Microtonal Wikiversity course?

If you don't know, now you know:
http://en.wikiversity.org/wiki/Wikiversity:Main_Page

Wikiversity is another Wikimedia resource, but this time, it aims to
organize information in virtual online "courses" or "lessons" or what
have you. It seems to be extremely flexible and I would assume can be
laid out however we want.

For a very "structured" example, here's a Wikiversity "Course" in
Electric Circuit Analysis:
http://en.wikiversity.org/wiki/Electric_Circuit_Analysis
And for a "looser" example, here's Wikiversity's "School of Music":
http://en.wikiversity.org/wiki/School:Music -- note the
theory/composition page.

Anyone interested in collaborating to get a Microtonal "course" set
up, made up of different "lessons"? Or maybe a microtonal "department"
with different "courses." Something where there are relevant sections
for all of the constituent fields you have to understand to finally
know what's going on - psychoacoustics, all of the math, some signal
processing, etc, which would lead up to a "class" on Rothenberg's
ideas, and ultimately regular mapping, or something like that. Some
structured approach to learning this stuff.

I am still very new to all of the math involved in regular mapping,
but would be more than willing to contribute for psychoacoustics,
signal processing, Fourier analysis, etc.

Any takers?

-Mike

On 1 August 2010 21:20, Mike Battaglia <battaglia01@...> wrote:

> Any takers?

I think it's a great idea, and we surely have the expertise here to
get it done. But we also have a history of not working together on
unified expositions, so it may be doomed.

There are other things I plan to do this summer. I can't promise I'll
be able to give attention to this. If the momentum gets going I can
comment on the things I know about and maybe work on some articles.

Graham

It would be great!!!!

François

I would contribute - probably at the intersection of 12 and micro

On Sun, Aug 1, 2010 at 4:20 PM, Mike Battaglia <battaglia01@...>wrote:

>
>
> If you don't know, now you know:
> http://en.wikiversity.org/wiki/Wikiversity:Main_Page
>
> Wikiversity is another Wikimedia resource, but this time, it aims to
> organize information in virtual online "courses" or "lessons" or what
> have you. It seems to be extremely flexible and I would assume can be
> laid out however we want.
>
> For a very "structured" example, here's a Wikiversity "Course" in
> Electric Circuit Analysis:
> http://en.wikiversity.org/wiki/Electric_Circuit_Analysis
> And for a "looser" example, here's Wikiversity's "School of Music":
> http://en.wikiversity.org/wiki/School:Music -- note the
> theory/composition page.
>
> Anyone interested in collaborating to get a Microtonal "course" set
> up, made up of different "lessons"? Or maybe a microtonal "department"
> with different "courses." Something where there are relevant sections
> for all of the constituent fields you have to understand to finally
> know what's going on - psychoacoustics, all of the math, some signal
> processing, etc, which would lead up to a "class" on Rothenberg's
> ideas, and ultimately regular mapping, or something like that. Some
> structured approach to learning this stuff.
>
> I am still very new to all of the math involved in regular mapping,
> but would be more than willing to contribute for psychoacoustics,
> signal processing, Fourier analysis, etc.
>
> Any takers?
>
> -Mike
>
>

I'm not heavy into the advanced math, but I can help out with the introductory material. What would the scope of the course be?

Andrew

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 1 August 2010 21:20, Mike Battaglia <battaglia01@...> wrote:
>
> > Any takers?
>
> I think it's a great idea, and we surely have the expertise here to
> get it done. But we also have a history of not working together on
> unified expositions, so it may be doomed.
>
> There are other things I plan to do this summer. I can't promise I'll
> be able to give attention to this. If the momentum gets going I can
> comment on the things I know about and maybe work on some articles.
>
>
> Graham
>

As Mike suggested, I glanced at the theory/composition page.

It's rudimentary.

The reason for that is that the vast majority of people who might make use of something like Wikiversity would be at a beginning level.

To write a decent introduction would require someone (probably not me) with a great deal of patience and the gift for clear explanation. (This latter qualification rules out most of the people who post here--they are well past being interested in basics. It's hard to talk about things you're not interested in. The yen for mathematical shorthand and mathematical elegance makes it difficult to remember how to explain things in simple terms--in terms of other well-known things.)

The basics are worth literally years or a lifetime of activity.

They include equal temperaments other than 12, and Just Intonation. (JI may well keep me busy the rest of my life.)

To get a handle on the basics requires hands-on, with ear training. It requires some kind of electronic keyboard or tone-generator.

Mike, what you're talking about is the post-graduate stuff. *Literally post-grad school stuff.*

For an example of what NOT to do, see this (awful) explanation of Miracle temperament:

http://en.wikipedia.org/wiki/Miracle_temperament.

"In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones."

Explanations need to give a generally-understandable idea, with a clear image, in the first sentence. This one tells you a lot you don't need to know, and gives you no basic understanding.

This is slightly better:

http://x31eq.com/miracle.htm

"Miracle scales fit a 10, rather than 7 or 12, note scale. It consists of 9 small and one slightly larger interval to the octave.

But this page is still full of writing that only insiders will understand, if they do.

Mike, you're unusually gifted, intelligent, educated, and motivated.

Don't underestimate all the things you already know, and tend to take for granted.

Caleb

On Aug 2, 2010, at 8:42 AM, Chris Vaisvil wrote:

> I would contribute - probably at the intersection of 12 and micro
>
>
> On Sun, Aug 1, 2010 at 4:20 PM, Mike Battaglia <battaglia01@...> wrote:
>
> If you don't know, now you know:
> http://en.wikiversity.org/wiki/Wikiversity:Main_Page
>
> Wikiversity is another Wikimedia resource, but this time, it aims to
> organize information in virtual online "courses" or "lessons" or what
> have you. It seems to be extremely flexible and I would assume can be
> laid out however we want.
>
> For a very "structured" example, here's a Wikiversity "Course" in
> Electric Circuit Analysis:
> http://en.wikiversity.org/wiki/Electric_Circuit_Analysis
> And for a "looser" example, here's Wikiversity's "School of Music":
> http://en.wikiversity.org/wiki/School:Music -- note the
> theory/composition page.
>
> Anyone interested in collaborating to get a Microtonal "course" set
> up, made up of different "lessons"? Or maybe a microtonal "department"
> with different "courses." Something where there are relevant sections
> for all of the constituent fields you have to understand to finally
> know what's going on - psychoacoustics, all of the math, some signal
> processing, etc, which would lead up to a "class" on Rothenberg's
> ideas, and ultimately regular mapping, or something like that. Some
> structured approach to learning this stuff.
>
> I am still very new to all of the math involved in regular mapping,
> but would be more than willing to contribute for psychoacoustics,
> signal processing, Fourier analysis, etc.
>
> Any takers?
>
> -Mike
>
>
>

Mike, the ideas you've suggested seem more like a "tuning" related course - perhaps leave the idea of a "microtonal" course for a someone with a more composition-based and music-based approach?

I only mention this because it's rare for a musician, who is curious about microtonal composition, to give a damn about psychoacoustics and "all of the math" etc.

Sweet idea though, I hope this goes somewhere. I don't understand most tuning concepts and this would help!

Sean

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> If you don't know, now you know:
> http://en.wikiversity.org/wiki/Wikiversity:Main_Page
>
> Wikiversity is another Wikimedia resource, but this time, it aims to
> organize information in virtual online "courses" or "lessons" or what
> have you. It seems to be extremely flexible and I would assume can be
> laid out however we want.
>
> For a very "structured" example, here's a Wikiversity "Course" in
> Electric Circuit Analysis:
> http://en.wikiversity.org/wiki/Electric_Circuit_Analysis
> And for a "looser" example, here's Wikiversity's "School of Music":
> http://en.wikiversity.org/wiki/School:Music -- note the
> theory/composition page.
>
> Anyone interested in collaborating to get a Microtonal "course" set
> up, made up of different "lessons"? Or maybe a microtonal "department"
> with different "courses." Something where there are relevant sections
> for all of the constituent fields you have to understand to finally
> know what's going on - psychoacoustics, all of the math, some signal
> processing, etc, which would lead up to a "class" on Rothenberg's
> ideas, and ultimately regular mapping, or something like that. Some
> structured approach to learning this stuff.
>
> I am still very new to all of the math involved in regular mapping,
> but would be more than willing to contribute for psychoacoustics,
> signal processing, Fourier analysis, etc.
>
> Any takers?
>
> -Mike
>

On 2 August 2010 15:31, sevishmusic <sevish@...> wrote:
> Mike, the ideas you've suggested seem more like
> a "tuning" related course - perhaps leave the idea of
> a "microtonal" course for a someone with a more
> composition-based and music-based approach?

