Yahoo Tuning Groups Ultimate Backup tuning Catching up on the Regular Margo Paradigm discussions

> OK, I don't think there's any way I can get through all of the
> discussion that's transpired since I had to dip out for the hurricane.

Dear Mike,

First, thank you so much for your comments here -- and I hope that you
and your family are OK!

> So to try and address the essential point to Margo, regarding our
> different notions of "temperament":

This is a very well-written and intuitive presentation, and a great
starting point for discussion!

> The things we call temperaments are just mappings. For instance, if
> we say something like "there are two generators, and one of them is
> 2/1 and the other is 3/2, and four of the 3/2's map to 5/1," that
> uniquely specifies 5-limit meantone temperament. This is the same
> thing as stating "81/80 vanishes", due to some mathematical
> identities that are very deep. Any set of vanishing commas can
> automagically be turned into a mapping and vice versa.

This is a good example. And there are the kinds of strange mappings,
or implementations, that Graham gets into also

For example, when I was getting started, people would sometimes talk
about "positive meantones" with the tempering of the fifth given as a
_negative_ syntonic comma value. For a totally hypothetical example,
suppose that the nearer fifths of George Secor's 17-tone
well-temperament at 707.220 cents were somehow described as "close to
-1/4-comma meantone"!

The obvious reply: "That's the wrong comma -- it's actually close to
1/5-Archyan comma, as shown by the small semitones or thirdtones
around 64 cents, very close to 28/27."

But I get the point that in theory, one _could_ call this -1/4-comma
meantone or whatever: the mapping defines the temperament, however
curious the implementation.

Are we on the same wavelength, here?

> Some people use the word "temperament" to mean something else, so
> to clarify these are the things I called "temperament families."
> But for now I'm going to call them "temperament classes"
> instead. This is because Gene has another object called a
> "temperament family" which is really like a "temperament family
> family" under the old notation; it tells how different temperament
> classes interrelate and evolve into one another. So the thing I
> called "temperament family" and also "abstract temperament," which
> we just call "temperament" sometimes, I will call "temperament
> class" from now on to make it clear. Again, note that it's not the
> same thing you called a temperament.

Actually, there's a possible example of what _might_ be a "temperament
family" (or maybe "temperament clan," recalling a term someone used
for a superfamily of galaxies in a novel called _Tau Zero_) that I've
been advocating for some years. Let's see if this is an example of
what Gene has in mind.

As I learned around the year 2000, the term "meantone" tends to
suggest specifically tempering out the 81:80, as opposed to the 64:63
or 896:891 whatever, as a way to map four 5ths up to 5/4 specifically,
rather than 9/7 or 14/11 or whatever.

Suppose we chose the term "eventone" as a more general temperament
family name for a set of classes where 4 fifths up represents some
kind of regular major third: 5/4, 9/7, 14/11, you name it. The classes
within that family could meantone for 5/4; superpyth (if I'm right)
for 9/7; parapyth for 14/11; and so forth.

In fact, this precisely ties in with Jake's original post about "Three
similar temperaments" for 5/4, 9/7, or 14/11 major thirds -- a similar
structure, but with different thirds and, more specifically, commas!
So my proposal would be to call that structure or mapping "eventone,"
and the GWS "temperament family" or group of classes the eventone
family.

> That being said, the really important point is that our
> "temperament classes" don't have a defined range of "valid
> tunings." For instance, the meantone temperament class isn't only
> valid for octaves around 1200 cents and generators around 696
> cents. You can tune the octaves to 1 cent and the fifths to
> 120837102873 cents if you want, so long as you make sure that one
> of these "octaves" is still being called 2/1, that one of these
> "fifths" is still being mapped as 3/2, and that four "fifths" is
> still being mapped to 5/1. So different temperament classes can be
> assigned different "tuning maps," and there is no limit on which
> tuning maps are mathematically acceptable. However, different
> tuning maps can be -rated- by how much "error" they have. So for
> instance, the tuning map corresponding to 1 cent octaves and
> 120837102873 cent fifths would have an absurdly high error - so
> high that nobody would ever use it.

Well, among other things, anyone who used it must have a taste for
really wide voice spacing, a bit beyond the usual 16th-century
part-writing conventions of Zarlino or Morley <grin>.

But I get the point of an abstract template or mapping where we can
pour in any values we want.

It's a bit like Gene's great example of pouring the POTE Parapyth
generators into the mapping of my MET-24 -- it could have been O3, or
Peppermint for that matter, and the result would have been the same.
This may be a bit different, though, because what MET-24 gave was the
2x12-MOS form -- we agree that a lattice is potentially infinite in
all of the relevant dimensions, e.g. 2x17-MOS, or 2x29-MOS, etc.

> You'd be better off setting the octaves to 1200 cents and the fifths
> to 696 cents, which has much lower error. You may ask, why the heck
> would we want to do it this way? That's musically useless, isn't it?
> Don't we want to assign temperament classes some kind of "valid
> tuning range?" And the answer is yes - mapping 120837102873 cents as
> 3/2 is almost certainly musically useless. But, if you think about
> it, it makes more sense to do it this way, because nobody can agree
> on what the valid tuning range is - so we'll just let anyone tune
> things how they like, see what the error is for each tuning, and
> make their own judgments.

We're much in agreement -- and Graham nicely addresses this too --
that it's good to separate defining a structure from debating what are
good, bad, marginal, or optimal parameters. Better a descriptive than
a prescriptive approach!

If we agree that descriptive rather than prescriptive is a good thing,
the one argument that occurs to me from another perspective is that
sometimes a bit of optional and friendly taxonomy might not hurt,
although there's the counterargument that taxonomy can quickly become
controversial at best and prescriptive in a very unpleasant way at
worst.

Suppose, for example, that someone tunes a rank-2 temperament at 707.8
cents and calls it "a syntonic comma temperament." Believe it or not,
some people came up with a theoretical system which did just about
exactly this, classifying any mapping from 7-EDO to 5-EDO with four
5ths up representing a major third as "syntonic."

Now a prescriptive approach would say, "That's a horrible attempt at a
meantone; you're totally outside the valid range, which surely ends
by the time we reach a just 3/2!"

I understand the RMP approach might say, "You've chosen values with a
very high error for a meantone or syntonic comma temperament, but yes,
you _can_ call it that as long as the mapping is there."

And a "gentle taxonomy" appraach might say, "You've chosen a fine
temperament where, like a meantone or syntonic temperament, we map a
major third to four 5ths up. But generally we call that a superpyth or
Archytan comma temperament, because here you seem to be tempering out
64/63 rather than 81/80. Note, for example, that your major second
from 2 fifths up is 215.6 cents, somewhere between 9/8 and 8/7.
In a meantone, we'd expect it to be between 10/9 and 9/8. So I wonder
if this is meant to be an unusual meantone, or a very fine superpyth."

In other words, if someone thinks they are speaking standard French
but what we hear is an excellent standard Spanish, a bit of friendly
taxonomy might be helpful, "That would be a rather curious French, but
sounds like the Spanish of a native of Ecuador!"

A compromise between these two descriptive viewpoints: how about a set
of sample tunings for each temperament class, including the POTE and
TOP, and also some others showing a variety of "styles' -- not to stop
people from doing something a bit different, but to give an idea of
the usual ranges of variation even while leaving the class itself
abstract, and not subject to any limits.

> For instance, there's mavila temperament,

> for instance, which says that four fifths minus two octaves maps to
> 6/5, not to 5/4 - a similar tuning as to what gamelan and balafon
> ensembles use. This leads to very flat fifths, around 675 cents.

Yes, a bit like pelog fifths. My experience is that a bit of
timbre manipulation can make these sound quite "3/2-ish" --
I haven't tried this with Mavila, but with 23-EDO, for example.
And 11-EDO seems to be right on the edge of possibility, at least
with my crude timbral techniques.

