Graylessness and limit (was: notation systems)

Thanks Paul, for the thing on Reiman's zeta. Thanks also for the concept of
graylessness. If I may paraphrase:

An n-tET (n-EDO) is grayless at the m-limit if all m-limit ratios are no
further than a quarter step (+-1/4n of an octave) away from some degree of
the n-tET. This is equivalent to being Hahn level 2 consistent at the m-limit.

Is that all correct?

Paul Erlich wrote:

>> The lowest "gray-less" ETs for various odd limits are:
>>
>> odd limit ET
>>
>> 7 31
>> 9 41
>> 11 72
>> 13 270
>> 15 494
>> 17 3395
>> 19-21 8539

So it's:

odd limit ET
3 2
5 3

I see now why you left them off.

So these are the only ETs worth basing JI notation systems on? 72 is
definitely a winner. Rather than call them 1/12th-tones, could we call them
sextants? If 1/100th of a semitone is a cent then 1/6th of a semitone is a
sextant.

David Finnamore replied:

>Oh, yeah. I had an uncle once who worked exclusively in 8,539 EDO.
>The guy was nuts about the 21 limit. "Gray-less?" he used to say,
>"You don't know from grayless!" He had a special 61,908 note
>keyboard built for it. Since his fingers were 79 feet long, with
>tips no more than a millimeter in circumference, this was not a
>problem. As long as he kept his nails clipped. ;-)

Thanks for a good laugh David. It does rather point up the ridiculousness
of claiming that some piece is >15 limit JI. Who could tell? Even 13 is
pushing things.

Before I get jumped-on, I accept the validity of stuff like LaMonte
Young's, with notes sustained for minutes. But this seems to rely on
difference tones between sine waves through a nonlinearity, in which case
the idea of "harmonic limit" pretty much loses its meaning.

I guess there's also a "gray area" between these high numbered things that
rely on difference tones and the more conventional lower numbered things
that rely on coinciding partials.

Regards,
-- Dave Keenan
http://dkeenan.com

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/15523

>
> Before I get jumped-on, I accept the validity of stuff like LaMonte
> Young's, with notes sustained for minutes. But this seems to rely on
> difference tones between sine waves through a nonlinearity, in
which case the idea of "harmonic limit" pretty much loses its
meaning.
>
> I guess there's also a "gray area" between these high numbered
things that rely on difference tones and the more conventional lower
numbered things that rely on coinciding partials.
>

This is an interesting point, and it certainly seems to me, upon
extended listening to La Monte Young's "high partial" static
structures... I'm thinking specifically of the "Dream House..." that
there is really very little relationship to the otonal harmonic
series evidenced in the composite sound...

Do the La Monte Young 'experts' here agree??

__________ ____ __ __ _
Joseph Pehrson

Joseph Pehrson wrote:
>
> --- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
>
> http://www.egroups.com/message/tuning/15523
>
> >
> > Before I get jumped-on, I accept the validity of stuff like LaMonte
> > Young's, with notes sustained for minutes.

Try YEARS. It ain't called the Theatre of Eternal Music for nothing Bub.

> But this seems to rely on
> difference tones between sine waves through a nonlinearity, in
> which case the idea of "harmonic limit" pretty much loses its
> meaning.

How can his music be non-linear if he sustains his music for long
periods of time? Eh?

> > I guess there's also a "gray area" between these high numbered
> things that rely on difference tones and the more conventional lower
> numbered things that rely on coinciding partials.
> >
>
> This is an interesting point, and it certainly seems to me, upon
> extended listening to La Monte Young's "high partial" static
> structures... I'm thinking specifically of the "Dream House..." that
> there is really very little relationship to the otonal harmonic
> series evidenced in the composite sound...
>
> Do the La Monte Young 'experts' here agree??

His tunings are nothing BUT overtones (otonality). If you think that
his tunings bear little relationship to the otonal series, well, you're
stuck in a 12tet/5-limit mind frame. There are few modern composers who
to push their way up the harmonic series to the outland while there's a
plethora of
folks standing in the corner, shuffling their feet, looking distracted
but acting focused while discussing equal temperaments.

db

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:

http://www.egroups.com/message/tuning/15533

>
> His tunings are nothing BUT overtones (otonality). If you think
that
> his tunings bear little relationship to the otonal series, well,
you're stuck in a 12tet/5-limit mind frame. There are few modern
composers who to push their way up the harmonic series to the outland
while there's a plethora of folks standing in the corner, shuffling
their feet, looking distracted but acting focused while discussing
equal temperaments.
>

Hi David!

