Another scale that you just "must" try out -- an interesting discovery

Hi everyone.

Don't know if this is another of the "already mentioned" scales or if it's
really something I've just invented. Anyway, the tuning I'm posting here is
a regular "linear" octave-reduced system whose generator is the 7th root of
6 (i.e. ~443.136428695 cents). What comes out may look like this:

! or7r6.scl
!
Octave-reduced 2d tuning with 6^(1/7) as generator
12
!
129.40929
258.81857
313.72714
388.22786
443.13643
572.54571
3/2
756.86357
886.27286
1015.68214
1145.09143
2/1

Petr

On 5/19/06, Petr Pařízek <p.parizek@chello.cz> wrote:
> Hi everyone.
>
> Don't know if this is another of the "already mentioned" scales or if it's
> really something I've just invented. Anyway, the tuning I'm posting here is
> a regular "linear" octave-reduced system whose generator is the 7th root of
> 6 (i.e. ~443.136428695 cents). What comes out may look like this:
>
> ! or7r6.scl
> !
> Octave-reduced 2d tuning with 6^(1/7) as generator
> 12
> !
> 129.40929
> 258.81857
> 313.72714
> 388.22786
> 443.13643
> 572.54571
> 3/2
> 756.86357
> 886.27286
> 1015.68214
> 1145.09143
> 2/1
>
> Petr

This is "semisixths" with an eigenmonzo of 3/2 (gosh darn I love that
word!). It's a lovely temperament; why don't you write something in
it?

Keenan

Keenan wrote:

> This is "semisixths" with an eigenmonzo of 3/2 (gosh darn I love that
> word!). It's a lovely temperament; why don't you write something in
> it?

I surely would have written if I had known that temperaments like these
existed at all. About 3 months ago, I didn't have a least notion that a
major sixth could be achieved with any other procedure than taking 3 fifths
and removing 2 octaves. When I first delved into the world of BP and found
that two 9/7s also make an acceptable major sixth (just about 14 cents
narrower), I suddenly started to think about totally different kinds of
tunings than I used to make. But still, until yesterday, I had no idea that
such intervals like fifths or octaves could also be found this way. At that
time, I simply began to think of a 3/1-reduced tuning which had pure octaves
in it. That meant I had to find a tuning combining both 3/1 and 2/1. That's
why I chose the factor of 6 as the starting point. Somehow intuitively, I
divided this interval into 7 equal parts. Taking this as a generator and the
3/1 as a formal octave, I made a 14-tone scale this way. Just to see what it
did, I normalized this scale to 2/1 to get rid of the octave duplications
and make it an octave-reduced system. What came out was a 12-tone scale.
Then, I tried to do the same procedure but with a 15-tone scale instead of
14. After normalizing this to 2/1, I also got a 12-tone scale. Still, when I
tried this with 16 or even 17 tones, again, I got the very same 12-tone
scale after normalizing to 2/1. I had to use 18 tones in the original scale
to get at least 13 tones after normalizing. Having seen this, I finally took
the 12-tone scale and played a few chords in it. I can admit this was the
very first time I heard a scale like this. I don't know when it was the last
time you experienced something which was unusual and totally new to you but
I think you can understand how surprised I was. So I just HAD to post the
scale.
BTW: Do you think there is something somewhere on the web about these
"semi-sixth" tunings?

Petr

On 5/20/06, Petr Pařízek <p.parizek@chello.cz> wrote:
> I surely would have written if I had known that temperaments like these
> existed at all. About 3 months ago, I didn't have a least notion that a
> major sixth could be achieved with any other procedure than taking 3 fifths
> and removing 2 octaves. When I first delved into the world of BP and found
> that two 9/7s also make an acceptable major sixth (just about 14 cents
> narrower), I suddenly started to think about totally different kinds of
> tunings than I used to make. But still, until yesterday, I had no idea that
> such intervals like fifths or octaves could also be found this way. At that
> time, I simply began to think of a 3/1-reduced tuning which had pure octaves
> in it. That meant I had to find a tuning combining both 3/1 and 2/1. That's
> why I chose the factor of 6 as the starting point. Somehow intuitively, I
> divided this interval into 7 equal parts. Taking this as a generator and the
> 3/1 as a formal octave, I made a 14-tone scale this way. Just to see what it
> did, I normalized this scale to 2/1 to get rid of the octave duplications
> and make it an octave-reduced system. What came out was a 12-tone scale.
> Then, I tried to do the same procedure but with a 15-tone scale instead of
> 14. After normalizing this to 2/1, I also got a 12-tone scale. Still, when I
> tried this with 16 or even 17 tones, again, I got the very same 12-tone
> scale after normalizing to 2/1. I had to use 18 tones in the original scale
> to get at least 13 tones after normalizing. Having seen this, I finally took
> the 12-tone scale and played a few chords in it. I can admit this was the
> very first time I heard a scale like this. I don't know when it was the last
> time you experienced something which was unusual and totally new to you but
> I think you can understand how surprised I was. So I just HAD to post the
> scale.