<http://www.bestII.com/~mschulter/RhapsodyForDanStearns1.mp3>

> Some people have no problem saying that those 675 cent fifths are
> 3/2, others insist that it's actually more like 28/19 or what have
> you. Who's right and who's wrong? The answer is, nobody's right and
> nobody's wrong. We trust that people can make their own decisions.

The best policy! I'm familiar with this debate about fifths around 675
cents, for example in relation to 16-EDO, where claims were sometimes
heard that Easley Blackwood was "wrong" about 675 cents being a
recognizable fifth. My own experience with 23-EDO tells me he was
right -- but a theory of temperament might want wisely to let people
agree on a framework and carry on the debates from there.

> You can map it as 3/2, and just note that that 3/2 is going to have
> really high error. Or, you can map it as 28/19, and it'll be lower
> error. It's up to you. The take home point is that we're not naming
> tunings, just mappings. So Margo, what are you naming,
> specifically? Specific scales of specific tunings of mappings, or
> something more abstract? Thanks, Mike

The answer is that most of the time, I'm simply naming a specific
scale, or maybe better gamut, the kind that a Scala file defines:
Peppermint (2/1 and 704.096-cent fifth courtesy of Kennan Pepper), O3,
MET-24, etc.

And understand, under the old paradigm that I grew up with, so to
speak, we wouldn't even ask the question, "What are you describing, a
tuning or a mapping or an abstract temperament." Back then, a tuning
was pretty much a tuning, and the presumption was that you were
describing something definable in a Scala file.

But, in retrospect, _mappings_ were often the essence of what was
going on.

For example, my e-based temperament in 2000 -- "temperament" meaning
at that time simply a tuning system (2/1, 704.607 cents) -- had
mappings which included four 5ths up as a regular major third at 14/11
(896/891); and 15 fifths up as a 7/4 minor seventh (14680064/14348907).
The last mapping didn't seem much on the radar screens at the time:
people might recommend 1/4-comma or Kornerup or whatever for 7, or
else 22-EDO or the like. "Temper the fifths at around 704.5-705.0
cents, and use 15 fifths up as 7/4," wasn't such a familiar idea, as
far as I could see at the time.

But's let's consider that 14/11 mapping.

Now the textbooks and articles on "multiple divisions" (EDO's in
today's parlsance) would address 46-EDO from a 2-3-5 perspective, but
just try to find even a brief mention in that literature, "One notable
property of the 46-tone division is that 14/11 is virtually just."

So, in retrospect, it's a new mapping. And this can make a difference
for newbies and others.

If a "major third" always means the best approximation of 5/4 in a
given system, then 46-EDO is "hard" for a newbie because of the
complex mapping; but, assuming the newbie is used to a major third at
around the range of 5/4-400 cents, say, finding a congenial style is
"easy" once the mapping becomes familiar (and maybe some comma pumps
are finessed or deliberately exploited).

If a "major third" in 46-EDO Gentle maps to 14/11, then the "newbie"
will find it easy to navigate the system -- no different than in
meantone, just a different major third and comma. However, to use RMP
terminology -- and this is my turn to practice being descriptive
rather than prescriptive -- someone wanting to play meantone-like
music premised on a major third of 5/4-400 cents may find this mapping
rather high in its level of error, let us say.

And then the question might be one of either manipulating timbre to
"tame" those wide major thirds (as Ivor Darreg did with an instrument
for 17-EDO), or looking into a different musical style where active
thirds will fit just fine -- e.g. Machaut rather than Palestrina or
even Bach.

But from an RMP perspective, I guess, we just name the possible
mappings and commas, and let people decide what fits. And, of course,
it's possible to have guides that give the abstract temperament and
then comments about experiences that different people have had.

Here I'm addressing your question of what I've generally meant by
these terms, not what the usage could or should be. And traditionally,
we might speak of "meantone temperament" generally -- but maybe most
naturally as referring to the process of designing or tuning a
meantone, with "meantone temperaments" as the actual tunings.
At least, that's my first impression of the old-style usage.

Best,

Margo

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 5:21 PM, Margo Schulter <mschulter@...> wrote:
>
> First, thank you so much for your comments here -- and I hope that you
> and your family are OK!

Yep, much better than expected! Philly was mostly spared the wrath of Sandy.

> But I get the point that in theory, one _could_ call this -1/4-comma
> meantone or whatever: the mapping defines the temperament, however
> curious the implementation.
>
> Are we on the same wavelength, here?

Right, exactly. It's probably better described as 1/5-comma superpyth
or something, but you could call it -1/4-comma meantone if you want.

> As I learned around the year 2000, the term "meantone" tends to
> suggest specifically tempering out the 81:80, as opposed to the 64:63
> or 896:891 whatever, as a way to map four 5ths up to 5/4 specifically,
> rather than 9/7 or 14/11 or whatever.

Right, exactly.

> Suppose we chose the term "eventone" as a more general temperament
> family name for a set of classes where 4 fifths up represents some
> kind of regular major third: 5/4, 9/7, 14/11, you name it. The classes
> within that family could meantone for 5/4; superpyth (if I'm right)
> for 9/7; parapyth for 14/11; and so forth.

So one thing is that RMP is agnostic on interval categories like
"major third"; the only thing it's mapping is ratios. What counts as a
major third, for instance? Is 13/10 a major third or a perfect fourth?
Is 16/13 a major third? Is major thirdness an inherently perceptual
thing or a culturally learned thing? From RMP's perspective, there's
nothing special that inherently groups 5/4, 9/7, and 14/11 together
(or 7/6 and 6/5 together, or 8/7 and 9/8 and 10/9 together, etc). In
fact, there are temperaments that perceptually group 7/6 and 8/7
together, and ones which distinguish abruptly between 7/6 and 6/5 .

I wouldn't mind calling temperaments where the optimal tuning yields
an LLsLLLs MOS "eventone." But my question is, how does this sort of
classification/family scheme tie over to other temperaments where the
generator might not even be 3/2, for instance? Many of the more
interesting temperaments aren't generated by 3/2, such as porcupine,
for instance. Others, such as hedgehog in 22-EDO (the scale is 3 3 2 3
3 3 2 3) so abruptly distinguish 7/6 and 6/5 that they don't
perceptually seem to aggregate into the same interval class at all,
and yet 11/9 and 6/5 now sound like intonational variants of the same
thing. How do we organize these types of temperaments into families?

> In fact, this precisely ties in with Jake's original post about "Three
> similar temperaments" for 5/4, 9/7, or 14/11 major thirds -- a similar
> structure, but with different thirds and, more specifically, commas!
> So my proposal would be to call that structure or mapping "eventone,"
> and the GWS "temperament family" or group of classes the eventone
> family.

So in this case, Gene's temperament families deal with higher-limit
extensions of temperament classes into new ones. For instance, there's
2.3.5 meantone, which eliminates 81/80. There are then a number of
7-limit "extensions": one which eliminates 126/125, so that C-A# is
now 7/4; one which eliminates 36/35, so that C-Bb is now 7/4; one
which eliminates 525/512 so that C-Bbb is 7/4, etc. Then for each of
those there are further 11-limit extensions and so on. All of these
are different nodes in the meantone family. I'm currently trying to
extend this so that instead of always going from 5-limit to 7-limit to
11-limit and so on, we can also go from the 2.3.7 subgroup to the full
2.3.5.7 limit as an extension, or from 2.3.11 to 2.3.5.11 to
2.3.5.7.11, or something like that.

> It's a bit like Gene's great example of pouring the POTE Parapyth
> generators into the mapping of my MET-24 -- it could have been O3, or
> Peppermint for that matter, and the result would have been the same.
> This may be a bit different, though, because what MET-24 gave was the
> 2x12-MOS form -- we agree that a lattice is potentially infinite in
> all of the relevant dimensions, e.g. 2x17-MOS, or 2x29-MOS, etc.