Well, in "shuffling my feet" I really wasn't intending to "step on
any toes!" Sorry... looks like a hit a "hot spot" again without even
realizing it!

OK. I'm willing to admit perhaps it's just my listening... maybe I'm
not HEARING the higher-integer OTONAL relationships yet... surely it
seems like the "difference tone" phenominon (which is more akin to
the "artificial" UTONAL set, correct??)is more what I'm immediately
getting....

Anybody else have any comments?? It's clear what David's position
is, and I'm assuming thereby most of the rest of the Young "camp"
feels similarly and strongly.

Very interesting. Sorry I'm so "naive" about the "politics."
(There's been a lot of THAT lately!)
____________ ____ __ _
Joseph Pehrson

Dave Keenan wrote,

>Is that all correct?

Yup!

>So it's:

> odd limit ET
> 3 2
> 5 3

>I see now why you left them off.

Yeah, graylessness is useless unless the ET is sufficiently accurate to
uniquely represent all the m-limit ratios.

>Rather than call them 1/12th-tones, could we call them
>sextants?

The Boston Microtonal Society calls them 1/12-tones, or "twelfths" for short
(potentially confusing with the interval of a twelfth).

>Thanks for a good laugh David. It does rather point up the ridiculousness
>of claiming that some piece is >15 limit JI. Who could tell? Even 13 is
>pushing things.

I don't see why you say that. Jon Catler's music is clearly in 13-limit JI.
ETs don't enter the picture. It's only if you wanted a highly unambiguous
linear commeasurability of 13-limit intervals that 270-tET might come into
the picture.

Joseph Pehrson wrote,

>surely it
>seems like the "difference tone" phenominon (which is more akin to
>the "artificial" UTONAL set, correct??)

No, sir. I've gone over this so many times . . . difference tones are
unequivocally an OTONAL phenomenon . . . shall I explain this once more?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15541

>I've gone over this so many times . . . difference tones
are unequivocally an OTONAL phenomenon . . . shall I explain this
once more?

Ummm... I guess you're going to have to... sorry! Am I confusing
this with a "sub tone??" maybe a touch of the math here would be
helpful...

The UTONAL series starts with the "guide tone" and then multiplies by
INVERSIONAL fractions, yes?? 1/2, 1/3, 1/4 X the guide tone
frequency... in a somewhat "unnatural" development.

The DIFFERENCE tone (which I assume is the same thing that is created
by a "ring modulator, yes?" MULTIPLIES the frequencies?? But ALSO in
a somewhat "unnatural" development (??)

Ok, I get lost here...

________ ___ __ _
JP

Me:

>> Before I get jumped-on, I accept the validity of stuff like LaMonte
>> Young's, with notes sustained for minutes.

David Beardsley:

>Try YEARS. It ain't called the Theatre of Eternal Music for nothing Bub.

"Bub"? So I get jumped on anyway? Oh well. :-) My point was that music
"like" this must sustain notes for _at_least_ minutes to achieve its effect.

>> But this seems to rely on
>> difference tones between sine waves through a nonlinearity, in
>> which case the idea of "harmonic limit" pretty much loses its
>> meaning.

>How can his music be non-linear if he sustains his music for long
>periods of time? Eh?

It's not the music that's non-linear. It's what is between the oscillators
and the listener, mostly the listeners ears. Apparently, "non-linear" means
something different to you than it does to me.

A non-linear function f of a single variable x is one that cannot be
represented as f(x) = mx + c, where m and c are arbitrary (complex)
constants. e.g. f(x) = x^2 or f(x) = e^x.

But you don't need to understand the maths. The important point is (as has
been explained many times by Paul Erlich and others) that a collection of
sine waves passing through a non-linear medium produce new sine waves at
frequencies which are the sum and difference of integer multiples of the
original frequencies. This doesn't happen with a linear medium. The
amplitudes of these various combination tones depends on the exact nature
of the non-linearity. Another name for non-linearity is "distortion".

Very few things in the real world are linear, but many are approximately
linear. With sound, one can get some idea if a non-linear effect is
involved by turning down the volume. If the effect diminishes rapidly as
you turn down the volume, chances are it's due to non-linearity, whether in
the equipment or your ear.