Mike was only starting the ball rolling. If you think someone should
do a certain course, maybe that someone is you.

Graham

On Mon, Aug 2, 2010 at 5:39 AM, Graham Breed <gbreed@...> wrote:
>
> On 1 August 2010 21:20, Mike Battaglia <battaglia01@...> wrote:
>
> > Any takers?
>
> I think it's a great idea, and we surely have the expertise here to
> get it done. But we also have a history of not working together on
> unified expositions, so it may be doomed.

We do indeed, and I take some of the blame for just being younger and
stupider and thinking I knew more than I did, back when we were trying
to organize the open source software stuff. I don't really want to
lead this because I don't think I have enough knowledge yet to really
see the big picture. I'm basically just expressing a willingness to
contribute as people feel I am best needed. I would hope that someone
like you or Carl or Gene would organize the "big picture" of it and
see how it all fits.

But at this point I think we've reached some kind of critical mass in
the knowledge we have here, and so maybe it would be good for the
community if we did finally collaborate on something like this. It
would certainly be good for me, as my current mode of education is to
read the conversations between you and Gene on tuning-math and try to
figure out what the hell everyone's talking about.

> There are other things I plan to do this summer. I can't promise I'll
> be able to give attention to this. If the momentum gets going I can
> comment on the things I know about and maybe work on some articles.

If you'd let us adapt some of the stuff on your x31eq.com pages, that
itself might be a good start.

-Mike

On 2 August 2010 17:50, Mike Battaglia <battaglia01@...> wrote:

> If you'd let us adapt some of the stuff on your x31eq.com pages, that
> itself might be a good start.

You can take what you like, but I doubt it would be that useful. If
you try to understand what's there and re-write it in your own words,
that could work.

Another way of starting is to choose the course prerequisites and
objectives. That is, what the students are expected to know when they
start the course and what we want them to learn by the end of it.
Then you'll probably find that what I've written isn't at all
suitable, because it isn't aimed at music students.

Graham

On Mon, Aug 2, 2010 at 9:17 AM, caleb morgan <calebmrgn@...> wrote:
>
> As Mike suggested, I glanced at the theory/composition page.
> It's rudimentary.
> The reason for that is that the vast majority of people who might make use of something like Wikiversity would be at a beginning level.

Sure. I was also thinking that it would be a good resource for us,
though, internally. Maybe we should set it up as a "post-formal
education" department and assume that people have an understanding of
basic music theory. Stuff like "jazz theory" (lol, i hate that term)
shows up here as a use for functional tetradic and pentadic harmony,
but would probably be better under the normal music theory page.

> To write a decent introduction would require someone (probably not me) with a great deal of patience and the gift for clear explanation.  (This latter qualification rules out most of the people who post here--they are well past being interested in basics.  It's hard to talk about things you're not interested in.  The yen for mathematical shorthand and mathematical elegance makes it difficult to remember how to explain things in simple terms--in terms of other well-known things.)

This is really where I hope Carl can get involved, as he is really
good at dumbing things down or explaining them to people who are still
learning. I've been sending him offlist questions for a while because
of it. At the very least, if he'd allow us to adapt (or even just copy
and paste) the condensed tuning-math outline, that would be a great
start.

> The basics are worth literally years or a lifetime of activity.
> They include equal temperaments other than 12, and Just Intonation. (JI may well keep me busy the rest of my life.)
> To get a handle on the basics requires hands-on, with ear training.  It requires some kind of electronic keyboard or tone-generator.
> Mike, what you're talking about is the post-graduate stuff.  *Literally post-grad school stuff.*

Hopefully what will become grad school stuff, though... :)

> For an example of what NOT to do, see this (awful) explanation of Miracle temperament:
> http://en.wikipedia.org/wiki/Miracle_temperament.
> "In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones."
> Explanations need to give a generally-understandable idea, with a clear image, in the first sentence.  This one tells you a lot you don't need to know, and gives you no basic understanding.
> This is slightly better:
> http://x31eq.com/miracle.htm
> "Miracle scales fit a 10, rather than 7 or 12, note scale. It consists of 9 small and one slightly larger interval to the octave.

I agree with this, although the fact that wikiversity gives us some
options with how to design individual "courses" or "lessons" might
make dealing with stuff like this easier. So when we come up with a
lesson on miracle temperament, it might be as part of a sequence that
assumes you've read the "regular mapping" lesson first.

Or, we could have another page that explains microtonal scales as
well, but assuming you haven't been run through the regular mapping
math wringer yet, so it doesn't go into that much detail at first.

> But this page is still full of writing that only insiders will understand, if they do.
> Mike, you're unusually gifted, intelligent, educated, and motivated.
> Don't underestimate all the things you already know, and tend to take for granted.
> Caleb

I appreciate your kind words, and likewise to you. Let's figure out
how we can best contribute here. :)

-Mike

On Mon, Aug 2, 2010 at 10:31 AM, sevishmusic <sevish@...> wrote:
>
> Mike, the ideas you've suggested seem more like a "tuning" related course - perhaps leave the idea of a "microtonal" course for a someone with a more composition-based and music-based approach?
>
> I only mention this because it's rare for a musician, who is curious about microtonal composition, to give a damn about psychoacoustics and "all of the math" etc.
>
> Sweet idea though, I hope this goes somewhere. I don't understand most tuning concepts and this would help!
>
> Sean

Haha, well, I personally am a musician who's interested in these
things... the reason I have written so few microtonal compositions is
mainly that I lack the instruments to do so fluidly, and I don't
understand it well enough yet to actually express ideas. Something
that I hope this project might help solve :)

I certainly wouldn't want to be in charge of writing some "Basic
Principles of Microtonal Composition" page though... that is something
I'd rather get someone else's perspective on, perhaps yours or Gene's
or Iglashion's or Herman's or Kraig's or really anyone else's but me
:)

-Mike

Mike B. wrote:

> This is really where I hope Carl can get involved,

If I get involved in any more stuff online, I'm going to
disappear down an ethernet cable.

> At the very least, if he'd allow us to adapt (or even just copy
> and paste) the condensed tuning-math outline, that would be a
> great start.

Please consider tctmo, and anything else of mine, released
under the Creative Commons Attribution Share Alike license:
http://creativecommons.org/licenses/by-sa/3.0/

-Carl

I really wasn't suggesting that anybody create a guide to musical composition. I believe a few of our community members are already working on this kind of practical topic.

Just sayin' that a distinction should be made between tuning topics and microtonality in a musical context. I get worried that tuning discussion is a turn off for potential new microtonal musicians. Those who are looking for a tuning course wouldn't be affected.

Sean

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Aug 2, 2010 at 10:31 AM, sevishmusic <sevish@...> wrote:
> >
> > Mike, the ideas you've suggested seem more like a "tuning" related course - perhaps leave the idea of a "microtonal" course for a someone with a more composition-based and music-based approach?
> >
> > I only mention this because it's rare for a musician, who is curious about microtonal composition, to give a damn about psychoacoustics and "all of the math" etc.
> >
> > Sweet idea though, I hope this goes somewhere. I don't understand most tuning concepts and this would help!
> >
> > Sean
>
> Haha, well, I personally am a musician who's interested in these
> things... the reason I have written so few microtonal compositions is
> mainly that I lack the instruments to do so fluidly, and I don't
> understand it well enough yet to actually express ideas. Something
> that I hope this project might help solve :)
>
> I certainly wouldn't want to be in charge of writing some "Basic
> Principles of Microtonal Composition" page though... that is something
> I'd rather get someone else's perspective on, perhaps yours or Gene's
> or Iglashion's or Herman's or Kraig's or really anyone else's but me
> :)
>
> -Mike
>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 1 August 2010 21:20, Mike Battaglia <battaglia01@...> wrote:
>
> > Any takers?
>
> I think it's a great idea, and we surely have the expertise here to
> get it done.