Right so, are MET-24, O3, and Peppermint the names of specific scales?
Or specific mappings? Also, is a tuning specified?

> If we agree that descriptive rather than prescriptive is a good thing,
> the one argument that occurs to me from another perspective is that
> sometimes a bit of optional and friendly taxonomy might not hurt,
> although there's the counterargument that taxonomy can quickly become
> controversial at best and prescriptive in a very unpleasant way at
> worst.

I wouldn't mind a taxonomy of the different tunings that a temperament
offers, though I'm not sure how to classify them in a way that applies
to all of the strange and exotic temperament classes that are out
there.

> Suppose, for example, that someone tunes a rank-2 temperament at 707.8
> cents and calls it "a syntonic comma temperament." Believe it or not,
> some people came up with a theoretical system which did just about
> exactly this, classifying any mapping from 7-EDO to 5-EDO with four
> 5ths up representing a major third as "syntonic."

Right, I'm familiar with this "syntonic temperament" name. That's a
good example of why I think it makes more sense to keep the naming of
mappings and tunings separate. And if you're going to name tunings,
don't name them after vanishing commas which imply mappings!

> And a "gentle taxonomy" appraach might say, "You've chosen a fine
> temperament where, like a meantone or syntonic temperament, we map a
> major third to four 5ths up. But generally we call that a superpyth or
> Archytan comma temperament, because here you seem to be tempering out
> 64/63 rather than 81/80. Note, for example, that your major second
> from 2 fifths up is 215.6 cents, somewhere between 9/8 and 8/7.
> In a meantone, we'd expect it to be between 10/9 and 9/8. So I wonder
> if this is meant to be an unusual meantone, or a very fine superpyth."

Right; I think that most people do this intuitively. In my head, for
instance, if the fifth gets sharp of 12-EDO it's superpyth, and if
it's flat of 12-EDO it's meantone. Once it's flat of 19-EDO it becomes
flattone, and once it gets flat of 7-EDO it's mavila. Once it goes
south of 16-EDO we're firmly into pelogic territory. But this is just
my intuition built up from years of studying these maps of optimal
tuning ranges; is there a systematic way of working this out for
arbitrary temperaments?

> A compromise between these two descriptive viewpoints: how about a set
> of sample tunings for each temperament class, including the POTE and
> TOP, and also some others showing a variety of "styles' -- not to stop
> people from doing something a bit different, but to give an idea of
> the usual ranges of variation even while leaving the class itself
> abstract, and not subject to any limits.

I agree that this is a good idea and any in-depth discussion of a
temperament should definitely do this. I do have a bit to say on the
different tunings for mavila and porcupine, two temperaments which I'm
fairly familiar with, just like we all have a bit to say on the
different tunings for meantone. The only reason that POTE is what's
listed on the wiki is that it can be easily computed and printed out
for any arbitrary temperament, whereas it probably takes some
experimentation from a real live human being to be able to write a bit
about the musical possibilities afforded by variations in tuning. The
Facebook Xenharmonic Alliance group has seen a good amount of in-depth
discussion like this, with a good amount of conversation about
different mavila tunings and such. I even did these retunings of
pieces of common music to different parts of the mavila spectrum:

9-EDO http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments-9-edo/
25-EDO http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments-25-edo/
16-EDO http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments-16-edo/
23-EDO http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments-23-edo/

My impression is that 25-EDO version was melodically "clearer" in a
way in terms of establishing new perceptual interval categories that
are "stable" and don't run over and interfere with one another - but
the tradeoff is that the intonation for 25-EDO is really bad. In
contrast, 23-EDO has better intonation, but it's sometimes hard for me
to tell between the interval of a ~522 cent fourth and that of three
stacked ~158 cent whole steps. 16-EDO is a nice tradeoff between the
two. I note the "clearness" of 25-EDO comes out most when I play it a
certain way myself though; in these particular examples I think I
found 16-EDO clearest.

> Yes, a bit like pelog fifths. My experience is that a bit of
> timbre manipulation can make these sound quite "3/2-ish" --
> I haven't tried this with Mavila, but with 23-EDO, for example.
> And 11-EDO seems to be right on the edge of possibility, at least
> with my crude timbral techniques.
>
> <http://www.bestII.com/~mschulter/RhapsodyForDanStearns1.mp3>

Right, exactly! There's a whole Hornbostel/armodue/mavila/pelog
spectrum just like there's a flattone/meantone/superpyth/etc spectrum.

> The best policy! I'm familiar with this debate about fifths around 675
> cents, for example in relation to 16-EDO, where claims were sometimes
> heard that Easley Blackwood was "wrong" about 675 cents being a
> recognizable fifth. My own experience with 23-EDO tells me he was
> right -- but a theory of temperament might want wisely to let people
> agree on a framework and carry on the debates from there.

If there's one thing that I hope to convey in this conversation, it's
that I think the question of what's a "recognizable fifth" is a really
deep, meaningful question that touches on some of the farthest and
most complex areas of music cognition yet imagined. What it is that
constitutes an interval's recognizable "identity," and does so in a
way that multiple ratios can fit into it, is a pretty deep and
possibly subjective question. My goal is to develop an approach for
establishing new "stable" sets of interval identities, whatever they
might be, because I have a hunch that this is where the real jackpot
is to be found in the "xenharmonic" ideal - though I'm still not sure
what the rules of such perceptual stability are.

So this is why I'm sometimes hesitant to further enshrine 12-EDO based
categories into any theory and why I'm so much more enamored with
scale structure than ratios for the moment; I need to find a nice
informational framework to establish a raw structure for music so that
my puny human brain can figure out what's going on when I hear things.
Or well, I need to find one other than 12-EDO, anyway.

> The answer is that most of the time, I'm simply naming a specific
> scale, or maybe better gamut, the kind that a Scala file defines:
> Peppermint (2/1 and 704.096-cent fifth courtesy of Kennan Pepper), O3,
> MET-24, etc.

OK, I see. So if MET-24 is a scale, is MET itself a temperament (with
a specific tuning)?

Do these three scales correspond to the same mapping, or do they have
different mappings? Maybe we could abstract them to Peppermint, O3,
and MET temperament classes.

> And understand, under the old paradigm that I grew up with, so to
> speak, we wouldn't even ask the question, "What are you describing, a
> tuning or a mapping or an abstract temperament." Back then, a tuning
> was pretty much a tuning, and the presumption was that you were
> describing something definable in a Scala file.
>
> But, in retrospect, _mappings_ were often the essence of what was
> going on.

Right, so I'm coming into this much later; I'm just happy to see if I
can connect the dots between various preexisting ideas a little bit. I
think we're converging now.

> For example, my e-based temperament in 2000 -- "temperament" meaning
> at that time simply a tuning system (2/1, 704.607 cents) -- had
> mappings which included four 5ths up as a regular major third at 14/11
> (896/891); and 15 fifths up as a 7/4 minor seventh (14680064/14348907).
> The last mapping didn't seem much on the radar screens at the time:
> people might recommend 1/4-comma or Kornerup or whatever for 7, or
> else 22-EDO or the like. "Temper the fifths at around 704.5-705.0
> cents, and use 15 fifths up as 7/4," wasn't such a familiar idea, as
> far as I could see at the time.
>
> But's let's consider that 14/11 mapping.

OK, so for 2.3.7.11, this works out to be this as of yet unnamed temperament:

http://x31eq.com/cgi-bin/rt.cgi?ets=29_46&limit=2.3.7.11

What should this be named, E-base temperament? Euler temperament?
(Looks like below you call it "gentle" temperament?)