I imagine that listening to the Dream House output at low levels, mixed
down to a set of headphones, would be completely pointless. The same is not
true of say an 11-limit JI piece using timbres rich in harmonics (as
opposed to sine waves).

>There are few modern composers who
>to push their way up the harmonic series to the outland while there's a
>plethora of
>folks standing in the corner, shuffling their feet, looking distracted
>but acting focused while discussing equal temperaments.

I'm not particularly interested in hero-worship. I'm personally more
interested in the lower levels of the physics and psychology of music,
which I have some chance of mathematically modelling. But there's no doubt
that such things as hero-worship do come into the appreciation of all art.

I'm only trying to point out a distinction that I think has gone largely
unrecognised.

The term "just intonation" is being used to describe two very different
phenomena, but they do blur into one another.

What I think of as "historical" JI is the elimination of beating and the
minimisation of roughness of chords that is due to partials not coinciding
closely enough or not being far enough apart (critical band or harmonic
entropy theory). This assumes timbres with significant amounts of partials
and is just as important at low volume levels as at high. Chords only need
to be sustained for half a second to appreciate the effect.

Then there's this new stuff based on very precise large integer ratios and
sine waves with long sustain. I'm not sure what to call it.

Think about the original meaning of "just intonation". As I understand it,
it referred to taking a note in an underlying scale, which would remain
melodically and harmonically intelligible despite variations in pitch of up
to a comma, and "justly intoning" it, e.g. moving one's finger slightly on
the violin, or altering the vocal pitch slightly, until the beats cancelled.

This idea doesn't seem to apply to the Dream House.

Regards,
-- Dave Keenan
http://dkeenan.com

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/15546

> A non-linear function f of a single variable x is one that cannot be
> represented as f(x) = mx + c, where m and c are arbitrary (complex)
> constants. e.g. f(x) = x^2 or f(x) = e^x.
>
> But you don't need to understand the maths. The important point is
(as has been explained many times by Paul Erlich and others) that a
collection of sine waves passing through a non-linear medium produce
new sine waves at frequencies which are the sum and difference of
integer multiples of the original frequencies. This doesn't happen
with a linear medium. The amplitudes of these various combination
tones depends on the exact nature of the non-linearity. Another name
for non-linearity is "distortion".
>

Thanks so much, Dave, for explaining this. Actually, in my own
defense, I don't think I was yet on the list when Paul was explaining
this over and over. Paul HAS explained things to me over and over...
but this, I don't believe, is one of them... :)

I'm gathering that a "classic ring modulator" which operates with sum
and difference tones, creates timbre-chords using this model (??)

Now, when we say the "sum and difference of integer multiples of the
original frequencies" does that mean one actually SUMS or SUBTRACTS
the frequencies of the multiples arithmetically, or is it a kind of
"ratio sum" which is actully multiplying?? Now I know Paul never
explained THAT to me...

>
> I imagine that listening to the Dream House output at low levels,
mixed down to a set of headphones, would be completely pointless.

So THAT'S why it's so d... loud. I was wondering. I guess that
would be important for the effect!

The same is not
> true of say an 11-limit JI piece using timbres rich in harmonics (as
> opposed to sine waves).
>

Got it.

_______ ___ __ _
Joseph Pehrson

Dave Keenan wrote,

>I imagine that listening to the Dream House output at low levels, mixed
>down to a set of headphones, would be completely pointless.

David, that's rather harsh. Maybe there are a lot of wonderful things about
the Dream House that have nothing to do with the exact intonation?

Joseph wrote,

>Thanks so much, Dave, for explaining this. Actually, in my own
>defense, I don't think I was yet on the list when Paul was explaining
>this over and over. Paul HAS explained things to me over and over...
>but this, I don't believe, is one of them... :)

>I'm gathering that a "classic ring modulator" which operates with sum
>and difference tones, creates timbre-chords using this model (??)

>Now, when we say the "sum and difference of integer multiples of the
>original frequencies" does that mean one actually SUMS or SUBTRACTS
>the frequencies of the multiples arithmetically, or is it a kind of
>"ratio sum" which is actully multiplying?? Now I know Paul never
>explained THAT to me...