I'm still trying to get a basic collection of Xenwiki articles finished.

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:

> For an example of what NOT to do, see this (awful) explanation of Miracle temperament:
>
> http://en.wikipedia.org/wiki/Miracle_temperament.
>
> "In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones."

> Explanations need to give a generally-understandable idea, with a clear image, in the first sentence. This one tells you a lot you don't need to know, and gives you no basic understanding.

It tells you what miracle is, which is pretty basic for a first sentence.

> This is slightly better:
>
> http://x31eq.com/miracle.htm
>
> "Miracle scales fit a 10, rather than 7 or 12, note scale. It consists of 9 small and one slightly larger interval to the octave.

Which tells you things about miracle, but not what it is. When, if ever, would you explain what miracle temperament actually is if you wrote an introductory article on it?

On Mon, Aug 2, 2010 at 4:43 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > "In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones."
>
> > Explanations need to give a generally-understandable idea, with a clear image, in the first sentence. This one tells you a lot you don't need to know, and gives you no basic understanding.
>
> It tells you what miracle is, which is pretty basic for a first sentence.
//
> > "Miracle scales fit a 10, rather than 7 or 12, note scale. It consists of 9 small and one slightly larger interval to the octave.
>
> Which tells you things about miracle, but not what it is. When, if ever, would you explain what miracle temperament actually is if you wrote an introductory article on it?

Right, well the beauty of this approach is that we don't have to
choose just one way to explain it. If miracle temperament is addressed
as part of some "Principles of Microtonal Composition" class, where
the mathematics haven't been explained yet, then we can simplify it
down a bit. If we're going to address it as part of a class on regular
mapping or MOS scales or what have you, then we can go with the first
definition. We can do both.

-Mike

The best way to answer your question, I think, is below--the numbered steps.

It's obvious I don't know what a Miracle scale is. I've tried looking it up twice, and the explanations are not easy to understand.

If I had to guess, we disagree because you think like a mathematician, and I don't. You value concise, formally accurate definitions up front, perhaps.

I'm more interested in this stuff than 95% of the educated musicians out there. For someone like me, what's needed is a picture or a gist, first.

Explanations need to give as simple a picture as possible first.

If 'Miracle' scales are all 10 notes, then that would be one of the first things you should say. That's basic.

Here's the order that would make it easy for me to understand. It would put the important things first, and leave the less important things
for the 'Optional' parts 8 through 10.

1) Show three miracle scales, in cents, on an X axis, against a grid of something more familiar for comparison, like 12ET.
2) Show that they all have 9 small and 1 larger interval. They do or don't repeat at the 2/1, the 'octave'. Show this.
3) Explain what the usefulness of this construct is. Why should people, or even interested microtonalists, care?
4) Show the internal relationships that make this a useful scale.
5) Explain, step by step, with no math more complicated than addition, subtraction, multiplication, and division, how to derive one.
6) Step 5) will show what a 'generator' is, but give more explanation.
7) Explain, in the process of showing the steps in 5) and 6) the part about the 15/14 and 16/15 semitones.

8) Optional: Show the weaknesses of these scales--what don't they do compared to more familiar scales.
9) Optional: Explain the absurd name 'miracle'.--why not call it 'Ten Note Nine Plus One'? What's so 'miraculous' about it?
10)Optional: Mention that George Secor invented it, and mention the 'secor'.

On Aug 2, 2010, at 4:43 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> > For an example of what NOT to do, see this (awful) explanation of Miracle temperament:
> >
> > http://en.wikipedia.org/wiki/Miracle_temperament.
> >
> > "In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones."
>
> > Explanations need to give a generally-understandable idea, with a clear image, in the first sentence. This one tells you a lot you don't need to know, and gives you no basic understanding.
>
> It tells you what miracle is, which is pretty basic for a first sentence.
>
> > This is slightly better:
> >
> > http://x31eq.com/miracle.htm
> >
> > "Miracle scales fit a 10, rather than 7 or 12, note scale. It consists of 9 small and one slightly larger interval to the octave.
>
> Which tells you things about miracle, but not what it is. When, if ever, would you explain what miracle temperament actually is if you wrote an introductory article on it?
>
>

Here's a good explanation, for starters.

Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scalecalled "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.

-------------
Combined with a listing of the blackjack and canasta scales, this would be all that's needed at first.

Note that neither one of the explanations I mentioned gave this basic, simple explanation first.

Why not?

Because it's so basic that whoever wrote the explanations simply forgot that everyone doesn't know that already.

This process of forgetting the basics happens all the time, and is one of the reasons I wouldn't want to teach a basic synthesizer course any more, unless that's all I could get, because I've moved on.

Caleb

On Aug 3, 2010, at 9:15 AM, caleb morgan wrote:

>
> The best way to answer your question, I think, is below--the numbered steps.
>
> It's obvious I don't know what a Miracle scale is. I've tried looking it up twice, and the explanations are not easy to understand.
>
> If I had to guess, we disagree because you think like a mathematician, and I don't. You value concise, formally accurate definitions up front, perhaps.
>
> I'm more interested in this stuff than 95% of the educated musicians out there. For someone like me, what's needed is a picture or a gist, first.
>
> Explanations need to give as simple a picture as possible first.
>
> If 'Miracle' scales are all 10 notes, then that would be one of the first things you should say. That's basic.
>
> Here's the order that would make it easy for me to understand. It would put the important things first, and leave the less important things
> for the 'Optional' parts 8 through 10.
>
> 1) Show three miracle scales, in cents, on an X axis, against a grid of something more familiar for comparison, like 12ET.
> 2) Show that they all have 9 small and 1 larger interval. They do or don't repeat at the 2/1, the 'octave'. Show this.
> 3) Explain what the usefulness of this construct is. Why should people, or even interested microtonalists, care?
> 4) Show the internal relationships that make this a useful scale.
> 5) Explain, step by step, with no math more complicated than addition, subtraction, multiplication, and division, how to derive one.
> 6) Step 5) will show what a 'generator' is, but give more explanation.
> 7) Explain, in the process of showing the steps in 5) and 6) the part about the 15/14 and 16/15 semitones.
>
>
> 8) Optional: Show the weaknesses of these scales--what don't they do compared to more familiar scales.
> 9) Optional: Explain the absurd name 'miracle'.--why not call it 'Ten Note Nine Plus One'? What's so 'miraculous' about it?
> 10)Optional: Mention that George Secor invented it, and mention the 'secor'.
>
>
>
> On Aug 2, 2010, at 4:43 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>
>> > For an example of what NOT to do, see this (awful) explanation of Miracle temperament:
>> >
>> > http://en.wikipedia.org/wiki/Miracle_temperament.
>> >
>> > "In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones."
>>
>> > Explanations need to give a generally-understandable idea, with a clear image, in the first sentence. This one tells you a lot you don't need to know, and gives you no basic understanding.
>>
>> It tells you what miracle is, which is pretty basic for a first sentence.
>>
>> > This is slightly better:
>> >
>> > http://x31eq.com/miracle.htm
>> >
>> > "Miracle scales fit a 10, rather than 7 or 12, note scale. It consists of 9 small and one slightly larger interval to the octave.
>>
>> Which tells you things about miracle, but not what it is. When, if ever, would you explain what miracle temperament actually is if you wrote an introductory article on it?
>>
>
>
>

On 3 August 2010 15:07, caleb morgan <calebmrgn@...> wrote:
>
>
> Here's a good explanation,  for starters.
> Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scalecalled "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.
>
> -------------
> Combined with a listing of the blackjack and canasta scales, this would be all that's needed at first.
> Note that neither one of the explanations I mentioned gave this basic, simple explanation first.
> Why not?