So then we can organize this into a temperament family by seeing what
extensions there are on larger subgroups, in this case 2.3.5.7.11. You
can also say, if you want, that 5/1 is 21 fifths, giving you Leapday
temperament as an extension:
http://x31eq.com/cgi-bin/rt.cgi?ets=29_46&limit=11. Or, you can say
that 5/1 is 8 fourths, making this an extension of schismatic
temperament, which also has no name (what should it be named?):
http://x31eq.com/cgi-bin/rt.cgi?ets=29_46c&limit=11. So these are both
extensions of the E-base temperament family.

> Now the textbooks and articles on "multiple divisions" (EDO's in
> today's parlsance) would address 46-EDO from a 2-3-5 perspective, but
> just try to find even a brief mention in that literature, "One notable
> property of the 46-tone division is that 14/11 is virtually just."
>
> So, in retrospect, it's a new mapping. And this can make a difference
> for newbies and others.

Do you mean that the literature at the time never thought to venture
beyond the 5-limit for something like 46-EDO?

> If a "major third" always means the best approximation of 5/4 in a
> given system, then 46-EDO is "hard" for a newbie because of the
> complex mapping; but, assuming the newbie is used to a major third at
> around the range of 5/4-400 cents, say, finding a congenial style is
> "easy" once the mapping becomes familiar (and maybe some comma pumps
> are finessed or deliberately exploited).

Do you mean the mapping of 14/11 to four fifths?

> If a "major third" in 46-EDO Gentle maps to 14/11, then the "newbie"
> will find it easy to navigate the system -- no different than in
> meantone, just a different major third and comma. However, to use RMP
> terminology -- and this is my turn to practice being descriptive
> rather than prescriptive -- someone wanting to play meantone-like
> music premised on a major third of 5/4-400 cents may find this mapping
> rather high in its level of error, let us say.

Sure, or they could alter their concept of major third to not always
be 5/4, the same as in 22-EDO. For instance, in 22-EDO, I don't expect
the sorts of major chords that are easily reachable by the circle of
fifths to be really concordant (because they're supermajor chords).
But I do expect that 22-EDO circle of fifth "major thirds" fit nicely
into 4:7:9 chords.

> And then the question might be one of either manipulating timbre to
> "tame" those wide major thirds (as Ivor Darreg did with an instrument
> for 17-EDO), or looking into a different musical style where active
> thirds will fit just fine -- e.g. Machaut rather than Palestrina or
> even Bach.

There's also something like 32-EDO, where the "major chords" become
10:13:15. 10:13:15 isn't much more complex than 10:12:15, and I
actually find it to be a bit relaxing - in a certain way they're less
harsh than 17-EDO or 22-EDO's "major" chords.

> But from an RMP perspective, I guess, we just name the possible
> mappings and commas, and let people decide what fits. And, of course,
> it's possible to have guides that give the abstract temperament and
> then comments about experiences that different people have had.

Well, in all fairness, people don't usually say that 46-EDO supports
meantone temperament, so it's all worked out so far :)

> Here I'm addressing your question of what I've generally meant by
> these terms, not what the usage could or should be. And traditionally,
> we might speak of "meantone temperament" generally -- but maybe most
> naturally as referring to the process of designing or tuning a
> meantone, with "meantone temperaments" as the actual tunings.
> At least, that's my first impression of the old-style usage.

Well, as long as we're all on the same page now about temperaments,
temperament classes, families, etc., I'm happy :)

But when people talk about "meantone temperament" in the literature,
they do mean the more abstract temperament class, right? Not just
specifically 1/4-comma meantone or something?

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Margo Schulter <mschulter@...>

Dear Mike,

Please forgive me for excerpting only a small part of my draft of
a reply in this message. This is mainly about Gentle temperament
class and its subclasses Parapyth and Euler -- in reply to some
questions you've asked.

>> First, thank you so much for your comments here -- and I hope that you
>> and your family are OK!

> Yep, much better than expected! Philly was mostly spared the wrath
> of Sandy.

This is good news to hear!

> Right, exactly. It's probably better described as 1/5-comma
> superpyth or something, but you could call it -1/4-comma meantone
> if you want.

Agreed. By the way, one issue with a truly "culturally impartial"
mapping system is either flexibly categorizing intervals based on
ratios or cents (as I often tend to do) or by primes -- even "abstract
primes" -- can be difficult to avoid. And you are right to make it an
explicit concern!

> OK, so for 2.3.7.11, this works out to be this as of yet unnamed
> temperament:
> [50]http://x31eq.com/cgi-bin/rt.cgi?ets=29_46&limit=2.3.7.11 What
> should this be named, E-base temperament? Euler temperament? (Looks
> like below you call it "gentle" temperament?)

Please let me go through this, if I can, both concisely and step by
step. First, here's a bit of documentation of what I think could very
appropriately be called Euler -- the abstract temperament or class, of
which the e-based tuning (2000) is one example, 63-EDO another, your
nearby POTE yet another, and 80-EDO still another at about the top of
the range.

<http://tech.groups.yahoo.com/groups/tuning/message/14361>
<http://tech.groups.yahoo.com/groups/tuning/message/20573>
<http://tech.groups.yahoo.com/groups/tuning/message/23881>
<http://tech.groups.yahoo.com/groups/tuning/message/24488>
<http://tech.groups.yahoo.com/groups/tuning/message/18098>

The key here is a rank-2 path to 7 through 13-14-15 generators.
Taking what I understand to be an RMP worldview, I'd say that Gentle
is a temperament class or possibly superclass or whatever with two
main members: Parapyth using a rank-3 mapping for 2.3.7.11.13; and
Euler using a rank-2 mapping.

Gentle class tempers out 896:891, 352:351, 364:363, and 10648:10647.
In either Parapyth or Euler, the two subclasses or whatever, taking
3/2 as the linear generator, 4 generators map to 14/11, and -3
generators to 13/11. Gentle class also tempers out 28672/28431,
so that 7-8-9-10 generators produce intervals at or near ratios
such as 14/13-21/13-63/52-189/104. This makes it possible to obtain
some neutral intervals within a single 12-note chain which are
supplemented by others from the spacing generator in Parapyth rank-3,
but serves as the main pathway in rank-2 Euler. Thus we have at
least as many neutral intervals in Gentle (Parapyth or Euler) as
we have schismatic intervals in a Pythagorean tuning of the same
size, or 7-based intervals in a meantone tuning around 1/4-comma
or 31-EDO.

Parapyth rank-3 typically ranges from around 29-EDO to 702.2 cents
say. By a pure 14/11 tuning (704.377 cents) or the almost identical
46-EDO, the rank-2 Euler strategy starts to get us close to 7/6 and
7/4, while we get a third from -13 generators at 443 cents or around
128/99, a really intriguing ratio.

In Parapyth, the spacing generator acts as the 28/27 step, among other
things. In Euler, the 12-note diesis is large enough to serve this
purpose -- as a quite narrow step around 33/32 around 46-EDO or a pure
14/11 tuning, but as a "natural spacing" very similar to Parapyth in
the main Euler range of around 704.6-705.0 cents.

Gentle class refers to an undecimal/tredecimal eventone with 14/11 and
13/11 as the common theme.

The term "ChristmasEve" might be used, not so much as the name of a
class, as an informal term for the area around 704.377 cents (or
46-EDO) used in a generally 2.3.7.11.13 manner, in contrast to the
13-limit Leapday.

So we have, in outline form:

I. Gentle class: 14/11, 13/11 from 4 and -3 generators
Commas tempered out: 896/891, 352/351, 364/363, 10648/10647,
28672/28431 (the last is important!)
Primes 2.3.7.11.13

A. Parapyth, mostly rank-3 (say 29-EDO to 704.2 cents)
These observe 169/168
(14/13, 7 generators; 13/12, -5 generators + spacing)
1. Peppermint (2/1, 704.096, 58.680)
2. O3 (2/1, 703.893, 57.148)
3. MET-24 (2/1, 703.711, 57.422)
4. POTE (2/1, 703.856, 58.339)

B. Euler, rank-2 (optimal range at 704.5-705.0 cents)
These temper out 169/168 (7 generators best 14/13, 13/12)
1. e-based tuning (2/1, 704.607)
2. POTE (2/1, 704.753)
3. Minimax with 7/6 eigenmonzo (2/1, 704.776)
4. 9/7 eigenmonzo (2/1, 704.994)

Tempering out the 28672/28431 to get neutral rather than 5-limit
schismatic thirds is rather like tempering out the 81/80 in
meantone to get 5/4 rather than 81/64: the idea of tempering is
to get something a bit different than we would with pure fifths.

