Yahoo Tuning Groups Ultimate Backup tuning-math The Regular Mapping Paradigm -- revised

Graham Breed wrote:
> I've updated the page at
> > http://x31eq.com/paradigm.html
> > I might still tweak it a bit, but it's largely in a state that I can > mention it on the main list. Note the glossary and the new section on > inconsistent temperaments. It might help if somebody can give me > alternative names for temperaments that could refer to the same pitches.
> > > Graham

Great stuff! I know just what you mean about changing the way you think about the music you'd like to write. My problem is I've been doing a lot more thinking about the music I'd _like_ to write and less actually _writing_ new music. :) But I can't imagine going back to the old way. Lately I've been using the computer keyboard as a crude generalized keyboard to give me ideas, trying different key mappings for various temperaments.... An appropriate key mapping can turn even something as unwieldy as wuerschmidt temperament into something quite usable....

I'm not sure what you mean by alternative names for temperaments that could refer to the same pitches. My favorite example of an inconsistent temperament is 64-ET, which can alternately be treated as a diminished and a meantone temperament. Or maybe you're referring to something like bug temperament, which if you carry it out to more than 9 notes metamorphoses into what I call superpelog temperament. Bug is mapped [1, 2, 3] [0, -2, -3], but superpelog is [1, 2, 1] [0, -2, 6]. ("Bug" is a great name for a temperament that metamorphoses so quickly, come to think of it!) Another example is father [1, 2, 2] [0, -1, 1] which in one part of its range metamorphoses to sensi / semisixths [1, -1, -1] [0, 7, 9].

Your comment about unifying three different paradigms brings to mind other paradigm shifts that have done similar things, most notably James Clerk Maxwell's 19th-century unification of electricity and magnetism with a set of equations that described how the electric and magnetic fields are related aspects of the same phenomenon. More recently, quantum mechanics has blurred the boundaries between chemistry and physics. I haven't read Kuhn's book, but I'm guessing he might have said something about these.

Do you really want to describe Partch's 43-note scale as "infamous"? That seems a little harsh.

One minor point: although hundreds of regular temperaments have been described, not all of these have been (yet) used for making music. Certanly many of them have been used, and I've played around with less often mentioned temperaments like "muggles" enough to see their potential usefulness for music, but I'd guess no more than a few dozen of these temperaments have seen actual use in music. Off the top of my head I can think of meantone, schismatic, pajara, porcupine, miracle, augmented, diminished, negri, hanson, orwell, mavila, ennealimmal, bug, superpelog, father, and lemba as having had some use. A few rank-3 temperaments like marvel and starling could be added to the list, and you could also count the tempered version of Bohlen-Pierce. But I don't hear a lot of music in sharptone, bipelog, vulture, or misty, to pick a few at random.

And a minor typo: "because it's easy do define the one in terms of the other" should be "because it's easy TO define the one in terms of the other". Also, "it's properties" should be "its properties".

Another minor point: the notation

<< 1, 1, 3, 3, 2],
< 0, 6, -7, -2, 15]]

makes it look like a wedgie. These are really just two distinct components of the mapping, so I'd think a regular square bracket notation would be better, something like this:

[< 1, 1, 3, 3, 2],
< 0, 6, -7, -2, 15]]

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Graham Breed <gbreed@gmail.com>

Herman Miller wrote:

> Great stuff! I know just what you mean about changing the way you think > about the music you'd like to write. My problem is I've been doing a lot > more thinking about the music I'd _like_ to write and less actually > _writing_ new music. :) But I can't imagine going back to the old way. > Lately I've been using the computer keyboard as a crude generalized > keyboard to give me ideas, trying different key mappings for various > temperaments.... An appropriate key mapping can turn even something as > unwieldy as wuerschmidt temperament into something quite usable....

Ah yes, but I've not written more than you over the past few years.

> I'm not sure what you mean by alternative names for temperaments that > could refer to the same pitches. My favorite example of an inconsistent > temperament is 64-ET, which can alternately be treated as a diminished > and a meantone temperament. Or maybe you're referring to something like > bug temperament, which if you carry it out to more than 9 notes > metamorphoses into what I call superpelog temperament. Bug is mapped [1, > 2, 3] [0, -2, -3], but superpelog is [1, 2, 1] [0, -2, 6]. ("Bug" is a > great name for a temperament that metamorphoses so quickly, come to > think of it!) Another example is father [1, 2, 2] [0, -1, 1] which in > one part of its range metamorphoses to sensi / semisixths [1, -1, -1] > [0, 7, 9].