What's a fifth? What useful properties? What do you mean by
"efficient"? What's the 11-limit? What's a temperament? What
purposes? What's 72-EDO?

Graham

Sorry, I didn't make myself clear.

This is from googling this:

http://xenharmonic.wikispaces.com/Regular+Temperaments

It was NOT my explanation.

Now, I understand your rhetoric, but do you really wish to argue that musicians won't know what a fifth is?

**The point was that it gave a basic crucial bit of information first.**

That is, a miracle temperament divides the 5th in 6 equal parts.

Is that not the case?

If so, there are three explanations of miracle temperament on the web, all incomprehensible or wrong.

However, I suspect that it's right.

My point is that any basic explanation needs to include this basic information *first.*

On Aug 3, 2010, at 10:58 AM, Graham Breed wrote:

> On 3 August 2010 15:07, caleb morgan <calebmrgn@...> wrote:
> >
> >
> > Here's a good explanation, for starters.
> > Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scalecalled "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.
> >
> > -------------
> > Combined with a listing of the blackjack and canasta scales, this would be all that's needed at first.
> > Note that neither one of the explanations I mentioned gave this basic, simple explanation first.
> > Why not?
>
> What's a fifth? What useful properties? What do you mean by
> "efficient"? What's the 11-limit? What's a temperament? What
> purposes? What's 72-EDO?
>
> Graham
>

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> Here's a good explanation, for starters.
>
> Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scalecalled "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.
>
> -------------
> Combined with a listing of the blackjack and canasta scales, this would be all that's needed at first.
>
> Note that neither one of the explanations I mentioned gave this basic, simple explanation first.
>
> Why not?
>
> Because it's so basic that whoever wrote the explanations simply forgot that everyone doesn't know that already.

No, because a scale is secondary. The most important information in the definition you quote is that it divides the fifth into six equal parts. That is similar to knowing the "generator" is between 16/15 and 15/14, which you objected to, but you've got to say something like that early on. Then comes the tuning information, 72et, which allows you to surmise the generator steps need to be around 7/72 of an octave, and the information that it is in some way "most efficient" for 11-limit, which is not defined so it's being assumed you know what that is. The least important data is about scales, since many scales are possible, and those listed are just two examples.

I agree, and think that this should come first, if this is what a Miracle temperament is.

(Then, some examples of Miracle scales spelled out.

Then the idea that 1/6th of 3/2 (702 cents) is around 117 cents, which is BETWEEN 15:14 (119.428 cents) and 16:15 (111.720 cents).

I objected to the ordering, the secor, and the "serves as". I had, and have, no idea what "serves as" means in this context.)

This rest of the quotation is *also* important:

A 21-note scale called "blackjack" and a 31-note scalecalled "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.

I'm saying: Use your words. "Fur" is good. Don't be terse and like, scienc-y. Give the simple image, (divides the 3/2 into 6 parts) up front.

That George Secor thought of it should go in a footnote, in a longer article.

-c

On Aug 3, 2010, at 12:59 PM, genewardsmith wrote:

> The most important information in the definition you quote is that it divides the fifth into six equal parts.

On 3 August 2010 16:17, caleb morgan <calebmrgn@...> wrote:
>
>
> Sorry, I didn't make myself clear.
> This is from googling this:
> http://xenharmonic.wikispaces.com/Regular+Temperaments
> It was NOT my explanation.

Oh, sorry.

> Now, I understand your rhetoric, but do you really wish to argue that musicians won't know what a fifth is?

You understand my point, so do we need to argue about the details?

> **The point was that it gave a basic crucial bit of information first.**
> That is,  a miracle temperament divides the 5th in 6 equal parts.

That's good, yes. It's a useful summary. But it does also depend on
understanding some technical language that you said you wanted to
avoid before. It all depends on where it comes in the course.
Perhaps miracle would be used to explain the 11-limit, so you wouldn't
want to scare people with that term up front. And, really, people can
easily get scared by terms they don't understand.

> Is that not the case?

It's the case.

> If so, there are three explanations of miracle temperament on the web, all incomprehensible or wrong.
> However, I suspect that it's right.

Come on, simmer down. I didn't say it was wrong, did I?

> My point is that any basic explanation needs to include this basic information *first.*

It's good to start with a summary, yes, and that's something I didn't do.

Not everybody agrees about the basic information, of course. Gene's
come in to say that scales are the least important thing whereas I
always explain them first. And I think working with melody only is a
good first exercise when you have a new temperament to learn.

You can also describe Miracle as a "quartertone" equivalent of
Slendric (5&21, formerly Wonder). That would make sense if you were
teaching Slendric first, which I can see making sense. So the
introduction might be something like "Miracle divides the Slendric
steps to give twice as many notes, and allows full 7-limit harmony, as
well as opening the doors to the 11-limit".

You can describe Miracle as being a combination of Slendric, Mohajira
(7&31), and Marvel (225:224 planar). Whether that's a practical way
of teaching I don't know.

Here's a draft course outline for regular temperaments:

- Meantone, 5-limit harmony, spiral of fifths.
- Schismatic, 7-limit harmony.
- Mavila.
- Slendric, generators that aren't fifths.
- Gamelans, inharmonic timbres.
- Pajara, periods that aren't octaves.
- Mohajira, neutral thirds and seconds.
- Magic, 9-limit harmony, Marvel?
- Orwell, 11-limit harmony.
- Miracle, 72-equal notation.
- Overview of higher limits and other classes.

That would give 11 lessons, and I think it would be a good progression
for music students who are going to understand fifths, notation,
harmony, and whatever. There are surely other ways to do it.

Graham

Caleb>" Miracle temperament divides the fifth into 6 equal steps. A 21-note
scale called "blackjack" and a 31-note scale called "canasta" have some useful
properties. It's the most efficient 11-limit temperament for many purposes with
a tuning close to 72-EDO."

Right, but within that comes the questions:
A) What, mathematically, is the fifth (some people don't 'even' know it's 3/2)
B) What do you mean by divide the fifth into 6 equal steps? (explain it's the
6th root of 3/2)
C) What is 72EDO (explain: all the notes formed by taking the 72nd root of the
2/1 octave...and that any EDO is generated this way)
D) What do you mean by 'efficient'? (I don't know either...dyadic, chord wise,
good for even or odd limit, etc.)
E) What is a tuning vs. a temperament?

Comes to think of it...I think ANY really good tuning/scale primer will
explain how scales are generated and notated first. Only after that is
explained, IMVHO, can someone easily comprehend properties (dyadic, chord-wise,
commas) which happen as an effect of such generation methods.

And it might be a good idea to give an overview what scales are generally
considered good at using more general musician terms (IE 24TET can be good for
blues and merging between major and minor scale feel in melody but generally is
unable to give many more options in terms of non-sour chords). Otherwise the
person reading may very well think "wow, this is making me impatient: LOTS of
math without much emotional/artistic reasoning...seems more like weird science
than coherent art...not my thing".

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:

> **The point was that it gave a basic crucial bit of information first.**
>
> That is, a miracle temperament divides the 5th in 6 equal parts.

That basic, crucial bit of information is closely related to the trivial, unimportant information that it uses an interval between 16/15 and 15/14 as an approximation to both, which you started this thread by objecting to. At some point to understand miracle you will need both of these facts.

> My point is that any basic explanation needs to include this basic information *first.*

Arguably the single most basic piece of information is that about an interval between 16/15 and 15/14, which is just what you didn't want to know. It's like knowing that the tone in meantone is between 10/9 and 9/8 and is used as an approximation for both, where you still need to deduce that an octave plus a tone is divided in half to get a flattened fifth, and two tones give a 5/4, and there is your 5-limit.

Once you know your generator is between 16/15 and 15/14 and is used as an approximation to both, you know that two of them give an approximate 8/7, and there is your approximation for 7. You still need to know that three of these 8/7s give a fifth, but that will not surprise you if you look at what is between six 16/15 at 670 cents and six 15/14 at 717 cents, or note that three 8/7s is 693.5 cents. From this you can deduce that taking a fifth up by 16/15 gives you an 8/5, and there is your approximation for 5. I'm not saying you should have to work all this out for yourself, just that you *could* work it out using the very information you rejected as unimportant.

