattone on the
> other side) and 12-ET on the other (dominant, which is equivalent to
> schismic in 12-ET).

You need to make some kind of badness bounds or it turns into a
complete dog's breakfast. This business is associated to what I was
calling a "nexus", since a nexus gives you one; and you can find as
many of those as you like, but only finitely many which make any
sense. Another slant on it can be gleaned from the posting I did
recently where I stepped through the Farey sequence near the fifth,
and all sorts of temperaments could be seen lurking in the underbrush.

Then the characteristic range of schismic could be
> defined by 41-ET, which is bound on the other side by the "number 53"
> temperament [<1, 2, 16, 14|, <0, -1, -33, -27|].

This I was calling "kwai" since it is bridgable; you could also call
it 41&70, or equivalently 65/111. If we believe Graham's theory that
slightly sharp fifths are better than slightly flat fifths, we might
go for this one, and it extends nicely to higher limits.

Both 12-ET and 29-ET
> are outside the typical range of schismic. Between schismic and
12-ET is
> the "number 114" temperament [<1, 2, -1, -8|, <0, -1, 8, 26|]...

I thought you wanted to name this grackle?

> Since all of these are consistent with 27/65-LT (if we ignore the
> inconsistency of 65-ET), it would be nice to be able to name all of
them
> in relation to 27/65. How about giving the range of generators in
> parentheses?

It's a little strange to base names for 7-limit temperaments on 65,
which as you say is not 7-limit consistent.

> 27/65-LT (-4..2) = [<1, 2, 4, 2|, <0, -1, -4, 2|] dominant
> 27/65-LT (-1..8) = [<1, 2, -1, 2|, <0, -1, 8, 2|] schism
> 27/65-LT (-1..14) = [<1, 2, -1, -3|, <0, -1, 8, 14|] schismic
> 27/65-LT (-1..26) = [<1, 2, -1, -8|, <0, -1, 8, 26|] number 114

Garibaldi (your schismic) and schism are related by a 12-et nexus, and
garibaldi and grackle (number 114) by another 12-nexus, leading to a
contorted 24-et nexus between schism and grackle. Dominant, however,
has a 12-et nexus with meantone and a 7-et nexus with flattone, which
shows where it belongs. So far as pontiac goes, schism has a 41-et
nexus, garibaldi a 53-et nexus, and grackle a 65-et nexus, with
dominant, no surprise, not having a nexus.

By the way, looking at my table of the 99th row of Farey, I find 58/99
in this sharp fifth region also; 29&70. It's basically 99-equal
arranged as a chain of fifths.

🔗Dave Keenan <d.keenan@bigpond.net.au>

You can see this idea visualised here, at the mountain peaks, for
the limited cases of generators which are fifths (in a very broad
sense) and one or two periods to the octave, 5-limit and 7-limit,
with specific limits on complexity.
http://dkeenan.com/Music/1ChainOfFifthsTunings.htm
http://dkeenan.com/Music/2ChainOfFifthsTunings.htm

So, one possibilty is that we agree on different complexity bands,
and for each band of complexity (and for each number of periods per
octave and each odd or prime limit) we divide the entire generator
spectrum up among the contending linear temperaments, with no gaps
and no overlaps. The boundaries (pivots) will always be at ETs.

Then we step up to the next complexity level, and do it all over
again, this time with more temperaments. Some temperaments at the
next level get to keep the same name, but with "complex" prepended.

This pushes the "dogs breakfast", that Graham mentioned, into the
problem of how to decide where each complexity band should start and
finish.

Just a thought.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> So, one possibilty is that we agree on different complexity bands,
> and for each band of complexity (and for each number of periods per
> octave and each odd or prime limit) we divide the entire generator
> spectrum up among the contending linear temperaments, with no gaps
> and no overlaps. The boundaries (pivots) will always be at ETs.
>
> Then we step up to the next complexity level, and do it all over
> again, this time with more temperaments. Some temperaments at the
> next level get to keep the same name, but with "complex" prepended.