The "something like bug" is what I'm looking for. Ideally where the optimal tunings for different mappings are very close (or even identical).

> Your comment about unifying three different paradigms brings to mind > other paradigm shifts that have done similar things, most notably James > Clerk Maxwell's 19th-century unification of electricity and magnetism > with a set of equations that described how the electric and magnetic > fields are related aspects of the same phenomenon. More recently, > quantum mechanics has blurred the boundaries between chemistry and > physics. I haven't read Kuhn's book, but I'm guessing he might have said > something about these.

Yes, it's a common theme in physics, at least, that a new theory should bring together old ones. Perhaps my presentation's biased by this idea. Another good example is Newton's universal gravitation, which unifies the force that makes things fall when you drop them with the force that makes planets move around.

> Do you really want to describe Partch's 43-note scale as "infamous"? > That seems a little harsh.

Yes, although only in a playful sense. He went on record as saying that the particular notes, and especially the number 43, aren't really that important. But it's always one of the first things people have to say about his music and often stands in for his whole theory. A trend I'm continuing, of course.

> One minor point: although hundreds of regular temperaments have been > described, not all of these have been (yet) used for making music. > Certanly many of them have been used, and I've played around with less > often mentioned temperaments like "muggles" enough to see their > potential usefulness for music, but I'd guess no more than a few dozen > of these temperaments have seen actual use in music. Off the top of my > head I can think of meantone, schismatic, pajara, porcupine, miracle, > augmented, diminished, negri, hanson, orwell, mavila, ennealimmal, bug, > superpelog, father, and lemba as having had some use. A few rank-3 > temperaments like marvel and starling could be added to the list, and > you could also count the tempered version of Bohlen-Pierce. But I don't > hear a lot of music in sharptone, bipelog, vulture, or misty, to pick a > few at random.

Okay, the offending passage now says "... some have even been used for making music." That's still a pretty impressive list for a class of things most people would consider obscure. And you left off magic.

> And a minor typo: "because it's easy do define the one in terms of the > other" should be "because it's easy TO define the one in terms of the > other". Also, "it's properties" should be "its properties".

fixed

> Another minor point: the notation
> > << 1, 1, 3, 3, 2],
> < 0, 6, -7, -2, 15]]
> > makes it look like a wedgie. These are really just two distinct > components of the mapping, so I'd think a regular square bracket > notation would be better, something like this:
> > [< 1, 1, 3, 3, 2],
> < 0, 6, -7, -2, 15]]

I think your right that it's wrong, because one index should be covariant. So I've changed it to

[< 1, 1, 3, 3, 2],
< 0, 6, -7, -2, 15]>

Is that satisfactory? It means you can find the size of an octave

[<1, 1, 3, 2, 2],<0, 6, -7, -2, 15]>[1, 0, 0, 0, 0>
= [<1, 1, 3, 2, 2][1, 0, 0, 0, 0>, <0, 6, -7, -2, 15][1, 0, 0, 0, 0>>
= [1, 0>

and it looks like an interval rather than a mapping. You can also find the mapping for a period

[<1, 1, 3, 2, 2],<0, 6, -7, -2, 15]><1, 0]
= <1, 1, 3, 2, 2]

and the operation isn't ambiguous in the 3-limit.

The wedgie looks like this:

<1, 1, 3, 2, 2]<0, 6, -7, -2, 15]
= <<6, -7, -2, 15, -25, -20, 3, 15, 59, 49]]

And multiplying it by an octave gives a different result:

(<1, 1, 3, 2, 2]<0, 6, -7, -2, 15])[1, 0, 0, 0, 0>
= <0, 6, -7, -2, 15]

Graham

🔗Graham Breed <gbreed@gmail.com>

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>I've updated the page at
>>
>>http://x31eq.com/paradigm.html
>>
>>I might still tweak it a bit, but it's largely in a state that I can >>mention it on the main list. > > It's looking good! A few nitpicks: I would say 72-et supports miracle,
> not that miralce supports 72-et. Also, timbres which correspond to a
> tuning seems to be to be an old idea going back to the Hammond organ.
> I posted a different example with Mysterious Mush.