Hi Caleb,

Let me first just say that there's no one best way to describe one temperament (like miracle) in a paragraph to someone who has no idea what regular temperaments are. Your issues with the given definitions of Miracle stem more from your unfamiliarity with the concept of regular temperaments than from anything else; if you had that one "basic" concept down, the given definitions would make a lot more sense.

This is why I believe that in teaching microtonality, jumping straight in to individual scales or temperaments is not the way to go. People need to be given FIRST a basic understanding of what regular temperaments are, how they are designed and evaluated, how they can give structure to music (i.e. the sorts of scales they can produce), etc. THEN, once those basics are understood, you can start looking at specific temperaments. That way, you don't need to cram a summary of how regular temperaments work into the descriptions of each temperament.

Honestly, Caleb, knowing that Miracle temperaments produce 10-note scales of a given shape is really NOT the most useful thing to know about them (though knowing you need to have at least 10 notes of one to have a decent number of good triads might be more useful). When you actually work with temperaments, probably the most useful thing to know is how many generators it takes to make a "useful" interval like a perfect fifth, a major third, or a 7/4 minor seventh. This is actually MORE important than just knowing how many scale-steps the temperament produces at its various "moments of symmetry". For instance, Orwell temperament produces a moment of symmetry scale at 9 notes, but it only has two triads with a perfect fifth in that scale; this is because it takes several Orwell generators to get to a perfect fifth. If you know how many generators it takes to produce various intervals, you can get a better idea of how many notes of the temperament you'll need to make music.

Of course, if you didn't know that you needed to know that about temperaments, you might THINK that knowing how many notes are in the scales produced by a temperament is most important. This is why you have to understand the concept of "mapping" that is at the core of all regular temperaments: all temperaments "map" the same intervals (depending on the limit of the temperament, they will map the 3rd, 5th, 7th, 11th, 13th, etc. harmonics, i.e. various prime harmonics) to some number of generators. What varies between temperaments is how many generators the harmonics are mapped to. In meantone or pythagorean temperament, the 3rd harmonic is mapped to 1 generator + 1 period (because the generator is a fifth, and the period is an octave). In magic, the 5th harmonic is mapped to 1 generator plus 2 periods (because the generator is a pure-ish major third, and the period is an octave).

So basically, regular temperaments are for folks who are interested in using those harmonics as a musical basis, but prefer to use scales produced by series' of generators (because these scales are more "regular" and "even" than JI and you don't have to go through the headache of multiplying and dividing ratios every time you move the tonic). Of course, they're also for folks who just like the sound of them, but the preceding is basically the purpose for which they were designed. If you understood this FIRST, then the thing you'd want to see listed at the top of every temperament's description is how many generators and periods it takes to get to each harmonic, how well those harmonics are approximated at that point, and THEN what the generator is, and THEN how many notes you need in a scale of that temperament to use its "good" properties. THEN you'd probably want to look in to the cents values of some scales produced by that temperament, to see what other kinds of intervals appear in it.

Of course, if you're not too interested in approximating prime harmonics, regular temperaments might not be of much interest to you. So if you knew that the concept of regular temperaments has "approximating prime harmonics" at its core, you'd know not to waste your time reading about them. If you'd rather look at scales in terms of how many notes they have, and THEN pick them apart by what kind of harmonies they produce, you'd be better served by looking at things like the Scale Tree and Moment-of-Symmetry scales in general. Or maybe you'd rather just see spreadsheets of various equal temperaments, the JI ratios they approximate (and how well they approximate them), and the various MOS scales possible in each temperament. Either way, it's best to know the rationale behind the various ways of describing (and creating) scales, as well as the benefits they offer, before you dive in and start trying to learn about them.

HTH!

-Igs

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

> You can also describe Miracle as a "quartertone" equivalent of
> Slendric (5&21, formerly Wonder). That would make sense if you were
> teaching Slendric first, which I can see making sense.

You'd have to explain that slendric is a no-fives rank two temperament (tempering out 1029/1024 and 243/242, in case anyone wants to know) which gets you into the whole area of subgroup temperaments. That's potentially a bit confusing, and of course you've also got the rank three temperament tempering out 1029/1024 and 243/242 lurking about. But I guess it would be one way to go.

> You can describe Miracle as being a combination of Slendric, Mohajira
> (7&31), and Marvel (225:224 planar).

Since mohajira tempers out 81/80, you had better NOT do any such thing.

On Mon, Aug 2, 2010 at 4:50 PM, Mike Battaglia <battaglia01@...> wrote:
>
> Right, well the beauty of this approach is that we don't have to
> choose just one way to explain it. If miracle temperament is addressed
> as part of some "Principles of Microtonal Composition" class, where
> the mathematics haven't been explained yet, then we can simplify it
> down a bit. If we're going to address it as part of a class on regular
> mapping or MOS scales or what have you, then we can go with the first
> definition. We can do both.

I'm glad that my thread has kicked off some discussion, but as I said
here ^^^^ - we don't have to "pick one way to explain miracle
temperament."

This entire discussion is a red herring because if we're going to
start a microtonal department with different courses and lessons and
stuff, we'd never just start writing about miracle temperament as
though it's the first thing that the reader is going to see. We'd put
it as part of a sequence that assumes they know how to understand the
words "generator," "EDO," "MOS," and so on. A more technical
definition is certainly fine for this purpose.

If we're going to also address miracle in some kind of "Intro to
Microtonality" course, or as part of a "Microtonal Composition" course
that assumes the reader hasn't delved into the principles of regular
mapping, then for that course we could define it in a more
"conceptual" way that leaves the math stuff out.

So why argue over which is "better?" There is enough room in a
wikiversity department for all of these approaches.

-Mike

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> So basically, regular temperaments are for folks who are interested in using those harmonics as a musical basis, but prefer to use scales produced by series' of generators (because these scales are more "regular" and "even" than JI and you don't have to go through the headache of multiplying and dividing ratios every time you move the tonic).

The reason I was saying scales were information of lesser importance is that I don't think this is true even for rank two temperaments, and clearly it isn't for higher ranks.

With a rank two temperament like valentine or pajara, we normally will say that vlaentine has MOS of size 15, 16 and 31, or pajara has MOS of size 10, 12 and 22, and leave it at that. This is more out of convenience than any belief it is a complete picture on my part, and I doubt I am alone in thinking that. In fact, Paul's favorite scales for pajara aren't MOS, because like any MOS with a period of 600 cents it's tonally ambiguous, and he generally preferred a scale with a stronger tonal sense.

For a rank three temperament like starling, we normally don't give any scales at all, since asking for a scale in this case is like asking for a 5-limit JI scale, and there's just too many choices to list. But consider this: if you aks for what would be a good edo to tune starling, you might get "77. 108 or 185" as your answer. But if you ask for a good edo to tune valentine, you might equally well expect to hear "77, 108 or 185". There's not much to be gained in tems of tuning accuracy by tempering out just 126/125 as opposed to both 126/125 and 1029/1024, which gives valentine. But if you take a 5-limit scale and temper it for starling, focusing on chains of minor thirds to let starling tempering have its full effect, and tune it using 185edo, you get a scale you can regard as in valentine temperament. But it will probably be a long, long way from being a MOS. Even myna temperament, with a minor third generator, will likely not give you a MOS. And yet, unless the scale is small (less than 12 notes, say) it is likely the 1029/1024 will also kick in, giving you authentic valentine tempering.