One thing which we could use to possibly make the complexity bands
unnecessary is that bounding by logflat badness produces a finite list
of possibilities, without regard to complexity. So, for instance, if
we chose a TOP badness of 100 in the 7-limit, as I was doing, you
would only get a finite list of temperaments with generator 3/2 and
period octave, another with generator 3/2 and period 1/2 octave,
another with generator 1/2 of a fifth and period an octave, and so forth.

🔗monz <monz@attglobal.net>

hi Herman,

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

> I wonder if the idea of a pivot ET could be used to "define"
> the characteristic ranges of the associated temperaments?
> It seems to work for meantone, with 19-ET as a pivot on one
> end (with flattone on the other side) and 12-ET on the other
> (dominant, which is equivalent to schismic in 12-ET). Then the
> characteristic range of schismic could be defined by 41-ET,
> which is bound on the other side by the "number 53" temperament
> [<1, 2, 16, 14|, <0, -1, -33, -27|]. Both 12-ET and 29-ET
> are outside the typical range of schismic. Between schismic
> and 12-ET is the "number 114" temperament [<1, 2, -1, -8|,
> <0, -1, 8, 26|], which sets the upper bound of typical schismic
> to 27/65; depending on how far you go in the chain of
> generators, 27/65-LT can support dominant [<1, 2, 4, 2|,
> <0, -1, -4, 2|], schismic [<1, 2, -1, -3|, <0, -1, 8, 14|],
> or "number 114". <etc. -- snip>

i love this.

i was talking to Chris today (my Tonalsoft partner) about
this pivot-ET idea.

there's an analogy to old-fashioned "common-practice"
harmony. there, composers used common-tones as the
basis of chord-progression, and common-chords as the
basis of tonal modulation.

in this modern age of polymicrotonality, where we are
aware of and can easily quantify the mathematics of our
tunings, we can use the pivot-ETs in the same way, to
transition between different tuning-families.

-monz

🔗monz <monz@attglobal.net>

hi Gene,

i'm glad that you have a term for this concept, because
it should have one. but altho "nexus" is entirely
appropriate, i vote that we use something else, because
Partch already established "numerary nexus" as an important
term in tuning theory, and it's quite different from your
term.

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

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> > I wonder if the idea of a pivot ET could be used to "define" the
> > characteristic ranges of the associated temperaments? It seems to work
> > for meantone, with 19-ET as a pivot on one end (with flattone on the
> > other side) and 12-ET on the other (dominant, which is equivalent to
> > schismic in 12-ET).
>
> You need to make some kind of badness bounds or it turns into a
> complete dog's breakfast. This business is associated to what I was
> calling a "nexus", since a nexus gives you one; and you can find as
> many of those as you like, but only finitely many which make any
> sense. Another slant on it can be gleaned from the posting I did
> recently where I stepped through the Farey sequence near the fifth,
> and all sorts of temperaments could be seen lurking in the underbrush.
>
> Then the characteristic range of schismic could be
> > defined by 41-ET, which is bound on the other side by the "number 53"
> > temperament [<1, 2, 16, 14|, <0, -1, -33, -27|].
>
> This I was calling "kwai" since it is bridgable; you could also call
> it 41&70, or equivalently 65/111. If we believe Graham's theory that
> slightly sharp fifths are better than slightly flat fifths, we might
> go for this one, and it extends nicely to higher limits.
>
> Both 12-ET and 29-ET
> > are outside the typical range of schismic. Between schismic and
> 12-ET is
> > the "number 114" temperament [<1, 2, -1, -8|, <0, -1, 8, 26|]...
>
> I thought you wanted to name this grackle?
>
> > Since all of these are consistent with 27/65-LT (if we ignore the
> > inconsistency of 65-ET), it would be nice to be able to name all of
> them
> > in relation to 27/65. How about giving the range of generators in
> > parentheses?
>
> It's a little strange to base names for 7-limit temperaments on 65,
> which as you say is not 7-limit consistent.
>
> > 27/65-LT (-4..2) = [<1, 2, 4, 2|, <0, -1, -4, 2|] dominant
> > 27/65-LT (-1..8) = [<1, 2, -1, 2|, <0, -1, 8, 2|] schism
> > 27/65-LT (-1..14) = [<1, 2, -1, -3|, <0, -1, 8, 14|] schismic
> > 27/65-LT (-1..26) = [<1, 2, -1, -8|, <0, -1, 8, 26|] number 114
>
> Garibaldi (your schismic) and schism are related by a 12-et nexus, and
> garibaldi and grackle (number 114) by another 12-nexus, leading to a
> contorted 24-et nexus between schism and grackle. Dominant, however,
> has a 12-et nexus with meantone and a 7-et nexus with flattone, which
> shows where it belongs. So far as pontiac goes, schism has a 41-et
> nexus, garibaldi a 53-et nexus, and grackle a 65-et nexus, with
> dominant, no surprise, not having a nexus.
>
> By the way, looking at my table of the 99th row of Farey, I find 58/99
> in this sharp fifth region also; 29&70. It's basically 99-equal
> arranged as a chain of fifths.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Gene,
> >
> >
> > i'm glad that you have a term for this concept, because
> > it should have one. but altho "nexus" is entirely
> > appropriate, i vote that we use something else, because
> > Partch already established "numerary nexus" as an important
> > term in tuning theory, and it's quite different from your
> > term.
>
> Hmm, "node" maybe? Anyone want to add to the jargon explosion?