Well, it may be a known temperament, but keep doing what you're doing.
If you only find temperaments we've already found, that lends support
to the theory, and if you find a good one that's totally new, that
would really be something.

> BTW: Do you think there is something somewhere on the web about these
> "semi-sixth" tunings?

There's an entry at Tonalsoft: http://tonalsoft.com/enc/s/semisixths.aspx

and Gene has some compositions in it by Mark Gould:
http://66.98.148.43/~xenharmo/coll.htm

Keenan

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:

> I surely would have written if I had known that temperaments like these
> existed at all.

You really should spend time on the tuning-math list.

What hardware and software do you use to hear your scales?

Cornell III, Howard M wrote:
> > What hardware and software do you use to hear your scales?

I don't know if this was addressed to anyone in particular (it would help if you included the context of what you're replying to), but as for me, I use primarily Scala (http://www.xs4all.nl/~huygensf/scala/) and Yamaha DX7II (http://www.vintagesynth.com/yamaha/dx7ii.shtml).

> > This is "semisixths" with an eigenmonzo of 3/2 (gosh darn I love
> > that word!).

I think we sometimes get carried away with the sound of a new word and
fail to notice that it merely obscures meaning for those not "in the
know".

I had never seen the word "eigenmonzo" before and could not figure out
what it meant from its parts, and so I could make no sense of the
above sentence. I went searching and found:
http://www.tonalsoft.com/enc/e/eigenmonzo.aspx
and was still none the wiser, until near the end when Gene finally
gives a clue to what he is on about after Carl says "You've lost me".

I can now tell you that the sentence quoted at the start of this
message could have been written as simply "This is 'semisixths' with a
just fifth", or "This is 'semisixths' with an unchanged interval of
2:3".

And now I'd like to propose the Obfuscation of the Year Award for
2005, for the definition of "eigenmonzo" on the above-linked page. ;-)

Here's my translation:

Every tuning of a linear temperament has one interval, other than the
octave, that is unchanged by the tempering. This is often a simple
rational interval such as the just major third (4:5) of 1/4-comma
meantone or the just minor third (5:6) of 1/3-comma meantone. This
unchanged interval or "eigeninterval" can be used to characterise a
specific tuning of a temperament, so for example we can speak
of "miracle with a just minor third (5:6)" versus "miracle with a just
minor seventh (5:9)".

---

Don't worry about it Gene, you're a genius with the math, we can't
expect everything from you.

A "monzo" or "prime exponent vector" is just one way of representing
an eigeninterval. The fact that you're really talking
about "eigenintervals" not "eigenmonzos" is attested by the fact that
you don't give them in vector form, but as ratios such as 5/4, 9/5.

----

By the way, "semisixths" is one of the few linear temperament names
from which you can actually figure out the temperament. i.e. its
generator is around half of a major sixth.

"Klesimic" is another, assuming you can look up what the kleisma is
and infer from its very small size that it is the vanishing comma of
the temperament (rather than its generator).

I note that the above-linked page
contains "catakleismic", "superkleismic" and "hemikleismic", but
not "kleismic". Why is that? -- he asks completely disingenuously.

-- Dave Keenan

Dave Keenan wrote:

> Every tuning of a linear temperament has one interval, other than the > octave, that is unchanged by the tempering. This is often a simple > rational interval such as the just major third (4:5) of 1/4-comma > meantone or the just minor third (5:6) of 1/3-comma meantone. This > unchanged interval or "eigeninterval" can be used to characterise a > specific tuning of a temperament, so for example we can speak > of "miracle with a just minor third (5:6)" versus "miracle with a just > minor seventh (5:9)".

Nice, but this isn't true of all tunings. Even non-equal tunings. Lucy Tuning, for example, leaves only octaves pure.