In fact, meantone and what is now called Gentle have curious
parallels. With meantone, the idea was to take Pythagorean
schismatic thirds and make them the regular thirds from 4 or -3
generators, so that as many thirds as possible would be smooth.

With Gentle, the idea is to have -8 and +9 generators produce
intriguing and rather complex thirds, different from 14/11 and
13/11 at +4 and -3 generators, but congenial in having an active
and forward-going quality, for example in medieval European
cadences. Here are two posts, the second addressing this point,
from the year 2000, with the curious terminology showing how this
part of the spectrum was sort of up for grabs. The discussion of
"alternative thirds" is in the second part:

</tuning/topicId_11096.html#11096>
</tuning/topicId_11108.html#11108>

And I'd just add for now that while the literature on 46-EDO did
sometimes discuss higher primes, the regular and near-pure 14/11
was rarely mentioned as any kind of asset: the idea that someone
might use a 4-generator mapping for the major third wasn't
considered. George Secor, in tuning the 8 more remote fifths
of his 17-WT (1978) at 704.377 cents for some pure 14/11 thirds,
was really setting a landmark -- and with a wonderful and peerless
example of this form of well-temperament.

With many thanks,

Margo

🔗Margo Schulter <mschulter@...>

HI, Mike.

>> First, thank you so much for your comments here -- and I hope that you
>> and your family are OK!

> Yep, much better than expected! Philly was mostly spared the wrath
> of Sandy.

This is good news to hear!

> Right, exactly. It's probably better described as 1/5-comma
> superpyth or something, but you could call it -1/4-comma meantone
> if you want.

Agreed. By the way, one issue with a truly "culturally impartial"
mapping system is that either flexibly categorizing intervals based on
ratios or cents (as I often tend to do) or by primes -- even "abstract
primes" -- can be difficult to do without some viewpoint, with one
person's viewpoint often being another's "detected bias." And you are
right to make it an explicit concern!

> So one thing is that RMP is agnostic on interval categories like
> "major third"; the only thing it's mapping is ratios.

Here we may be running into a paradox: for the RMP or my "Neoclassical
Mapping Paradigm" as I might call it, not to mention lots of other
approaches, biases may not be so easy to avoid.

For example, 29-EDO has something very close to 10:13:15 (as you
mention below for 32-EDO also). Viewing 248 cents as a quite
inaccurate 8/7 or 7/6 might be less conducive to the spirit of the
tuning than viewing it as 15/13, or maybe just 248 cents, or as a
"hemifourth" (which it is, literally, in this system). And "I want an
appropriate or exact hemifourth" is a lot more congenial mapping
concept here than "I want an inaccurate representation of prime 7."

So the Neoclassical approach, if that's the term, has the strength
that it's as easy to talk about seeking 15/13 or more generally a
hemifourth as it is to talk about seeking 7, but the disadvantage that
it does involve some interval category concepts, even if open to
revision to fit this or that cultural scheme.

The RMP, in contrast, by using primes, may imply certain expectations
simply by the choice of primes. And I suspect that there may be a
certain Partchian worldview involved at some level, Partch being a
great musician and theorist, but definitely one not afraid to make
value judgments about what was or was not an apt tuning!

For example, consider what I might term a Partchian Anomalous
Saturated Suspension (PASS), more specifically 12:14:18:21.
Evidently Paul Erlich and Graham and whoever else developed this PASS
concept noted the interesting fact that this routine 2.3.7.9 sonority
doesn't contain 5, and yet can't have 5 added while having each
interval stay within the 9-odd limit (7:4, 3:2, 3:2, 7:6, 9:7, 7:6).

Mathematically interesting, but not so "anomalous" if one takes
2.3.7.9 or 2.3.7.9.11.13 for granted, where it could be a close
analogous of a dominant seventh chord in major/minor harmony. So PASS
could simply be a recognition of a curious quirk of 12:14:18:21 from a
certain Partchian perspective, or in Paul Erlich's decatonic harmony;
but the "anomalous" part can sound a bit curious when used frequently
as a term for something which is totally an everyday reality from a
2.3.7.9 perspective -- maybe a "subgroup subculture" like mine <grin>.
The PASS name makes it clear that the "anomaly" is in the eyes of a
Partchian perspective, making the term both amusing and edifying from
anyone's viewpoint, I think.

And some of the RMP and xenwiki terminology, I should emphasize, is
wonderful, like "Swiss tetrad" for 12:14:18:21. This is one of my
favorite sonorities for a cadence, which I now call a Bernese cadence,
where the outer 7/4 contracts by stepwise contrary motion to a 3/2,
and the 7/6 thirds to unisons, while the middle 9/7 third expands to a
fifth. Bernese cadences are tricolor: they may be intensive (ascending
28/27 semitones or thirdtones), remissive (descending 28/27 steps), or
equable (steps close to 14/13 and 13/12 in Parapyth, or equal and
around 132-135 cents in Euler, where 169:168 is tempered out).

But "Swiss tetrad" and "Bernese cadence" have now become part of my
standard vocabulary. Although it might have been intoned closer to
Pythagorean than to septimal, there's actually an example of the
progression I'm describing near the end of the first section of
Perotin's _Sederunt Principes_. A melodic phrase gets repeated twice,
the second time with strong cadence where the sonority E-G-B-D
resolves to the fifth F-C: that's the Bernese cadence, apart from the
question of septimal intonation. The melodic phrase in the audible
upper melodic line is C-D-E-F D-E-C-B D-D C. (Some performers might
have used a Bb inflection giving us the more tense tritonic sonority
E-G-Bb-D resolving to F-C, as distinct from the Bernese form.)

That maybe illustrates another possible difference in viewpoints when
people discuss tunings. The RMP is often out to do really
groundbreaking things: every other new tuning might be a new "abstract
temperament" or whatever. In contrast, I'm exploring intonational
nuances and variations for a cadence 800 years old. In fact, a 12-MOS
of MET-24 is meant to temper as little as possible while meeting other
goals of the system (e.g. 14/13 within three cents of just), so that
the fifths and fourths aren't too far from the ideal of medieval
Pythagorean tuning. From that perspective, it's more Neoclassical than
radical.

> What counts as a major third, for instance? Is 13/10 a major third
> or a perfect fourth? Is 16/13 a major third? Is major thirdness an
> inherently perceptual thing or a culturally learned thing? From
> RMP's perspective, there's nothing special that inherently groups
> 5/4, 9/7, and 14/11 together (or 7/6 and 6/5 together, or 8/7 and
> 9/8 and 10/9 together, etc). In fact, there are temperaments that
> perceptually group 7/6 and 8/7 together, and ones which distinguish
> abruptly between 7/6 and 6/5 .

The question about 16/13 is an excellent one; my own perception is
that 16/13 defines about the top of the "central neutral range," with
21/17 or 26/21 as a large neutral or "submajor" third, which could be
described as a special kind of "majorness" different from that of 5/4,
14/11, or 9/7. Even around 16/13, the fact that some Arab theorists
have spoken of Rast as "the Arab major scale" suggests that you have a
point! But I wouldn't put it under the heading of prime 5, any more
than I would call it 14/11 or 9/7, for example. And that's where
mapping notation can make a difference.