I would recommend that the course be labeled a course on Microtonal temperaments as opposed to microtones in general

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> If you don't know, now you know:
> http://en.wikiversity.org/wiki/Wikiversity:Main_Page
>
> Wikiversity is another Wikimedia resource, but this time, it aims to
> organize information in virtual online "courses" or "lessons" or what
> have you. It seems to be extremely flexible and I would assume can be
> laid out however we want.
>
> For a very "structured" example, here's a Wikiversity "Course" in
> Electric Circuit Analysis:
> http://en.wikiversity.org/wiki/Electric_Circuit_Analysis
> And for a "looser" example, here's Wikiversity's "School of Music":
> http://en.wikiversity.org/wiki/School:Music -- note the
> theory/composition page.
>
> Anyone interested in collaborating to get a Microtonal "course" set
> up, made up of different "lessons"? Or maybe a microtonal "department"
> with different "courses." Something where there are relevant sections
> for all of the constituent fields you have to understand to finally
> know what's going on - psychoacoustics, all of the math, some signal
> processing, etc, which would lead up to a "class" on Rothenberg's
> ideas, and ultimately regular mapping, or something like that. Some
> structured approach to learning this stuff.
>
> I am still very new to all of the math involved in regular mapping,
> but would be more than willing to contribute for psychoacoustics,
> signal processing, Fourier analysis, etc.
>
> Any takers?
>
> -Mike
>

Kraig,

My idea was to start a Microtonal "Department," of which there'd be a
lot of different courses.

-Mike

On Tue, Aug 3, 2010 at 5:24 PM, Kraig Grady <kraiggrady@...> wrote:
>
> I would recommend that the course be labeled a course on Microtonal temperaments as opposed to microtones in general
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > If you don't know, now you know:
> > http://en.wikiversity.org/wiki/Wikiversity:Main_Page
> >
> > Wikiversity is another Wikimedia resource, but this time, it aims to
> > organize information in virtual online "courses" or "lessons" or what
> > have you. It seems to be extremely flexible and I would assume can be
> > laid out however we want.
> >
> > For a very "structured" example, here's a Wikiversity "Course" in
> > Electric Circuit Analysis:
> > http://en.wikiversity.org/wiki/Electric_Circuit_Analysis
> > And for a "looser" example, here's Wikiversity's "School of Music":
> > http://en.wikiversity.org/wiki/School:Music -- note the
> > theory/composition page.
> >
> > Anyone interested in collaborating to get a Microtonal "course" set
> > up, made up of different "lessons"? Or maybe a microtonal "department"
> > with different "courses." Something where there are relevant sections
> > for all of the constituent fields you have to understand to finally
> > know what's going on - psychoacoustics, all of the math, some signal
> > processing, etc, which would lead up to a "class" on Rothenberg's
> > ideas, and ultimately regular mapping, or something like that. Some
> > structured approach to learning this stuff.
> >
> > I am still very new to all of the math involved in regular mapping,
> > but would be more than willing to contribute for psychoacoustics,
> > signal processing, Fourier analysis, etc.
> >
> > Any takers?
> >
> > -Mike

caleb morgan wrote:
> Here's a good explanation, for starters.
> > Miracle temperament divides the fifth into 6 equal steps. A 21-note
> scale called "blackjack" and a 31-note scalecalled "canasta" have
> some useful properties. It's the most efficient 11-limit temperament
> for many purposes with a tuning close to 72-EDO.
> > ------------- Combined with a listing of the blackjack and canasta
> scales, this would be all that's needed at first.

It's a good start to bring up the fact that a fifth is divided into 6 equal steps. Similarly you could say that porcupine temperament divides the fourth into 3 equal steps, etc. These are features of the temperament that relate directly to their definitions. It's also useful to get an idea of the sizes of typical scales you might use with them (e.g. the 15-note scale of porcupine).

As far as being "the most efficient 11-limit temperament for many purposes", you get into the tradeoffs of accuracy vs. complexity. One idea I've found useful is the idea of "first class" temperaments (or "gold medal" temperaments as I've called them), each of which is more accurate than any other temperament of the same or lower complexity. (You can always find more accurate temperaments that are more complex.) The contents of this list will vary depending on how you measure error and complexity, but a list of "first class" 11-limit temperaments might look something like this:

augene [<3, 5, 7, 8, 10], <0, -1, 0, 2, 2]>
porcupine [<1, 2, 3, 2, 4], <0, -3, -5, 6, -4]>
pajara [<2, 3, 5, 6, 8], <0, 1, -2, -2, -6]>
orwell [<1, 0, 3, 1, 3], <0, 7, -3, 8, 2]>
valentine [<1, 1, 2, 3, 3], <0, 9, 5, -3, 7]>
miracle [<1, 1, 3, 3, 2], <0, 6, -7, -2, 15]>
wizard [<2, 1, 5, 2, 8], <0, 6, -1, 10, -3]>
luna [<1, 4, 2, 2, 7], <0, -15, 2, 5, -22]>
unidec [<2, 5, 8, 5, 6], <0, -6, -11, 2, 3]>

You could extend this list in either direction by including temperaments less accurate than augene or more complex than unidec, but these are the ones with common names. So you might say that miracle has "one of the best tradeoffs of error vs. complexity among 11-limit temperaments", or something along those lines. It's also good as a 7-limit temperament, although the 5-limit version (ampersand) isn't very remarkable.

> Note that neither one of the explanations I mentioned gave this
> basic, simple explanation first.
> > Why not?
> > Because it's so basic that whoever wrote the explanations simply
> forgot that everyone doesn't know that already.
> > This process of forgetting the basics happens all the time, and is
> one of the reasons I wouldn't want to teach a basic synthesizer
> course any more, unless that's all I could get, because I've moved
> on.
> > Caleb

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

> augene [<3, 5, 7, 8, 10], <0, -1, 0, 2, 2]>
> porcupine [<1, 2, 3, 2, 4], <0, -3, -5, 6, -4]>
> pajara [<2, 3, 5, 6, 8], <0, 1, -2, -2, -6]>
> orwell [<1, 0, 3, 1, 3], <0, 7, -3, 8, 2]>
> valentine [<1, 1, 2, 3, 3], <0, 9, 5, -3, 7]>
> miracle [<1, 1, 3, 3, 2], <0, 6, -7, -2, 15]>
> wizard [<2, 1, 5, 2, 8], <0, 6, -1, 10, -3]>
> luna [<1, 4, 2, 2, 7], <0, -15, 2, 5, -22]>
> unidec [<2, 5, 8, 5, 6], <0, -6, -11, 2, 3]>

Here's a gold medal list concocted using Graham's cangwu badness, which looks rather different. This sort of thing is likely to be pretty variable, but at least miracle is still on the list. In fact, miracle, pajara, augene, and orwell are on both lists. Mine include some that are either too complex or too inaccurate to make Herman's list, and curiously enough has opossum rather than porcupine, and has 11-limit meantone and myna, but no valentine, unidec or wizard.

Gold medals
Hemiennealimmal <<36 54 36 18 2 -44 -96 -68 -145 -74||
Miracle <<6 -7 -2 15 -25 -20 3 15 59 49||
<<0 0 0 1 0 0 2 0 3 3||
<<102 210 216 222 96 56 -1 -88 -211 -124||
Pajara <<2 -4 -4 -12 -11 -12 -26 2 -14 -20||
Octoid <<24 32 40 24 -5 -4 -45 3 -55 -71||
Augene <<3 0 -6 -6 -7 -18 -20 -14 -14 4||
Orwell <<7 -3 8 2 -21 -7 -21 27 15 -22||
Meantone <<1 4 10 18 4 13 25 12 28 16||
Father <<1 -1 3 4 -4 2 3 10 13 1||
Beep <<2 3 1 -2 0 -4 -10 -6 -15 -9||
Myna <<10 9 7 25 -9 -17 5 -9 27 46||
Jamesbond <<0 0 7 0 0 11 0 16 0 -24||
<<0 2 2 2 3 3 3 -1 -2 -1||
<<2 1 3 5 -3 -1 1 4 8 4||
Injera <<2 8 8 12 8 7 12 -4 0 6||
Meanadecal <<1 4 10 6 4 13 6 12 0 -18||
Diminished <<4 4 4 0 -3 -5 -14 -2 -14 -14||
Dicot <<2 1 6 5 -3 4 1 11 8 -7||
Opossum <<3 5 9 4 1 6 -4 7 -8 -20||

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Of course, if you're not too interested in approximating prime
> harmonics, regular temperaments might not be of much interest to
> you. So if you knew that the concept of regular temperaments
> has "approximating prime harmonics" at its core, you'd know not
> to waste your time reading about them.