> ---
> > Don't worry about it Gene, you're a genius with the math, we can't > expect everything from you.
> > A "monzo" or "prime exponent vector" is just one way of representing > an eigeninterval. The fact that you're really talking > about "eigenintervals" not "eigenmonzos" is attested by the fact that > you don't give them in vector form, but as ratios such as 5/4, 9/5.

How about "unchanged intervals"?

Graham

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

> How about "unchanged intervals"?

It's useful to know that tunings can be viewed as projection operators
and that linear algebra can help to understand the theory, however.

On 5/28/06, Dave Keenan <d.keenan@bigpond.net.au> wrote:
> I think we sometimes get carried away with the sound of a new word and
> fail to notice that it merely obscures meaning for those not "in the
> know".
[...]
> A "monzo" or "prime exponent vector" is just one way of representing
> an eigeninterval. The fact that you're really talking
> about "eigenintervals" not "eigenmonzos" is attested by the fact that
> you don't give them in vector form, but as ratios such as 5/4, 9/5.
[...]

These objections are absolutely valid, but I still love the word all
the same. =P

Keenan Pepper

Keenan Pepper wrote:

> These objections are absolutely valid, but I still love the word all
> the same. =P
>

I love every occurrence of "eigen" in english. It infallibly appears in contexts that I can make nothing of, so the first meaning I think of is the (probably) last one in the dictionaries, that of "funny-peculiar".

klaus

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
>
> > How about "unchanged intervals"?
>
> It's useful to know that tunings can be viewed as projection
operators
> and that linear algebra can help to understand the theory, however.
>

You're absolutely right. For math types who already knew about eigen-
vectors it was great that you saw that connection and pointed it out.
And it was rather unfair of me to complain, since as far as I can
tell, the term was only used on tuning-math.

-- Dave Keenan

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Dave Keenan wrote:
>
> > Every tuning of a linear temperament has one interval, other
than the
> > octave, that is unchanged by the tempering. This is often a
simple
> > rational interval such as the just major third (4:5) of 1/4-
comma
> > meantone or the just minor third (5:6) of 1/3-comma meantone.
This
> > unchanged interval or "eigeninterval" can be used to
characterise a
> > specific tuning of a temperament, so for example we can speak
> > of "miracle with a just minor third (5:6)" versus "miracle with
a just
> > minor seventh (5:9)".
>
> Nice, but this isn't true of all tunings. Even non-equal
tunings. Lucy
> Tuning, for example, leaves only octaves pure.

That's true from a purely mathematical perspective, but we know that
the distinction between rational and irrational intervals is not
audible or measurable and so even for Lucy Tuning there is a
rational interval (in fact an infinite number of rational intervals)
that is within measurement accuracy of being unchanged by the
temperament.

Of course even the simplest such "unchanged" interval may well be so
complex as to be of no interest whatsoever, but I thought it best to
avoid cluttering my tuning list "translation" of Gene's work, with
such details.

I agree that for non-math types the term "unchanged interval" is way
better than "eigeninterval". Apparently even german speakers are
mystified by the english usage of "eigen-".

-- Dave Keenan

On second thoughts, all I have to do to make it true is to add the
word rational at the start like this:

Every rational tuning of a linear temperament has one interval, other
than the octave, that is unchanged by the tempering. This is often a
simple rational interval such as the just major third (4:5) of 1/4-
comma meantone or the just minor third (5:6) of 1/3-comma meantone.
This unchanged interval or "eigeninterval" can be used to characterise
a specific tuning of a temperament, so for example we can speak
of "miracle with a just minor third (5:6)" versus "miracle with a just
minor seventh (5:9)".

Good point Graham.

I hasten to add that Gene already had this covered in his original
version.

-- Dave Keenan

Dave Keenan wrote:
> On second thoughts, all I have to do to make it true is to add the > word rational at the start like this:
> > Every rational tuning of a linear temperament has one interval, other > than the octave, that is unchanged by the tempering. This is often a > simple rational interval such as the just major third (4:5) of 1/4-
> comma meantone or the just minor third (5:6) of 1/3-comma meantone. > This unchanged interval or "eigeninterval" can be used to characterise > a specific tuning of a temperament, so for example we can speak
> of "miracle with a just minor third (5:6)" versus "miracle with a just
> minor seventh (5:9)".

Um, yes, but ... what's a rational tuning? It isn't a tuning with rational number frequency ratios, which is the first thing I expect. It is one with rational prime-power coefficients, or algebraic number frequency ratios. But neither of those are very digestible concepts.