The grouping together of 7/6 and 8/7 is very important in
understanding a Near Eastern modality such as Hijaz as practiced by
some Turkish musicians, or likewise Persian Esfahan: there's a large
middle step in the tetrachord, 8/7-ish for some and 7/6-ish for
others, with myriad gradations and hemifourth interpretations right
around 250 cents or so as well. The Arab symbol of 5 quartertones, if
taken as Amine Beyhom does to mean 230-270 cents or so, really does
express the category quite well.

A humorous story is that one excellent writer on 13th-century Near
Eastern theory noted that some sources said a mode called Buzurg
(somewhat analogous in its lower tetrachord to modern Persian Esfahan)
was in this tetrachord 14:13-8:7-13:12, while another gave
12:11-7:6-22:21. This author thought that the first version might be
just a different notation for the second tuning, used by a theorist
who didn't favor a 7:6 chromatic step as part of his modal theory!

But measurements of modern Turkish Hijaz or Persian Esfahan suggest
that both theorists may have been right: "somewhere between around 225
and 275 cents" or whatever may have been the practice then, as now.

Your point about 6/5 and 7/6 as not necessarily equivalent is also a
good one! Some books on historical European temperaments assert that
it is impossible for a minor third to become a "wolf" -- in a meantone
system or well-temperament, it will always be acceptably close to
either 6/5 or 7/6.

But in the 16th century, even someone as radical as Vicentino with his
taste for neutral thirds around 11/9 (his estimate of the approximate
size of this irrational meantone interval) found "minimal thirds,"
probably close to 7/6, rather leaning to dissonance, and to be used
sparingly. Praetorius called them "wolves." As a lover of 7/6, I need
to keep in mind that the enthusiasm of Huygens was not a universal
view in the 16th-17th centuries!

> I wouldn't mind calling temperaments where the optimal tuning
> yields an LLsLLLs MOS "eventone." But my question is, how does this
> sort of classification/family scheme tie over to other temperaments
> where the generator might not even be 3/2, for instance?

Good question, and that's an area where Graham's ideas and those of
others involved in the RMP could be really helpful.

> Many of the more interesting temperaments aren't generated by 3/2,
> such as porcupine, for instance. Others, such as hedgehog in 22-EDO
> (the scale is 3 3 2 3 3 3 2 3) so abruptly distinguish 7/6 and 6/5
> that they don't perceptually seem to aggregate into the same
> interval class at all, and yet 11/9 and 6/5 now sound like
> intonational variants of the same thing. How do we organize these
> types of temperaments into families?

It might be difficult.

> Right so, are MET-24, O3, and Peppermint the names of specific
> scales? Or specific mappings? Also, is a tuning specified?

Since this was the main question I addressed in my first reply last
night, which in a nutshell confirms that these are all simply tunings,
what I'd like to do here is look at why this kind of question would be
very natural to ask from an RMP perspective, and rather surprising
from a Neoclassical perspective like mine where it would go without
saying that they are tunings unless I call them "tuning families" or
"tuning styles" or the like.

From Graham's writings and the RMP literature generally, I sense you
are in what Thomas Kuhn might call the musical equivalent of a
scientific revolution, as the term "paradigm" implies. In such a
revolutionary situation, any new tuning (or "temperament," as we would
often call it, e.g. 2/7-comma meantone, or Kornerup, etc.) might also
represent a "new temperament" in your sense of a new mapping scheme.
Your literature lists scores or even hundreds of these, and you aspire
to find many more.

For me, it's mostly Kuhn's normal science in "the 704-cent
neighborhood," now known as Gentle (i.e. Parapyth and Euler). The
territory is familiar, indeed my everyday habitation as it has been
for over a decade (apart from some diversions with modified meantone
circles, another story), so it's mostly fine-tuning and exploring
variations on medieval European polyphony or maqam music, etc.

True, I did have a mildly revolutionary period, around 2000-2002. That
was like a brief Kuhnian revolution -- and then back to normal
science! The revolution basically had two parts.

The first part, with medieval European Pythagorean intonation as my
starting point as an ardent medievalist, happened in 2000: the e-based
tuning (2/1, 704.607), with regular thirds near 14/11 and 13/11 (not
so radically different from 81/64 and 32/27), and 15 fifths up as 7/4.

Actually, the most "revolutionary" part of the e-based tuning (and
various others in that region) wasn't so much the relatively subtle
shift in the color of regular thirds to 2.3.7.11.13 rather than
Pythagorean ratios, but what happens at -8 and 9 generators! Rather
than schismatic 5-limit thirds, we have neutral ones, or more
specifically large and small neutral or submajor/supraminor, "reverse
Pythagorean thirds" as I termed them, because like 14/11 and 13/11
they are rather complex and active. The system thus had a kind of
consistent ethos, with both the regular and "alternative" thirds
nicely supporting medieval European cadences from _imperfectly_
concordant and somewhat tense sonorities involving thirds and/or
sixths to stable intervals like fifths or ideally a complete 2:3:4.

Now from my perspective, the idea of announcing an "abstract
temperament" and then deciding on one more tunings for it seems very
curious: for me, the tuning, e.g. e-based, comes first, and then maybe
at some point a realization that this might be different enough that
it could define a new genre or family. I tend often to proceed by
pragmatic induction, starting with tunings and looking for larger
patterns from there.

Getting back to the "velvet revolution" I experienced in 2000-2002,
the e-based tuning in 24 notes (with 13-14-15 generators for
9/7-7/6-7/4) had a feature that, in a sense, led naturally to the
second phase of the revolution.

Mapped to my 24-note arrangement with two 12-note MIDI keyboards,
e-based had the feel of (2/1, 704.607, 55.283). Only the first two
generators are independent, the third being defined as the 12-note
diesis, but it looked and felt and sounded like a spacing parameter.
And the "natural" spacing served as 28/27, among other things.

In an article in the early 2001 on the e-based tuning, I noted that it
was possible, for example, to place two 12-note chains of 17-EDO at
the arbitrary distance of 55.106 cents to obtain pure 7/6 thirds. In
fact, it's in the last footnote to the first post (March 30, 2001) in
this series on the e-based tuning:

<http://tech.groups.yahoo.com/groups/tuning/message/20573>
<http://tech.groups.yahoo.com/groups/tuning/message/23881>
<http://tech.groups.yahoo.com/groups/tuning/message/24488>

But my learning of George Secor's 17-tone well-temperament during the
summer of 2001, and our actual contact near the end of that summer,
was the catalytic ingredient for the second step, with George's 29-HTT
as a model and "how-to." More specifically, the 2.3.7.11.13 subset of
his tuning had a rank-3 structure of (2/1, 703.579, 58.090).

In addition to learning of that model for what I would soon do, I also
learned with his mentoring and through our collaboration about things
like the equable divisions, we called them, of 12:13:14 or 11:12:13
(harmonic, as here, or arithmetic) -- as well as the wonderful world
of Near Eastern music.

So it was a revolution, but not a disorienting one: a rich neomedieval
outlook became yet richer by taking into account Near Eastern as well
as medieval European musics! No paradigm was broken: but my universe
indeed expanded.

And this led by June of 2002 to Peppermint: simply taking a 12-note
chain of Pepper/Wilson (2/1, 704.096), and tuning a second identical
chain at what we now call a spacing parameter of 58.680 cents for a
pure 7/6 and a virtually just 12:13:14 division.

At that point, the deed had been done. I thought of Peppermint, a very
specific tuning (to me, temperament usually means the same thing), as
maybe representing a new genre of neomedieval tuning because of the
high accuracy for 2.3.7.11.13 -- with George Secor's influence leading
me often to think in terms of these ratios.

So Peppermint, the second stage and completion of the "revolution,"
involved the elements of the 704-cent neighborhood, ratios of
2.3.7.11.13 with my own idiomatic focus on superparticular neutral
seconds used in divisions of a 7/6 or 13/11 minor third (e.g. 14:13:12
or 13:12:11); and a focus on Near Eastern as well as medieval European
music.