You could say it either way round. But miracle supports all of 72-et whereas 72-et only supports one specific tuning of miracle. So it's more true to say that miracle supports 72-et than that 72-et supports miracle.

Wherever did I say that timbres corresponding to a tuning was a new idea? Gamelans are tuned that way. It's much older than Hammond organs. What would be new is allowing the tuning and timbre to change together.

> Note the glossary and the new section on > >>inconsistent temperaments. It might help if somebody can give me >>alternative names for temperaments that could refer to the same pitches.
> > Not sure what you want; obiously 12-et, meantone, and dominant can all
> refer to the same pitches.

Yes, and dominant's my example, isnt it? I could use the name there. But dominant will typically have sharper fifths than meantone. I want an example where the typical tunings are identical.

Graham

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Herman Miller <hmiller@IO.COM>

Graham Breed wrote:
> Herman Miller wrote:
>> I'm not sure what you mean by alternative names for temperaments that >> could refer to the same pitches. My favorite example of an inconsistent >> temperament is 64-ET, which can alternately be treated as a diminished >> and a meantone temperament. Or maybe you're referring to something like >> bug temperament, which if you carry it out to more than 9 notes >> metamorphoses into what I call superpelog temperament. Bug is mapped [1, >> 2, 3] [0, -2, -3], but superpelog is [1, 2, 1] [0, -2, 6]. ("Bug" is a >> great name for a temperament that metamorphoses so quickly, come to >> think of it!) Another example is father [1, 2, 2] [0, -1, 1] which in >> one part of its range metamorphoses to sensi / semisixths [1, -1, -1] >> [0, 7, 9].
> > The "something like bug" is what I'm looking for. Ideally where the > optimal tunings for different mappings are very close (or even identical).

Well, TOP bug is 260.26, 1200.00, but that's the 5-limit version. In 5-limit, superpelog is just a double mavila scale with the TOP tuning having a 260.76 cent generator and 1206.55 cent period.

If you take the 7-limit version, the common 7-limit version of bug is beep [<1, 2, 3, 3], <0, -2, -3, -1]] with a TOP tuning of 254.90, 1194.64, which isn't all that close to superpelog. The 7-limit mapping of superpelog is [<1, 2, 1, 3], <0, -2, 6, -1]], so the only difference is the mapping of 5/1. But the TOP beep tuning isn't within the acceptable range for superpelog.

Keemun and catakleismic might be a better example. TOP keemun is 317.83, 1203.19 and TOP catakleismic is 316.91, 1200.54. Also in the range of keemun is parakleismic, with TOP tuning 315.11, 1199.74. Keemun covers a larger range than the more complex tunings, so it can metamorphose into either one or the other depending on the tuning.

keemun [<1, 0, 1, 2], <0, 6, 5, 3]]
catakleismic [<1, 0, 1, -3], <0, 6, 5, 22]]
parakleismic [<1, 5, 6, 12], <0, -13, -14, -35]]

Actually with the TOP parakleismic tuning, beyond keemun you can use the mapping [<1, 5, 6, 7], <0, -13, -14, -16]], which doesn't appear to have a name as far as I can tell, before you get to parakleismic.

As tends to be the case, the tuning of the more complex temperaments is more sensitive. Other cases of complex temperaments in the middle of a range usually associated with less complex temperaments:

garibaldi [<1, 2, -1, -3], <0, -1, 8, 14]] 498.12, 1200.76
kwai [<1, 2, 16, 14], <0, -1, -33, -27]] 495.25, 1199.68

cynder/mothra [<1, 1, 0, 3], <0, 3, 12, -1]] 232.52, 1201.70
guiron [<1, 1, 7, 3], <0, 3, -24, -1]] 233.99, 1200.49

> Okay, the offending passage now says "... some have even been used for > making music." That's still a pretty impressive list for a class of > things most people would consider obscure. And you left off magic.

Right, there are probably a few others, but magic is one I should have thought of.

> I think your right that it's wrong, because one index should be > covariant. So I've changed it to
> > [< 1, 1, 3, 3, 2],
> < 0, 6, -7, -2, 15]>
> > Is that satisfactory? It means you can find the size of an octave

Well, I'm not sure that I really understand this, but it looks less confusing than the other way. I don't know whether an "interval" of mappings makes more sense than a "mapping" of mappings, but at least it can't be confused with a wedgie.