Good write-up, Igs. Just one nitpick here: regular temperaments,
because of their regularity, approximate all of just intonation
(at the given prime limit), not just the bare primes.

-Carl

You put your finger right on it--I didn't know what regular temperaments are, and I felt defensive.

On Aug 3, 2010, at 1:29 PM, cityoftheasleep wrote:

> Hi Caleb,
>
> Let me first just say that there's no one best way to describe one temperament (like miracle) in a paragraph to someone who has no idea what regular temperaments are. Your issues with the given definitions of Miracle stem more from your unfamiliarity with the concept of regular temperaments than from anything else; if you had that one "basic" concept down, the given definitions would make a lot more sense.
>
> This is why I believe that in teaching microtonality, jumping straight in to individual scales or temperaments is not the way to go. People need to be given FIRST a basic understanding of what regular temperaments are, how they are designed and evaluated, how they can give structure to music (i.e. the sorts of scales they can produce), etc. THEN, once those basics are understood, you can start looking at specific temperaments. That way, you don't need to cram a summary of how regular temperaments work into the descriptions of each temperament.
>
> Honestly, Caleb, knowing that Miracle temperaments produce 10-note scales of a given shape is really NOT the most useful thing to know about them (though knowing you need to have at least 10 notes of one to have a decent number of good triads might be more useful). When you actually work with temperaments, probably the most useful thing to know is how many generators it takes to make a "useful" interval like a perfect fifth, a major third, or a 7/4 minor seventh. This is actually MORE important than just knowing how many scale-steps the temperament produces at its various "moments of symmetry". For instance, Orwell temperament produces a moment of symmetry scale at 9 notes, but it only has two triads with a perfect fifth in that scale; this is because it takes several Orwell generators to get to a perfect fifth. If you know how many generators it takes to produce various intervals, you can get a better idea of how many notes of the temperament you'll need to make music.
>
> Of course, if you didn't know that you needed to know that about temperaments, you might THINK that knowing how many notes are in the scales produced by a temperament is most important. This is why you have to understand the concept of "mapping" that is at the core of all regular temperaments: all temperaments "map" the same intervals (depending on the limit of the temperament, they will map the 3rd, 5th, 7th, 11th, 13th, etc. harmonics, i.e. various prime harmonics) to some number of generators. What varies between temperaments is how many generators the harmonics are mapped to. In meantone or pythagorean temperament, the 3rd harmonic is mapped to 1 generator + 1 period (because the generator is a fifth, and the period is an octave). In magic, the 5th harmonic is mapped to 1 generator plus 2 periods (because the generator is a pure-ish major third, and the period is an octave).
>
> So basically, regular temperaments are for folks who are interested in using those harmonics as a musical basis, but prefer to use scales produced by series' of generators (because these scales are more "regular" and "even" than JI and you don't have to go through the headache of multiplying and dividing ratios every time you move the tonic). Of course, they're also for folks who just like the sound of them, but the preceding is basically the purpose for which they were designed. If you understood this FIRST, then the thing you'd want to see listed at the top of every temperament's description is how many generators and periods it takes to get to each harmonic, how well those harmonics are approximated at that point, and THEN what the generator is, and THEN how many notes you need in a scale of that temperament to use its "good" properties. THEN you'd probably want to look in to the cents values of some scales produced by that temperament, to see what other kinds of intervals appear in it.
>
> Of course, if you're not too interested in approximating prime harmonics, regular temperaments might not be of much interest to you. So if you knew that the concept of regular temperaments has "approximating prime harmonics" at its core, you'd know not to waste your time reading about them. If you'd rather look at scales in terms of how many notes they have, and THEN pick them apart by what kind of harmonies they produce, you'd be better served by looking at things like the Scale Tree and Moment-of-Symmetry scales in general. Or maybe you'd rather just see spreadsheets of various equal temperaments, the JI ratios they approximate (and how well they approximate them), and the various MOS scales possible in each temperament. Either way, it's best to know the rationale behind the various ways of describing (and creating) scales, as well as the benefits they offer, before you dive in and start trying to learn about them.
>
> HTH!
>
> -Igs
>
>

I'll be terse: no, twice, not trivial, just incomprehensible at first.

On Aug 3, 2010, at 1:29 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> > **The point was that it gave a basic crucial bit of information first.**
> >
> > That is, a miracle temperament divides the 5th in 6 equal parts.
>
> That basic, crucial bit of information is closely related to the trivial, unimportant information that it uses an interval between 16/15 and 15/14 as an approximation to both, which you started this thread by objecting to. At some point to understand miracle you will need both of these facts.
>
> > My point is that any basic explanation needs to include this basic information *first.*
>
> Arguably the single most basic piece of information is that about an interval between 16/15 and 15/14, which is just what you didn't want to know. It's like knowing that the tone in meantone is between 10/9 and 9/8 and is used as an approximation for both, where you still need to deduce that an octave plus a tone is divided in half to get a flattened fifth, and two tones give a 5/4, and there is your 5-limit.
>
> Once you know your generator is between 16/15 and 15/14 and is used as an approximation to both, you know that two of them give an approximate 8/7, and there is your approximation for 7. You still need to know that three of these 8/7s give a fifth, but that will not surprise you if you look at what is between six 16/15 at 670 cents and six 15/14 at 717 cents, or note that three 8/7s is 693.5 cents. From this you can deduce that taking a fifth up by 16/15 gives you an 8/5, and there is your approximation for 5. I'm not saying you should have to work all this out for yourself, just that you *could* work it out using the very information you rejected as unimportant.
>
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Good write-up, Igs. Just one nitpick here: regular temperaments,
> because of their regularity, approximate all of just intonation
> (at the given prime limit), not just the bare primes.
>

Thanks, Carl. I kinda figured they did, but I never see mappings given for anything but the primes, so I wasn't sure how regular temperaments deal with other ratios. I mean, if the primes are well-approximated, then the products of the primes should be approximated as well, right? Like if you've got a good 5/1, 3/1, and 2/1, you've almost certainly got a pretty good 15/8, 8/5, 5/4, 5/3, 4/3, and 3/2 (etc.) as well, though they may be quite complex in a given temperament...right? But of course, from the sound of Herman's posts about "gold medal" temperaments, temperaments are *usually* evaluated on how well they approximate the primes (and how complex they make them), not how well they do random ratios like 9/7, 11/6, or 13/5. I'm sure they *could* be looked at that way, but in discussion of regular temps, they never are. So reading the "literature" on regular temperaments may not be very helpful to those who are not too interested in the "bare primes".

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Thanks, Carl. I kinda figured they did, but I never see mappings given for anything but the primes, so I wasn't sure how regular temperaments deal with other ratios.

The reason for the focus on prime numbers is mostly the Fundamental Theorem of Arthmetic, which tells us that for any positive rational number q, there is a unique expression of the form

q = 2^e2 3^e3 ... p^ep

where the primes are taken in ascending order up to the largest prime, p, which divides either the numerator or denominator of q in reduced form, where the exponents are (positive or negative) integers, and where 1 is considered to have a null prime factorization.

However, unlike the situation for positive integers, primes are not the only possible choices for rational numbers which could be used in factorizations like this. For instance, instead of writing the 5-limit in terms of 2, 3 and 5, we can equally well write it in terms of 9/8, 10/9 and 16/15.

Another reason for the interest in primes is that some ways of measuring complexity, such as Tenney height, treat 15, 3/5, 5/3 and 1/15 as all equally complex, which allows us to consider complexity as a weighted sum of the complexity of the prime factors. This makes life easier mathematically, and gives good results, but it makes you treat 2 like any other prime, and hence such things as TOP tuning.