In fact, the division of the fourth 33:36:39:42:44 expresses to me
much of the essence of Peppermint and later related tunings like O3
and MET-24; and in Secorian style, one could also look at the hexad
4:6:7:9:11:13.

Here's a post reporting this "mini-revolution" marked by Peppermint --
again simply the name of the tuning.

<http://tech.groups.yahoo.com/groups/tuning/message/38440>

From there on, it's mostly Kuhn's normal science, with O3 and MET-24
simply as refinements, like yet another new and "optimal" shade of
meantone!

A strange analogy: but our dialogue seems a bit like that between a
group of researchers on the cutting edge of supercollider research in
particle physics, and an 18th-century architect. The first is a
pursuit leading to radical new discoveries, while the second involves
applying and fine-tuning mostly familiar principles.

In fact, as far as I know, the term meantone emerged sometime around
the 17th or 18th century: but in the late 15th and 16th centuries, it
might be called simply _participatio_ or _systema participata_ (the
latter term occurring in the 17th century, at any rate): basically, a
system for "sharing out" commas, or more specifically the 81/80.

Likewise, with the e-based tuning or Peppermint and the like, I look
at it as a style or ethos of temperament: "the 704-cent neighborhood,"
or neomedieval tuning, or two 12-note chains at a tempered 28/27
apart, etc. Left to myself, I probably wouldn't think of giving the
genre a proper name.

However, since learning of what we now call the RMP project, I have
wanted the e-based and Peppermint tunings, or more generally the
styles of neomedieval tuning they represent, to be a part of the RMP
catalogue. That it's happening now is very exciting, because it means
a broader scope for your paradigm, and also an opportunity for people
to learn about a fuller range of options.

> I wouldn't mind a taxonomy of the different tunings that a
> temperament offers, though I'm not sure how to classify them in a
> way that applies to all of the strange and exotic temperament
> classes that are out there.

Maybe these are two distinct but complementary projects, like theories
of generative or universal grammar and descriptive studies of the
details of a specific language. At this point Parapyth is close to
being my mothertongue, as it were -- or more precisely, a tuning in
which I can speak my compositional mothertongue, 13th-14th century
European polyphony, easily and fluently (by my standards, not
necessarily that high as musicianship goes!).

But it's like my settled literary language, with modified meantone
(Zest-24) as a second language, while RMP people often seem to delight
in writing each new novel or even short story in a different language!

> Right, I'm familiar with this "syntonic temperament" name. That's a
> good example of why I think it makes more sense to keep the naming
> of mappings and tunings separate. And if you're going to name
> tunings, don't name them after vanishing commas which imply
> mappings!

We're agreed there!

> Right; I think that most people do this intuitively. In my head,
> for instance, if the fifth gets sharp of 12-EDO it's superpyth, and
> if it's flat of 12-EDO it's meantone. Once it's flat of 19-EDO it
> becomes flattone, and once it gets flat of 7-EDO it's mavila. Once
> it goes south of 16-EDO we're firmly into pelogic territory. But
> this is just my intuition built up from years of studying these
> maps of optimal tuning ranges; is there a systematic way of working
> this out for arbitrary temperaments?

This kind of map of the spectrum -- not to be confused with an RMP
mapping -- is something I can relate to! I love timelines and the
like, sometimes associating street numbers with years or historical
events.

> I agree that this is a good idea and any in-depth discussion of a
> temperament should definitely do this. I do have a bit to say on the
> different tunings for mavila and porcupine, two temperaments which
> I'm fairly familiar with, just like we all have a bit to say on the
> different tunings for meantone. The only reason that POTE is what's
> listed on the wiki is that it can be easily computed and printed out
> for any arbitrary temperament, whereas it probably takes some
> experimentation from a real live human being to be able to write a
> bit about the musical possibilities afforded by variations in
> tuning.

Certainly the POTE is useful as one example, and the Parapyth one is
very helpful, a fine tuning of this class. In fact, the POTE adds to
something like O3 (with an almost identical generator), because it
shows how spacing can be done in different ways, with different
intervals favored (e.g. 9/7 in the POTE, a really fine choice for
2-3-7-9 generally).

With Mavila, by the way, your TOP or POTE at rank-2 is an interesting
variation on the Wilson-Grady approach of using a regular temperament
to approximate a much freer form of tuning -- much as Parapyth might
very imperfectly approximate the irregular tunings favored for Persian
tar or santur. It might be good to give Chopi tunings also that people
have emulated.

> Right, exactly! There's a whole Hornbostel/armodue/mavila/pelog
> spectrum just like there's a flattone/meantone/superpyth/etc
> spectrum.

True! Hornbostel wrote some fascinating things about interval
perception, and there's the playful quip about how people are still
debating whether his hypothesis about the possible acoustical origins
of these small fifths might be a bit overblown :). But he certainly
focused on a vital facet of many musical traditions.

> OK, I see. So if MET-24 is a scale, is MET itself a temperament (with
> a specific tuning)?

> Do these three scales correspond to the same mapping, or do they
> have different mappings? Maybe we could abstract them to
> Peppermint, O3, and MET temperament classes.

As has by now likely become clear, that would be a bit like different
temperament classes for 1/4-comma meantone, Kornerup's Golden
Meantone or the POTE, and 2/7-comma meantone.

A humorous way to put this is that you seem to be crediting me with a
lot more originality than I deserve!

It's sort of like a biologist discovering three new species -- or
possibly rediscovering them, since George Secor's 2.3.7.11.13 portion
of 29-HTT pretty much provided a model for Peppermint -- and then
being asked, "Are those three new species or three new phyla?"

> Right, so I'm coming into this much later; I'm just happy to see if
> I can connect the dots between various preexisting ideas a little
> bit. I think we're converging now.

Yes. And part of that convergence may be a realization that different
terminologies may reflect different experiences and research programs,
so to speak.

With RMP, a new tuning might well be a new abstract temperament as
well. With a more traditional agenda -- and mine likely is -- a new
tuning is likely to be mostly normal science, with the temperament of
the fifth rising or falling like hemlines in fashion.

A quick recap of my first response last night would be that Euler
class or subclass -- with Gentle as a larger class including both
rank-2 Euler and rank-3 Parapyth -- is a very appropriate name.
And e-based temperament is simply an Euler class tuning.

What I might just add here is that from a keyboardist's point of view
-- or at least mine, with two 12-note MIDI keyboards rather than a
generalized keyboard -- Euler and Parapyth have the same key
mappings, with the e-based tuning, for example, the equivalent of
(2/1, 704.607, 55.283). The difference is only that the spacing isn't
an independent generator in Euler, but is defined by the 12-tone
diesis.

So the keyboard map in Euler (e.g. e-based tuning, 2000) is the same
as in Parapyth (e.g. Peppermint, 2002).

For 7/6, it's tone plus spacing (e.g. D-E*). For 11/8, it's fourth
plus spacing (e.g. C-F*). And in either Euler or Parapyth, we can get
a nice Rast with either B-C#-Eb-E on either keyboard, or C*-D*-E-F*.
The difference is that in Euler, a rank-2 system with full
transposibility within the range of the 24-note chain of fifths, these
two forms of Rast are identical (e.g. 0-209-363-495 cents in e-based);
in Parapyth, the first has a brighter neutral third than the second
(e.g. 0-207-370-496 or 0-207-357-497 in MET-24).

So with Parapyth rank-3, we save almost a cent of tempering, and get
more shadings of neutral intervals. But Euler has its own advantages,
and not just for newbies or oldsters who might prefer a somewhat
simpler and more fully transposible system, itself, of course, a
legitimate consideration.