A better way is to follow your other post and add the words "near enough":

Every tuning of a linear temperament has one interval, other than the
octave, that is near enough unchanged by the tempering. This is often a
simple rational interval such as the just major third (4:5) of 1/4-
comma meantone or the just minor third (5:6) of 1/3-comma meantone.
This unchanged interval or "eigeninterval" can be used to characterise
a specific tuning of a temperament, so for example we can speak
of "miracle with a just minor third (5:6)" versus "miracle with a just
minor seventh (5:9)".

A problem with this as a definiton is that different ideas of "near enough" will lead to different results. So let's translate it to the active voice:

Consider tuning a linear temperament so that one interval, other than
the octave, is unchanged by the tempering. You may choose a
simple rational interval such as the just major third (4:5) of 1/4-
comma meantone or the just minor third (5:6) of 1/3-comma meantone.
We may characterise the specific tuning of the temperament by this unchanged interval or "eigeninterval". For example we can speak
of "miracle with a just minor third (5:6) "versus "miracle with a just
minor seventh (5:9)".

Graham

Message 7 From: "Dave Keenan" d.keenan@bigpond.net.au
Date: Sun May 28, 2006 6:38pm(PDT) Subject: Obfuscation award

>> > > This is "semisixths" with an eigenmonzo of 3/2 (gosh darn I love >> > > that word!).
>> I think we sometimes get carried away with the sound of a new word and fail to notice that it merely obscures meaning for those not "in the know".

I had never seen the word "eigenmonzo" before and could not figure out what it meant from its parts, and so I could make no sense of the above sentence. I went searching and found:
http://www.tonalsoft.com/enc/e/eigenmonzo.aspx
and was still none the wiser, until near the end when Gene finally gives a clue to what he is on about after Carl says "You've lost me".

I can now tell you that the sentence quoted at the start of this message could have been written as simply "This is 'semisixths' with a just fifth", or "This is 'semisixths' with an unchanged interval of 2:3".

semi sixth? how about half sixth, i took semi to mean part, not necessarily an equal part.
Half says exactly what it is

And now I'd like to propose the Obfuscation of the Year Award for 2005, for the definition of "eigenmonzo" on the above-linked page. ;-) Here's my translation:

Every tuning of a linear temperament has one interval, other than the octave, that is unchanged by the tempering. This is often a simple rational interval such as the just major third (4:5) of 1/4-comma meantone or the just minor third (5:6) of 1/3-comma meantone. This unchanged interval or "eigeninterval" can be used to characterise a specific tuning of a temperament, so for example we can speak of "miracle with a just minor third (5:6)" versus "miracle with a just minor seventh (5:9)".

Unchanged interval? how about generator. and doesn't Paul Erlich like many indonesian scales models have temperaments which stretch or shrinks the octave.
also there are temperaments where the interval does change in varied sizes within a certain range

---

Don't worry about it Gene, you're a genius with the math, we can't expect everything from you.

A "monzo" or "prime exponent vector" is just one way of representing an eigeninterval. The fact that you're really talking about "eigenintervals" not "eigenmonzos" is attested by the fact that you don't give them in vector form, but as ratios such as 5/4, 9/5.

----

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Message 14 From: "Dave Keenan" d.keenan@bigpond.net.au
Date: Mon May 29, 2006 2:54am(PDT) Subject: Re: Obfuscation award

That's true from a purely mathematical perspective, but we know that the distinction between rational and irrational intervals is not audible or measurable

of course it is, if not directly , often the smallest degree of error pops up and shows it ugly head.
This was the assumption behind 768 ET as a midi standard, a momentous mistake that we are still dealing with and probably why microtones did not catch on as we might have hoped.
best not to have such assumptions and not end up in the same place again.

and so even for Lucy Tuning there is a rational interval (in fact an infinite number of rational intervals) that is within measurement accuracy of being unchanged by the temperament. Of course even the simplest such "unchanged" interval may well be so complex as to be of no interest whatsoever, but I thought it best to avoid cluttering my tuning list "translation" of Gene's work, with such details. I agree that for non-math types the term "unchanged interval" is way better than "eigeninterval". Apparently even german speakers are mystified by the english usage of "eigen-". -- Dave Keenan
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

still no one will have any idea what is meant by miracle, and eigen is not useful. at all
nor does anyone understand what is meant by linear, as the linear chain formed by the repetition of the same interval

Consider tuning a linear temperament so that one interval, other than
the octave, is unchanged by the tempering. You may choose a
simple rational interval such as the just major third (4:5) of 1/4-
comma meantone or the just minor third (5:6) of 1/3-comma meantone.
We may characterise the specific tuning of the temperament by this unchanged interval or "eigeninterval". For example we can speak
of "miracle with a just minor third (5:6) "versus "miracle with a just
minor seventh (5:9)".