i love "node".

(
brings to mind a view of my favorite plant ...

offtopic:

http://www.cannabis.com/growing/

http://www.sweetleaf.co.uk/grow_cuttings.php

something related to Paul's "floragrams" too, i bet ...
)

write up a good 1/2 "for dummies" (and please try as
hard as you can to make them *real* dummies) and
1/2 technical math jargon definition for me.

and use a separator line.

----------

if it's not too much trouble, include links to recent
posts on the topic.

(i was just writing one that you could have pointed to,
but it accidentally got zapped. i'll try to recall it.)

-monz

🔗monz <monz@attglobal.net>

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

> i was talking to Chris today (my Tonalsoft partner) about
> this pivot-ET idea.
>
> there's an analogy to old-fashioned "common-practice"
> harmony. there, composers used common-tones as the
> basis of chord-progression, and common-chords as the
> basis of tonal modulation.
>
> in this modern age of polymicrotonality, where we are
> aware of and can easily quantify the mathematics of our
> tunings, we can use the pivot-ETs in the same way, to
> transition between different tuning-families.

that is, "transition between different tuning-families"
*within the same piece*.

i had written some good stuff here in a post that
unfortunately got zapped before i finished it,
and i don't remember it now.

but here's where i left off ...

i was illustrating a sequence of tuning modulations
via various ETs.

one example:

(start)
v
17edo as 5-limit schismic
v
12edo as 5-limit schismic > 12edo as 5-limit meantone
v
31edo as 5-limit meantone > 31edo as 11-limit miracle
v
72edo as 11-limit miracle

maybe someone else out there would like to join in
the fun and figure out how to transition back

at first i called this "tuning modulation", then
thought of calling it "tuning transition", with the
idea of really using only "transition" with the
idea of shifting between tunings understood.

but i prefer to add to the jargon explosion and come up
with a single-word term for this, because i think it's
important enough to warrant its own term.

my initial favorite was "surf" ... and it still
like it, but its modern meaning is so ingrained as
internet browsing that i decided not to use it.

so i spent some time searching the thesaurus

http://thesaurus.reference.com/search?q=transition

and decided on transfer (my preference) or swing
(more colorful).

another, more "dissonant" procedure, is to assign
each instrumental or vocal part a different tuning,
grouping different parts according to similarity
of tuning-family and/or similarity of ET-cardinality.