For example, Parapyth has more neutral shadings but also, in its
24-note version, fewer locations for each shading. We don't quite have
enough generators in a single chain for something as simple as a
submajor flavor of Rast with standard steps for inflections: large
neutral sixth, major sixth, minor seventh, and large neutral seventh,
say 104/63, 22/13, 39/22, and 13/7 in MET-24. The problem is that to
get both 22/13 and 104/63, we need mappings of 3 and -9 generators,
but we only have 11 in a chain! So there are creative kludges like
using 12/7 in place of 22/13, or a lower 13/8 in place of 104/63.

But in Euler, such kludges are unnecessary: a single 24-note chain is
long enough to get a bright Rast without cutting corners.

> [51]http://x31eq.com/cgi-bin/rt.cgi?ets=29_46&limit=11. Or, you can
> say that 5/1 is 8 fourths, making this an extension of schismatic
> temperament, which also has no name (what should it be named?):

Whatever I might say about this, odds are that someone will do it,
because it's been done for the 353-cent third in 17-EDO, and that's 33
cents from 5/4, or almost as far as you can get from either 5/4 or 6/5
within the 25/24 region that separates them!

This isn't a new phenomenon, either: in 1555, Vicentino wrote about
how people often confused a "proximate minor third," his term for a
third a meantone diesis larger than minor and associated with an
approximate ratio of 11/9, with a major third, meaning the usual
meantone version at 5/4. It's a point of view, and whoever wants to
take it could design and name a system based on this view of Euler.

To share my own perspective, I'll repeat this link, pointing people to
the discussion of "alternative thirds":

</tuning/topicId_11108.html#11108>

> So then we can organize this into a temperament family by seeing
> what extensions there are on larger subgroups, in this case
> 2.3.5.7.11. You can also say, if you want, that 5/1 is 21 fifths,
> giving you Leapday temperament as an extension:

Actually, in Euler, Leapday gives major thirds ranging from around
1/8-comma meantone quality in e-based (396.745 cents) to an even 400
cents in 63-EDO, very close to the 2.3.7.9 minimax with a 7/6
eigenmonzo or the slightly lower POTE; and 405 cents at 80-EDO or the
top of the region (just 9/7). Just find thrice the size of the
apotome, and you have the size of this third. The 5/4 eigenmonzo is at
704.110 cents, so we're far beyond this point by the time we reach the
Euler POTE.

But the really accurate strategy is -25 generators, with the "double
diesis" for the e-based tuning (twice 55.283 cents, or 110.566 cents)
serving as 16/15, and a third at 384.825 cents. In a 46-MOS, say, this
could be a very effective strategy, and as a curiosity I should add
that the e-based tuning will circulate in 109 notes. At the POTE, or
(2/1, 704.753 cents) we have 381.175 cents, if I'm correct, just a bit
smaller than 7/22 octave.

> Do you mean that the literature at the time never thought to
> venture beyond the 5-limit for something like 46-EDO?

What I mean, referring both to the traditional literature oriented
around 2-3-5 and some discussions focusing on higher primes also, it
seemed generally assumed that "a major third" would generally mean 5/4
rather than 14/11, even in a system like 46-EDO where it was the
virtually just regular third. And whether the ratio of 14/11 was an
acoustically recognizable dyad, a wonderful emblematic ratio
regardless of the answer to the first question, or whatever, was a
question that did get some attention for time to time.

People can be enthusiastic about 4:5:6:7 or even 4:5:6:7:9:11:13 and
not necessarily be attuned to medievalish polyphony, say, with 14/11
and 13/11 as the regular, everyday thirds.

I may be overgeneralizing: but this was the way it seemed around the
year 2000 -- before I knew of George Secor's 17-WT and 29-HTT which he
had designed in 1978! And in a different direction, of course, the
Miracle Tuning was also a rediscovery of the Secor tuning of 1975.

> Do you mean the mapping of 14/11 to four fifths?

What I meant is that if the mapping is to the approximate 5/4, then
that couid be a bit complicated for a newbie, but would fit styles
familiar to lots of people with a classical European background, say.
The comma pumps are another issue, of course.

> Sure, or they could alter their concept of major third to not
> always be 5/4, the same as in 22-EDO. For instance, in 22-EDO, I
> don't expect the sorts of major chords that are easily reachable by
> the circle of fifths to be really concordant (because they're
> supermajor chords). But I do expect that 22-EDO circle of fifth
> "major thirds" fit nicely into 4:7:9 chords.

In a medieval setting, of course, this is easier, because anything
more complex than 3/2 or 4/3 is unstable, and expected to have some
degree of tension and striving to resolve. But I much agree that in
22-EDO, 6:7:9, 4:6:7:9, 12:14:18:21, etc., are all options, for use in
a medieval kind of polyphony or otherwise. And in 22-EDO, since the
64/63 is tempered out, you get lots and lots of them!

In Euler and even more so in Parapyth, we observe the 64/63 and get
higher accuracy -- but with the need for larger tuning sets to get a
reasonable number. That's why a tuning size of 24 is so attractive, or
2x12-MOS, even though 24 itself isn't a MOS like 17.

For 9/7-7/6-7/4-21/16 we need 13-14-15-16 generators, so 24 notes
means we get eight locations for a tempered version of the 1-3-7-9
hexany (1/1-9/8-7/6-21/16-7/4-2/1), and nine locations for the Swiss
tetrad or 12:14:18:21. This is just about right, although 34 notes
(2x17-MOS) would be yet richer with these septimal sonorities.

With 17, we've just begun to get these wonderful intervals: so 24 is a
good size.

> There's also something like 32-EDO, where the "major chords" become
> 10:13:15. 10:13:15 isn't much more complex than 10:12:15, and I
> actually find it to be a bit relaxing - in a certain way they're
> less harsh than 17-EDO or 22-EDO's "major" chords.

My experience with these is in 24-EDO and 29-EDO, for example, and I
agree, with something like 26:30:39 quite suave and unique. Of course,
I wouldn't really look for a "major chord" in the sense of a stable
triad except in something like meantone context: I tend to avoid these
terms in a medieval context, because they can suggest an expectation
of restfulness and conclusiveness that just isn't there. But whatever
one's point of view, the different alternatives and shadings are
interesting.

> Well, in all fairness, people don't usually say that 46-EDO
> supports meantone temperament, so it's all worked out so far :)

True enough :). I don't recall that ever being a controversy here, and
there have been a fair number over the years.

> Well, as long as we're all on the same page now about temperaments,
> temperament classes, families, etc., I'm happy :)

Yes, I'd say "temperament class" or "abstract temperament"
disambiguates nicely on all sides. I guess my own approach might be to
find a concrete temperament in the old sense, and let analysis or
experience suggest where that specific tuning should fit in the larger
scheme of things. But the very nature of the RMP project may lead you
to more frequent instances of really new abstract temperaments -- not
just new tunings and tweakings of established mappings.

> But when people talk about "meantone temperament" in the
> literature, they do mean the more abstract temperament class,
> right? Not just specifically 1/4-comma meantone or something?

Yes, except for some purists who will say: "There's really only one
meantone, and that's 1/4-comma, where the mean-tone is precisely the
mean of 9/8 and 10/9." In my usage, we could speak of either "meantone
temperament" as a general category, or "meantone temperaments" as
specific instances of this category. So "meantone temperament class"
fits nicely.

Best,

Margo

Catching up on the Regular Margo Paradigm discussions

🔗Mike Battaglia <battaglia01@...>

🔗Graham Breed <gbreed@...>

🔗cityoftheasleep <igliashon@...>

🔗Mike Battaglia <battaglia01@...>

🔗cityoftheasleep <igliashon@...>

🔗Margo Schulter <mschulter@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗Margo Schulter <mschulter@...>

🔗Margo Schulter <mschulter@...>

🔗Margo Schulter <mschulter@...>

🔗Margo Schulter <mschulter@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗Margo Schulter <mschulter@...>