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

> Here's my translation:
>
> Every tuning of a linear temperament has one interval, other
> than the octave, that is unchanged by the tempering. This is
> often a simple rational interval such as the just major
> third (4:5) of 1/4-comma meantone or the just minor third (5:6)
> of 1/3-comma meantone. This unchanged interval or
> "eigeninterval" can be used to characterise a specific tuning
> of a temperament, so for example we can speak of "miracle with
> a just minor third (5:6)" versus "miracle with a just minor
> seventh (5:9)".

I hope monz is watching.

*Every tuning*? Really?

> By the way, "semisixths" is one of the few linear temperament names
> from which you can actually figure out the temperament. i.e. its
> generator is around half of a major sixth.

If you have the secret decoder ring. I had no idea what this
temperament was until Keenan gave this answer to Petr.
Even if we look past the fact that diatonic nomenclature is
completely inappropriate here, what exactly is the size range
of a "sixth"? I'll take eigenmonzo anyday, thanks.

> "Klesimic" is another, assuming you can look up what the
> kleisma is and infer from its very small size that it is
> the vanishing comma of the temperament (rather than its
> generator).

Do you have a proposal for finding canonical generators
for temperaments, then?

-Carl

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@...> wrote:
>
> Keenan Pepper wrote:
>
> > These objections are absolutely valid, but I still love the word all
> > the same. =P
> >
>
> I love every occurrence of "eigen" in english. It infallibly appears in
> contexts that I can make nothing of, so the first meaning I think of is
> the (probably) last one in the dictionaries, that of "funny-peculiar".

I went looking on Wikipedia to see how it explains things, and Keenan
absolutely *must* read this article:

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

It has "eigenfaces"! Yes, you heard that right. Pictures of them, too.
Plus a distorted version of the Mona Lisa, and an animated graphic image.

Here's the article on eigenfaces, which I for one had never heard of:

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

Given all that, I don't think "eigenmonzo" is that bad, but you might
prefer "eigeninterval" instead. It makes me wonder, though, about
producing eigentunes instead of eigenfaces.

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

> You're absolutely right. For math types who already knew about eigen-
> vectors it was great that you saw that connection and pointed it out.
> And it was rather unfair of me to complain, since as far as I can
> tell, the term was only used on tuning-math.

Keenan Pepper likes it, which has led to it being mentioned elsewhere.
What can I say? I kind of like eigenfaces myself.

On 5/29/06, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> http://en.wikipedia.org/wiki/Eigenface

Yeah, I heard about that! Weird, huh?

> Given all that, I don't think "eigenmonzo" is that bad, but you might
> prefer "eigeninterval" instead. It makes me wonder, though, about
> producing eigentunes instead of eigenfaces.

Eigentunes? Get outta here! =P

Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> still no one will have any idea what is meant by miracle...

If you don't know what miracle is by now, it's because you haven't
paid attention. Your choice.

> > still no one will have any idea what is meant by miracle...
>
> If you don't know what miracle is by now, it's because you haven't
> paid attention. Your choice.

I think Kraig was just pointing out that by Dave's logic, one
of the things Dave named has an inappropriate name. He also
took a stab at "linear", which AFAIK comes from Erv.

-Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> Um, yes, but ... what's a rational tuning? It isn't a tuning with
> rational number frequency ratios, which is the first thing I
expect. It
> is one with rational prime-power coefficients, or algebraic number
> frequency ratios. But neither of those are very digestible
concepts.
>

You're absolutely right. Must have been too late at night. Scratch
that idea.

> A better way is to follow your other post and add the words "near
enough":
>

Right, but in the cases that are interesting it is exact, so "either
exactly or near enough".