You looked into the archives, then, I take it? Good! (...maybe someone
should pull together a list of references to few choice archives and put
them into a FAQ of some sort...) Anyway, to calculate the frequencies of
non-linear combination tones, one actually SUMS or SUBTRACTS the frequencies
arithmetically. So, if all the tones are in an otonal chord, all the sums or
subtractions will result in frequencies that are in the same otonal series,
since they are also integer multiples of the original fundamental, i.e.,
harmonics of the original fundamental.

For a subharmonic series and its guide tone, this kind of agreement doesn't
happen at all.

This didn't seem to make it out (sorry if duplicated) . . .

Joseph wrote,

>Thanks so much, Dave, for explaining this. Actually, in my own
>defense, I don't think I was yet on the list when Paul was explaining
>this over and over. Paul HAS explained things to me over and over...
>but this, I don't believe, is one of them... :)

>I'm gathering that a "classic ring modulator" which operates with sum
>and difference tones, creates timbre-chords using this model (??)

>Now, when we say the "sum and difference of integer multiples of the
>original frequencies" does that mean one actually SUMS or SUBTRACTS
>the frequencies of the multiples arithmetically, or is it a kind of
>"ratio sum" which is actully multiplying?? Now I know Paul never
>explained THAT to me...

You looked into the archives, then, I take it? Good! (...maybe someone
should pull together a list of references to few choice archives and put
them into a FAQ of some sort...) Anyway, to calculate the frequencies of
non-linear combination tones, one actually SUMS or SUBTRACTS the frequencies
arithmetically. So, if all the tones are in an otonal chord, all the sums or
subtractions will result in frequencies that are in the same otonal series,
since they are also integer multiples of the original fundamental, i.e.,
harmonics of the original fundamental.

For a subharmonic series and its guide tone, this kind of agreement doesn't
happen at all.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15560

>>
> You looked into the archives, then, I take it? Good! (...maybe
someone should pull together a list of references to few choice
archives and put them into a FAQ of some sort...)

Hi Paul...

I'm pretty certain of what's in the archives since December 1999 when
I came "on board" since I managed to review most of it at one time or
another.

My suspicion is that this stuff is somewhere between the origin of
the list in Dec. 1998 and the next year. Unfortunately, I have not
yet had time to "comb through" all that stuff... although I did start,
and would like to...

In any case, thanks again for the help!

__________ ___ __ __ _
Joseph Pehrson

Joseph Pehrson wrote:

> Hi David!
>
> Well, in "shuffling my feet" I really wasn't intending to "step on
> any toes!" Sorry... looks like a hit a "hot spot" again without even
> realizing it!

Just ranting. Wasn't really intending to jump on your case.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

David C Keenan wrote:

> Apparently, "non-linear" means
> something different to you than it does to me.

Seems to be the case.

> I imagine that listening to the Dream House output at low levels, mixed
> down to a set of headphones, would be completely pointless.

Then it wouldn't be a realisation of his composition.

>
> >There are few modern composers who
> >to push their way up the harmonic series to the outland while there's a
> >plethora of
> >folks standing in the corner, shuffling their feet, looking distracted
> >but acting focused while discussing equal temperaments.
>
> I'm not particularly interested in hero-worship.

You obviously missed the point I was making about his music.

> I'm personally more
> interested in the lower levels of the physics and psychology of music,
> which I have some chance of mathematically modelling. But there's no doubt
> that such things as hero-worship do come into the appreciation of all art.
>
> I'm only trying to point out a distinction that I think has gone largely
> unrecognised.
>
> The term "just intonation" is being used to describe two very different
> phenomena, but they do blur into one another.
>
> What I think of as "historical" JI is the elimination of beating and the
> minimisation of roughness of chords that is due to partials not coinciding
> closely enough or not being far enough apart (critical band or harmonic
> entropy theory). This assumes timbres with significant amounts of partials
> and is just as important at low volume levels as at high. Chords only need
> to be sustained for half a second to appreciate the effect.
>
> Then there's this new stuff based on very precise large integer ratios and
> sine waves with long sustain. I'm not sure what to call it.
>
> Think about the original meaning of "just intonation". As I understand it,
> it referred to taking a note in an underlying scale, which would remain
> melodically and harmonically intelligible despite variations in pitch of up
> to a comma, and "justly intoning" it, e.g. moving one's finger slightly on
> the violin, or altering the vocal pitch slightly, until the beats cancelled.
>
> This idea doesn't seem to apply to the Dream House.