🔗Graham Breed <gbreed@gmail.com>

Herman Miller wrote:

> Well, TOP bug is 260.26, 1200.00, but that's the 5-limit version. In > 5-limit, superpelog is just a double mavila scale with the TOP tuning > having a 260.76 cent generator and 1206.55 cent period.
> > If you take the 7-limit version, the common 7-limit version of bug is > beep [<1, 2, 3, 3], <0, -2, -3, -1]] with a TOP tuning of 254.90, > 1194.64, which isn't all that close to superpelog. The 7-limit mapping > of superpelog is [<1, 2, 1, 3], <0, -2, 6, -1]], so the only difference > is the mapping of 5/1. But the TOP beep tuning isn't within the > acceptable range for superpelog.

It looks like one of the temperament classes I'm not working on now (but used for my MMM Day piece) could be either beep or superpelog then, in that I don't use the 5/1 mapping.

> Keemun and catakleismic might be a better example. TOP keemun is 317.83, > 1203.19 and TOP catakleismic is 316.91, 1200.54. Also in the range of > keemun is parakleismic, with TOP tuning 315.11, 1199.74. Keemun covers a > larger range than the more complex tunings, so it can metamorphose into > either one or the other depending on the tuning.
> > keemun [<1, 0, 1, 2], <0, 6, 5, 3]]
> catakleismic [<1, 0, 1, -3], <0, 6, 5, 22]]
> parakleismic [<1, 5, 6, 12], <0, -13, -14, -35]]

Thanks to you and Gene for diggin these things out. But I don't think I'll use them in the web page. On reflection it'd probably confuse the readers more than it enlightens them.

>>I think your right that it's wrong, because one index should be >>covariant. So I've changed it to
>>
>>[< 1, 1, 3, 3, 2],
>> < 0, 6, -7, -2, 15]>
>>
>>Is that satisfactory? It means you can find the size of an octave
> > Well, I'm not sure that I really understand this, but it looks less > confusing than the other way. I don't know whether an "interval" of > mappings makes more sense than a "mapping" of mappings, but at least it > can't be confused with a wedgie.

It's a function that takes an interval and returns another interval. Hence it's a mapping from intervals to intervals, which is correct. The constituents < 1, 1, 3, 3, 2] and < 0, 6, -7, -2, 15] are functions that take an interval and return a scalar. So they aren't really regular temperament mappings and I can see an argument for calling them "vals" instead. You could also call them equal temperament wedgies, so a mapping is an interval of equal temperament wedgies.

I'd prefer to say that, for example, <12, 19, 28] is the mapping of an equal temperament and [<12, 19, 28]> is the mapping of a rank 1 regular temperament. The logic being that we know that an equal temperament can be indexed by a single integer, but a rank 1 temperament has to use a list of a single generator to be on an equal footing with temperaments of higher rank.

Perhaps distinguishing two previously equivalent terms is worse than inventing a new term. I can certainly see the sense of that if you're going to talk about the distinction a lot. For now, I think it's best to avoid drawing attention to it. If we want to use <12, 19, 28] as the mapping for an equal temperament, it's already apparent that equal temperament mappings are written differently to those of other regular temperaments. So if you want equal temperaments and rank 1 temperaments to be the same thing, the mapping has to be [<12, 19, 28]>. But that's needlessly ugly. So I'll talk about <12, 19, 28] as being an equal temperament mapping, and avoid talking about rank 1 regular temperament mappings.

Alternatively, let's say that strictly speaking <12, 19, 28] is not an equal temperament mapping, but when we're not being strict about the algebra we'll talk about it as being an equal temperament mapping regardless.

Graham

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

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

> > 72-et is a tuning for miracle, therefore 72-et supports miracle.
> > Miracle is not a tuning for 72-et, and hence does not support it.

> Miracle is a mapping for 72-et, therefore miracle supports 72-et.
> 72-et is not a mapping for miracle, and hence does not support it.