-monz

🔗monz <monz@attglobal.net>

🔗Carl Lumma <ekin@lumma.org>

>> So, one possibilty is that we agree on different complexity bands,
>> and for each band of complexity (and for each number of periods per
>> octave and each odd or prime limit) we divide the entire generator
>> spectrum up among the contending linear temperaments, with no gaps
>> and no overlaps. The boundaries (pivots) will always be at ETs.
>>
>> Then we step up to the next complexity level, and do it all over
>> again, this time with more temperaments. Some temperaments at the
>> next level get to keep the same name, but with "complex" prepended.
>
>One thing which we could use to possibly make the complexity bands
>unnecessary is that bounding by logflat badness produces a finite list
>of possibilities, without regard to complexity. So, for instance, if
>we chose a TOP badness of 100 in the 7-limit, as I was doing, you
>would only get a finite list of temperaments with generator 3/2 and
>period octave, another with generator 3/2 and period 1/2 octave,
>another with generator 1/2 of a fifth and period an octave, and so
>forth.

I thought the whole point of logflat badness was that it returned
an *infinite* number of temperaments.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> >One thing which we could use to possibly make the complexity bands
> >unnecessary is that bounding by logflat badness produces a finite list
> >of possibilities, without regard to complexity. So, for instance, if
> >we chose a TOP badness of 100 in the 7-limit, as I was doing, you
> >would only get a finite list of temperaments with generator 3/2 and
> >period octave, another with generator 3/2 and period 1/2 octave,
> >another with generator 1/2 of a fifth and period an octave, and so
> >forth.
>
> I thought the whole point of logflat badness was that it returned
> an *infinite* number of temperaments.

It produces an infinite list, but just barely. If you place the
further restriction on it that the generator has to be mapped from
some JI interval such as a fifth, it no longer gives you an infinite
list. Hence it can serve as a way of obtaining all reasonable systems
with a fifth as generator, where "reasonable" is defined relative to
alternative systems. Grackle may not be such a great temperament,
though as we have seen it is useful for producing circulating
temperaments, but you are going to find only a short list of
temperaments which are better relative to their complexity and have a
fifth as generator. The same would be true for 6/5, 44/39, etc. as a
generator.

I think the fact that there are, in some sense, only a finite number
of temperaments with any given kind of generator which are
"interesting" is, well, interesting. Dave may not think much of
pontiac as a temperament, but if you use it to survey 3/2 as a
generator this method tells you you *must* include it, but do not need
to keep adding ever more complex temeraments. Pontiac is, in some
sense which does not rely very much on subjective notions or theories
about which complexity ranges are worth looking at, an interesting
temperament with the fifth as a generator; and it is nice to know we
do *not* need to rely on ad hoc measures, moats with or without
crocodiles, or complexities beyond which we will not go to get a
finite list.

🔗Carl Lumma <ekin@lumma.org>

>> >One thing which we could use to possibly make the complexity bands
>> >unnecessary is that bounding by logflat badness produces a finite list
>> >of possibilities, without regard to complexity. So, for instance, if
>> >we chose a TOP badness of 100 in the 7-limit, as I was doing, you
>> >would only get a finite list of temperaments with generator 3/2 and
>> >period octave, another with generator 3/2 and period 1/2 octave,
>> >another with generator 1/2 of a fifth and period an octave, and so
>> >forth.
>>
>> I thought the whole point of logflat badness was that it returned
>> an *infinite* number of temperaments.
>
>It produces an infinite list, but just barely. If you place the
>further restriction on it that the generator has to be mapped from
>some JI interval such as a fifth, it no longer gives you an infinite
>list.