> A problem with this as a definiton is that different ideas
of "near
> enough" will lead to different results. So let's translate it to
the
> active voice:
>
> Consider tuning a linear temperament so that one interval, other
than
> the octave, is unchanged by the tempering. You may choose a
> simple rational interval such as the just major third (4:5) of 1/4-
> comma meantone or the just minor third (5:6) of 1/3-comma meantone.
> We may characterise the specific tuning of the temperament by this
> unchanged interval or "eigeninterval". For example we can speak
> of "miracle with a just minor third (5:6) "versus "miracle with a
just
> minor seventh (5:9)".

Fine.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
> semi sixth? how about half sixth, i took semi to mean part, not
necessarily an equal part.
> Half says exactly what it is

Feel free to call it that, but the root meaning of semi- (and hemi-
and demi-) is "half", and it certainly has that meaning in its
musical usages, in semitone, semiquaver, semibrieve. But maybe
americans don't use those terms.

> Unchanged interval? how about generator.

No, the unchanged interval is not the generator. All meantones are
generated by a tempered fifth (or fourth), but the unchanged
interval of 1/4-comma is a 4:5 major third and the unchanged
interval of 1/3-comma meantone is a 5:6 minor third.

All Miracle temperaments are generated by a "secor", a tempered
minOR SECond of around 116.6 cents or 7/72-oct, but it has varieties
with unchanged intervals of just minor thirds 5:6 or just major
seconds 9:10 (and their octave inversions and extensions).

>and doesn't Paul Erlich like many indonesian scales models have
temperaments which stretch or shrinks the octave.

Yes. Those are good. And my version of Gene's statement would have
to be tweaked to accomodate them, but I thought it was better to get
the basic idea out there without it being swamped by all these
caveats, even if it isn't 100% accurate. It's like when you tell
people that atoms have electrons orbiting around a nucleus rather
than get into quantum mechanics right away.

> also there are temperaments where the interval does change in
varied sizes within a certain range
>

Yes. I think Graham dealt with that.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
> From: "Dave Keenan" d.keenan@...
>
> That's true from a purely mathematical perspective, but we know
that the
> distinction between rational and irrational intervals is not
audible or
> measurable
>
> of course it is, if not directly , often the smallest degree of
error
> pops up and shows it ugly head.
> This was the assumption behind 768 ET as a midi standard, a
momentous
> mistake that we are still dealing with and probably why microtones
did
> not catch on as we might have hoped.
>
> best not to have such assumptions and not end up in the same
place again.
>

Hi Kraig,

You should know, from the Sagittal project, that I am not proposing
any particular ET as "close enough to JI", and certainly not
anything as coarse as 768-ET.

However, you should also know that for any irrational number there
is a rational number so close to it that if they were both realised
as musical intervals there would not be a single beat between them
in your lifetime, or before the sun burns out, or before the
universe becomes uninhabitable by hearing beings, or whatever
stringent condition you wish to apply, short of infinity.

That is all I mean when I say that "The distinction between rational
and irrational intervals is not audible or measurable". Perhaps I
should say, "is not (in general) audible or measurable."

Of course many rational intervals (particularly the simpler ones)
are audibly very different from irrational intervals that are not
close to them (particularly the noble numbers with the smaller
coefficients).

-- Dave Keenan

Hi all,

From: Gene Ward Smith on Mon May 29, 2006:
[snip]
> I went looking on Wikipedia to see how it explains things, and Keenan
> absolutely *must* read this article:
>
> http://en.wikipedia.org/wiki/Eigenvalue
>
> It has "eigenfaces"! Yes, you heard that right. Pictures of them, too.
> Plus a distorted version of the Mona Lisa, and an animated graphic image.
>
> Here's the article on eigenfaces, which I for one had never heard of:
>
> http://en.wikipedia.org/wiki/Eigenface

Because the technique is more generally useful
than just (!) for recognising faces, some authors
prefer the term 'eigenimage'.

Still, it's possible to specify almost any human
face as a linear combination of a fairly small set
of eigenfaces eg 23% of Face A + 17% of Face B
+ 11% of Face c + ...

> Given all that, I don't think "eigenmonzo" is that bad, but you might
> prefer "eigeninterval" instead. It makes me wonder, though, about
> producing eigentunes instead of eigenfaces.

Smart thought! A long time back, I wondered
whether it would be practical to use computers
help recognise the kinds of pattern that exist
in a given musical style, then use those patterns
to automate creating new music in that style.
At the time (60s), space and speed constraints on
computers mandated the representation of music
in highly abstract forms, eg representing all notes
as being from a limited set of pitch classes - the
forerunners of MIDI. But given today's compu-
ting power and capacity, perhaps such a program
might begin by directly processing either -
a) images of musical scores,
or -
b) digitally recorded music.