Hi Igs,

> > Good write-up, Igs. Just one nitpick here: regular temperaments,
> > because of their regularity, approximate all of just intonation
> > (at the given prime limit), not just the bare primes.
>
> Thanks, Carl. I kinda figured they did, but I never see
> mappings given for anything but the primes, so I wasn't sure how
> regular temperaments deal with other ratios.

Ah, that's the whole point of regular temperaments! They
mimic the structure (are a homomorphism) of JI.

> I mean, if the primes are well-approximated, then the products
> of the primes should be approximated as well, right?

Generally, yes. If you use weighted error (where error is
deemed worse if it happens to simple intervals like the octave)
you can get it to work out directly. That's what TOP does.
As the error in 2 & 3 compound in 6/5, so does the complexity,
so the weighted error is the same (or less)!

> Like if you've got a good 5/1, 3/1, and 2/1, you've almost
> certainly got a pretty good 15/8, 8/5, 5/4, 5/3, 4/3, and 3/2

Yes, but the unweighted errors in the primes could be in
opposite directions, so compound intervals can come out worse
than any prime. Again, TOP weighting takes care of this.

-Carl

genewardsmith wrote:
> > --- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> >> augene [<3, 5, 7, 8, 10], <0, -1, 0, 2, 2]>
>> porcupine [<1, 2, 3, 2, 4], <0, -3, -5, 6, -4]>
>> pajara [<2, 3, 5, 6, 8], <0, 1, -2, -2, -6]>
>> orwell [<1, 0, 3, 1, 3], <0, 7, -3, 8, 2]>
>> valentine [<1, 1, 2, 3, 3], <0, 9, 5, -3, 7]>
>> miracle [<1, 1, 3, 3, 2], <0, 6, -7, -2, 15]>
>> wizard [<2, 1, 5, 2, 8], <0, 6, -1, 10, -3]>
>> luna [<1, 4, 2, 2, 7], <0, -15, 2, 5, -22]>
>> unidec [<2, 5, 8, 5, 6], <0, -6, -11, 2, 3]>
> > Here's a gold medal list concocted using Graham's cangwu badness, which looks rather different. This sort of thing is likely to be pretty variable, but at least miracle is still on the list. In fact, miracle, pajara, augene, and orwell are on both lists. Mine include some that are either too complex or too inaccurate to make Herman's list, and curiously enough has opossum rather than porcupine, and has 11-limit meantone and myna, but no valentine, unidec or wizard.
> > Gold medals
> Hemiennealimmal <<36 54 36 18 2 -44 -96 -68 -145 -74||
> Miracle <<6 -7 -2 15 -25 -20 3 15 59 49||
> <<0 0 0 1 0 0 2 0 3 3||
> <<102 210 216 222 96 56 -1 -88 -211 -124||
> Pajara <<2 -4 -4 -12 -11 -12 -26 2 -14 -20||
> Octoid <<24 32 40 24 -5 -4 -45 3 -55 -71||
> Augene <<3 0 -6 -6 -7 -18 -20 -14 -14 4||
> Orwell <<7 -3 8 2 -21 -7 -21 27 15 -22||
> Meantone <<1 4 10 18 4 13 25 12 28 16||
> Father <<1 -1 3 4 -4 2 3 10 13 1||
> Beep <<2 3 1 -2 0 -4 -10 -6 -15 -9||
> Myna <<10 9 7 25 -9 -17 5 -9 27 46||
> Jamesbond <<0 0 7 0 0 11 0 16 0 -24||
> <<0 2 2 2 3 3 3 -1 -2 -1||
> <<2 1 3 5 -3 -1 1 4 8 4||
> Injera <<2 8 8 12 8 7 12 -4 0 6||
> Meanadecal <<1 4 10 6 4 13 6 12 0 -18||
> Diminished <<4 4 4 0 -3 -5 -14 -2 -14 -14||
> Dicot <<2 1 6 5 -3 4 1 11 8 -7||
> Opossum <<3 5 9 4 1 6 -4 7 -8 -20||

And yet another gold medal list, this one based on criteria I've been developing for my Zireen music theory. Miracle tends to show up on many of these lists, as does orwell. Interestingly opossum is showing up on many of these lists as well. I ought to give that one a try one of these days.

5&12 (variety of dominant) [<1, 2, 4, 2, 1], <0, -1, -4, 2, 6]>
15&25 (variety of blacksmith) [<5, 8, 12, 14, 18], <0, 0, -1, 0, -2]>
opossum [<1, 2, 3, 4, 4], <0, -3, -5, -9, -4]>
porcupine [<1, 2, 3, 2, 4], <0, -3, -5, 6, -4]>
keemun [<1, 0, 1, 2, 4], <0, 6, 5, 3, -2]>
pajara [<2, 3, 5, 6, 8], <0, 1, -2, -2, -6]>
semififth / mohajira [<1, 1, 0, 6, 2], <0, 2, 8, -11, 5]>
meantone [<1, 2, 4, 7, 11], <0, -1, -4, -10, -18]>
meanpop [<1, 2, 4, 7, -2], <0, -1, -4, -10, 13]>
magic [<1, 0, 2, -1, 6], <0, 5, 1, 12, -8]>
orwell [<1, 0, 3, 1, 3], <0, 7, -3, 8, 2]>
hitchcock / amity [<1, 3, 6, -2, 6], <0, -5, -13, 17, -9]>
miracle [<1, 1, 3, 3, 2], <0, 6, -7, -2, 15]>
catakleismic [<1, 0, 1, -3, 9], <0, 6, 5, 22, -21]>
wizard [<2, 1, 5, 2, 8], <0, 6, -1, 10, -3]>
unidec [<2, 5, 8, 5, 6], <0, -6, -11, 2, 3]>
harry [<2, 4, 7, 7, 9], <0, -6, -17, -10, -15]>

And the generator mappings for those other temperaments in Graham's cangwu badness list, for reference:

hemiennealimmal [<18, 28, 41, 50, 62], <0, 2, 3, 2, 1]>
<<0 0 0 1 0 0 2 0 3 3|| (no name) [<1, 2, 3, 3, 4], <0, 0, 0, 0, 1]>
342&480 [<6, 11, 17, 20, 24], <0, -17, -35, -36, -37]>
octoid [<8, 13, 19, 23, 28], <0, -3, -4, -5, -3]>
5&8d (variety of father) [<1, 2, 2, 4, 5], <0, -1, 1, -3, -4]>
pentoid (beep) [<1, 2, 3, 3, 3], <0, -2, -3, -1, 2]>
myna [<1, -1, 0, 1, -3], <0, 10, 9, 7, 25]>
septimal (jamesbond) [<7, 11, 16, 20, 24], <0, 0, 0, -1, 0]>
2&4e [<2, 3, 5, 6, 7], <0, 0, -1, -1, -1]>
4e&7 (variety of dicot) [<1, 1, 2, 2, 2], <0, 2, 1, 3, 5]>
injera [<2, 3, 4, 5, 6], <0, 1, 4, 4, 6]>
meanenneadecal [<1, 2, 4, 7, 6], <0, -1, -4, -10, -6]>
diminished [<4, 6, 9, 11, 14], <0, 1, 1, 1, 0]>
sharp (variety of dicot) [<1, 1, 2, 1, 2], <0, 2, 1, 6, 5]>

genewardsmith wrote:

> However, unlike the situation for positive integers, primes are not
> the only possible choices for rational numbers which could be used in
> factorizations like this. For instance, instead of writing the
> 5-limit in terms of 2, 3 and 5, we can equally well write it in terms
> of 9/8, 10/9 and 16/15.

I've considered alternatives, for instance, the superparticular intervals with primes in the numerator (i.e., 2/1, 3/2, 5/4, 7/6, 11/10), or ratios between adjacent primes (2, 3/2, 5/3, 7/5, 11/7). 7-limit temperaments could use 5/4, 6/5, 7/6, and 8/7, which has a certain appeal. Ultimately, I think that using smaller intervals in place of primes can have interesting and useful results, but the choice of which intervals to use and how to weight them is likely a matter of taste. You'll find the same temperaments whichever way you go, from what I've found, but the "badness" evaluations may vary significantly.