It's a matter of precise diction, is all. "Support" is being used
metaphorically, deriving from its primary meaning of bearing the
weight of from below. 72-et can bear the metaphorical weight of
miracle, since it is a good miracle tuning; hence it supports miracle.
"Support" also means to provide for, and 72-et provides for miracle by
supplying it with the means to be heard. What we have here is a notion
that a tuning supports a temperament mapping, and 72 is used
equivocally to mean both tuning and temperament. We even have separate
Wikipedia articles these days.

You want to have "support" mean "has a homomorphism onto", as far as I
can make out; so that a temperament of higher rank supports one of
lower rank which it is mapped to. If we follow this logic out, JI
supports everything. I don't see any good metaphorical meaning here.

🔗Graham Breed <gbreed@gmail.com>

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>>72-et is a tuning for miracle, therefore 72-et supports miracle.
>>>Miracle is not a tuning for 72-et, and hence does not support it.
> > >>Miracle is a mapping for 72-et, therefore miracle supports 72-et.
>>72-et is not a mapping for miracle, and hence does not support it.
> > > It's a matter of precise diction, is all. "Support" is being used
> metaphorically, deriving from its primary meaning of bearing the
> weight of from below. 72-et can bear the metaphorical weight of
> miracle, since it is a good miracle tuning; hence it supports miracle.
> "Support" also means to provide for, and 72-et provides for miracle by
> supplying it with the means to be heard. What we have here is a notion
> that a tuning supports a temperament mapping, and 72 is used
> equivocally to mean both tuning and temperament. We even have separate
> Wikipedia articles these days.

How can a single tuning with a finite number of notes bear the metaphorical weight of an infinite number of tunings with an infinite number of notes?

Miracle provides for 72-et. What it provides is *music*. Do you remember music? You can write music in miracle and play it in 72-et. If you write music in 72-et there's a good chance you won't be able to play it in another miracle tuning. It can also provide a notation and a keyboard layout.

> You want to have "support" mean "has a homomorphism onto", as far as I
> can make out; so that a temperament of higher rank supports one of
> lower rank which it is mapped to. If we follow this logic out, JI
> supports everything. I don't see any good metaphorical meaning here.

Great, so you don't know what I mean, but you've decided I must be wrong anyway.

Graham

🔗monz <monz@tonalsoft.com>

Hi Graham,

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> I've updated the page at
>
> http://x31eq.com/paradigm.html
>
> I might still tweak it a bit, but it's largely in a state
> that I can mention it on the main list. Note the glossary
> and the new section on inconsistent temperaments. It might
> help if somebody can give me alternative names for temperaments
> that could refer to the same pitches.

Wow, this is a great page! Kudos.

Some typos (besides those mentioned by Herman):

------

under "Adaptive Tuning"
http://x31eq.com/paradigm.html#compatadap

"particuler" should be "particular"

------

under "Inconsistent Temperaments"
http://x31eq.com/paradigm.html#incompatinco

"For example, in quarter comma meantone an augmented fourth
(e.g. C-A#) is a good approximation to the just interval 7:4"

"augmented fourth" should be "augmented sixth" (and of course
i prefer to write it "augmented-6th")

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Carl Lumma <ekin@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

monz wrote:
> Hi Graham,
> > --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>I've updated the page at
>>
>>http://x31eq.com/paradigm.html
>>
>>I might still tweak it a bit, but it's largely in a state
>>that I can mention it on the main list. Note the glossary
>>and the new section on inconsistent temperaments. It might
>>help if somebody can give me alternative names for temperaments
>>that could refer to the same pitches.
> > > > Wow, this is a great page! Kudos.
> > Some typos (besides those mentioned by Herman):

Thanks! I should upload the fixed version when I send this. Herman's typos should have been fixed as well. Did I miss some?

Graham

> > ------
> > under "Adaptive Tuning"
> http://x31eq.com/paradigm.html#compatadap
> > "particuler" should be "particular"
> > ------
> > under "Inconsistent Temperaments"
> http://x31eq.com/paradigm.html#incompatinco
> > "For example, in quarter comma meantone an augmented fourth
> (e.g. C-A#) is a good approximation to the just interval 7:4"
> > "augmented fourth" should be "augmented sixth" (and of course
> i prefer to write it "augmented-6th")
> > > > -monz
> http://tonalsoft.com
> Tonescape microtonal music software
> > > > > > > > > > > > > Yahoo! Groups Links
> > > > > > >