I'd considered that. I could see this if you allow infinitely many
generator-size regions, but it sounds like you're carving pretty
broad swaths, and finitely many of them. Can you show why it should
be finite with techniques similar to those used in the seminar
Paul linked to:

http://www.maths.bris.ac.uk/~maadb/research/seminars/online/fgfut/fgfut00.html

If not, I'll just trust you. :)

>I think the fact that there are, in some sense, only a finite number
>of temperaments with any given kind of generator which are
>"interesting" is, well, interesting. Dave may not think much of
>pontiac as a temperament, but if you use it to survey 3/2 as a
>generator this method tells you you *must* include it, but do not need
>to keep adding ever more complex temeraments. Pontiac is, in some
>sense which does not rely very much on subjective notions or theories
>about which complexity ranges are worth looking at, an interesting
>temperament with the fifth as a generator; and it is nice to know we
>do *not* need to rely on ad hoc measures, moats with or without
>crocodiles, or complexities beyond which we will not go to get a
>finite list.

Agreed, surely.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> I'd considered that. I could see this if you allow infinitely many
> generator-size regions, but it sounds like you're carving pretty
> broad swaths, and finitely many of them.

A fifth is just 3/2, so it doesn't carve any kind of swath. Of course,
you need to count it more than once--first as octave and fifth, then
as half-octave and fifth etc. You can also check for half-fifth, etc.

As you go up in complexity, the mistunings of the fifth or whatever
other generator you are looking at which you will allow become
increasingly small. Any regular temperament has generators which are
approximately something or other so far as the JI limit it maps from
are concerned, and by not considering anything too freakish you in
effect put a complexity bound on your temperaments.

Here are some less-familiar generators for some 7-limit temperaments:

pajara: 7/5 and 3/2
hemiwuerschmidt: 28/25
blackwood: 3/2 and 5/4
sensi (semisixths): 9/7
hemififths: 49/40
amity: 105/64
waage: 3/2 and 6/5
negri: 15/14 and 16/15 (secor)
valentine: 21/20
unidec: 567/400 and 10/9
porcupine: 10/9
octacot: 21/20

🔗Carl Lumma <ekin@lumma.org>

>> I'd considered that. I could see this if you allow infinitely many
>> generator-size regions, but it sounds like you're carving pretty
>> broad swaths, and finitely many of them.
>
>A fifth is just 3/2, so it doesn't carve any kind of swath.

3/2 is 3/2, a fifth can be a lot of things.

>As you go up in complexity, the mistunings of the fifth or whatever
>other generator you are looking at which you will allow become
>increasingly small.

Aha! But clearly you're admitting a region for the fifth here.
It just vanishes. So in effect you are carving generator space
down to something like the reals.

>Any regular temperament has generators which are
>approximately something or other so far as the JI limit it maps from
>are concerned, and by not considering anything too freakish you in
>effect put a complexity bound on your temperaments.

Yes indeedie.

>Here are some less-familiar generators for some 7-limit temperaments:
>
>pajara: 7/5 and 3/2
>hemiwuerschmidt: 28/25
>blackwood: 3/2 and 5/4
>sensi (semisixths): 9/7
>hemififths: 49/40
>amity: 105/64
>waage: 3/2 and 6/5
>negri: 15/14 and 16/15 (secor)
>valentine: 21/20
>unidec: 567/400 and 10/9
>porcupine: 10/9
>octacot: 21/20

Hmm, what do these have to do with the price of tea in China?

-Carl

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> > Sorry about that; I added "225/224" to the above. The
> > name "schismic" has been depreciated in favor of "schismatic",
> > and I made that change in the above also.
>
>
> hmm ... i missed that. so then are we using "schismatic"
> for the family of temperaments, and "schismic" for the
> c.1400's type of pythagorean tuning which used Gb=F# etc.?
> that's what i would propose.

If you look up "schismic temperament" on Wikipedia, you get redirected
to "schismatic temperament". If you wonder why that is, you will find
this first sentence:

``Schismatic temperament is the temperament which results from
tempering the schisma of 32805:32768 to a unison. ("Schismic" is an
idiosyncratic spelling or abbreviation, promulgated in the late 1990s
by Graham Breed, and having no precedent in the literature.)''

Possibly that should go on the discussion page, or at least not in the
first sentence, but so it reads now.