Long live the eigenminuetandrondo!

Regards,
Yahya

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> Long live the eigenminuetandrondo!

Eigenmusic is an interesting idea. Best to work on
a note-based format like MIDI, since you'd have to
extract it from sound anyway to get a representation
of what humans hear (and this is an unsolved problem).

Even so, I suspect eigenmusic will be harder than
eigenfaces, because of the temporal nature of music.
Images are more meaningful excerpts of video than,
say, chords, I should think.

-Carl

> > Long live the eigenminuetandrondo!
>
> Eigenmusic is an interesting idea. Best to work on
> a note-based format like MIDI, since you'd have to
> extract it from sound anyway to get a representation
> of what humans hear (and this is an unsolved problem).
>
> Even so, I suspect eigenmusic will be harder than
> eigenfaces, because of the temporal nature of music.
> Images are more meaningful excerpts of video than,
> say, chords, I should think.
>
> -Carl

Maybe I should have posted this on SpecMus. Yahya, are
you a member?

-C.

Hi Carl,

On Tue May 30, 2006, Carl Lumma wrote:
>
> > > Long live the eigenminuetandrondo!
> >
> > Eigenmusic is an interesting idea. Best to work on
> > a note-based format like MIDI, since you'd have to
> > extract it from sound anyway to get a representation
> > of what humans hear (and this is an unsolved problem).
> >
> > Even so, I suspect eigenmusic will be harder than
> > eigenfaces, because of the temporal nature of music.
> > Images are more meaningful excerpts of video than,
> > say, chords, I should think.
> >
> > -Carl
>
> Maybe I should have posted this on SpecMus. Yahya, are
> you a member?
>
> -C.

Nope, never heard of it. Should I've?

And how could I possibly keep up with
yet ANOTHER list? ;-) (I'm already
days behind with tuning-math.)

Regards,
Yahya

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On 5/31/06, Yahya Abdal-Aziz <yahya@melbpc.org.au> wrote:
>
> Hi Carl,
>
> On Tue May 30, 2006, Carl Lumma wrote:
> >
> > > > Long live the eigenminuetandrondo!
> > >
> > > Eigenmusic is an interesting idea. Best to work on
> > > a note-based format like MIDI, since you'd have to
> > > extract it from sound anyway to get a representation
> > > of what humans hear (and this is an unsolved problem).
> > >
> > > Even so, I suspect eigenmusic will be harder than
> > > eigenfaces, because of the temporal nature of music.
> > > Images are more meaningful excerpts of video than,
> > > say, chords, I should think.
> > >
> > > -Carl
> >
> > Maybe I should have posted this on SpecMus. Yahya, are
> > you a member?
> >
> > -C.
>
> Nope, never heard of it. Should I've?
>
> And how could I possibly keep up with
> yet ANOTHER list? ;-) (I'm already
> days behind with tuning-math.)
>
> Regards,
> Yahya
>
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>
>
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
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> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>
>
>
>
>
>

Whoops, excuse the empty message.

> > Maybe I should have posted this on SpecMus. Yahya, are
> > you a member?
> >
> > -C.
>
> Nope, never heard of it. Should I've?
>
> And how could I possibly keep up with
> yet ANOTHER list? ;-) (I'm already
> days behind with tuning-math.)

It's a very low-volume list.

-Carl

On Wed May 31, 2006 Carl Lumma wrote:
>
> > > Maybe I should have posted this on SpecMus. Yahya, are
> > > you a member?
> >
> > Nope, never heard of it. Should I've?
> >
> > And how could I possibly keep up with
> > yet ANOTHER list? ;-) (I'm already
> > days behind with tuning-math.)
>
> It's a very low-volume list.

Hi Carl,

Just checked out:
/SpecMus/

As you said, very low volume - almost moribund,
I think! At such a low volume, would it be possible
to maintain a conversation, I wonder ...

But yes, from its description, its seems to maybe
straddle the divide between pure TUNING theory
and MAKING micro music - as well as potentially
address other aspects of music-making besides
tuning.

That might be the place for me to start asking
questions about "micro-times" - for if any aspects
of music have remained subject to the tyranny of
those who won't count beyond 2 and 3, I suspect
they may be musical metre and rhythm.

Regards,
Yahya

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