Just Intonation is a tuning system where intervals are tuned by whole
number ratios.
The definition is that simple and covers all of the above. A simple
concept that you're
trying to make more complicated.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

Joseph Pehrson wrote,

>My suspicion is that this stuff is somewhere between the origin of
>the list in Dec. 1998 and the next year.

Likely true, though Dec. '98 was merely the time the list moved over to the
onelist server. Before that, it had been going on for about five years on
the Mills server -- and you'll find more on this in _those_ archives
(wherever they are), too . . .

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15570

> Joseph Pehrson wrote,
>
> >My suspicion is that this stuff is somewhere between the origin of
> >the list in Dec. 1998 and the next year.
>
> Likely true, though Dec. '98 was merely the time the list moved
over to the onelist server. Before that, it had been going on for
about five years on the Mills server -- and you'll find more on this
in _those_ archives (wherever they are), too . . .

I hear tell that Joe Monzo and Carl Lumma have a lot of these posts.
Do you think it would be possible someday to save these posts in a
text file and store it in the ONELIST files area??

Oh... a search did show some activity related to "difference tones"
after the first year on Onelist, but it was related to Kraig Grady's
utonal discussion regarding his music, and not so specifically
"spelled out" in the otonal realm...

________ ____ __ _
Joseph Pehrson

Joseph Pehrson wrote,

>Oh... a search did show some activity related to "difference tones"
>after the first year on Onelist, but it was related to Kraig Grady's
>utonal discussion regarding his music, and not so specifically
>"spelled out" in the otonal realm...

I'm sure this would be relevant . . . Kraig does favor otonal as more
"consonant" than utonal, unlike Partch . . . Kraig has used difference tones
in his consonance calculations . . . and this is not just a coincidence.

I earlier (mistakenly) wrote:

>>Thanks for a good laugh David. It does rather point up the
>>ridiculousness
>>of claiming that some piece is >15 limit JI. Who could tell? Even 13 is
>>pushing things.

Carl Lumma and Paul Erlich objected. Thanks to them I realise I made an
unwarranted conclusion from the minimum grayless ET data, and retract the
above, except for the "thanks for a good laugh" bit. :-)

Me:

>>I imagine that listening to the Dream House output at low levels, mixed
>>down to a set of headphones, would be completely pointless.

Paul Erlich:

>David, that's rather harsh.

I only "imagine"d that. I'm happy to be corrected. But David Beardsley
agreed with me.

>Maybe there are a lot of wonderful things about
>the Dream House that have nothing to do with the exact intonation?

I don't think I suggested otherwise. I've never heard it, by the way. I
wish I could. I was just saying that I imagine it has to be loud to get the
important (nonlinear) effects. Joe Pehrson confirmed that it _is_ loud.

Joe Pehrson:

>I'm gathering that a "classic ring modulator" which operates with sum
>and difference tones, creates timbre-chords using this model (??)

Yes. I believe a ring modulator (approximately) multiplies the
instantaneous values of the waves fed into its two inputs. This is a very
nonlinear thing to do. Linear only allows you to _add_ the instantaneous
values of the waves. The only multiplication that is allowed in the linear
case is to multiply by a constant, i.e. change the amplitude (or "volume")
of the waves.

But multiplying the instantaneous values results in adding and subtracting
the frequencies (and integer multiples thereof).

There are two different ways of looking at a wave. Time domain (waveform)
and frequency domain (spectrum). Multiplication in the time domain looks
like addition and subtraction in the frequency domain.

David Beardsley:

>Just Intonation is a tuning system where intervals are tuned by whole
>number ratios.
>The definition is that simple and covers all of the above. A simple
>concept that you're
>trying to make more complicated.

As people have tried to explain many times before, this definition is so
simple it is useless. It includes all possible tuning systems, including
12-tET. It is unfortunate that some people get very emotional about this,
when it is simply an observable fact of maths and physics. The real world
(even electronic instruments) has limited precision. As long as the
temperature is above absolute zero there will always be noise to interfere.
And of course the precision of the human ear-brain system is way more
limited than that. Therefore one can always find large enough whole numbers
to make a ratio that is for all practical purposes indistinguishable from
any irrational number.

Is the following a JI tuning system?

12
!
1059463/1000000
561231/500000
1189207/1000000
1259921/1000000
33371/25000
707107/500000
1498307/1000000
1587401/1000000
1681793/1000000
1781797/1000000
1887749/1000000
2/1

It deviates from 12-tET by less than 0.0006 of a cent, or one beat every 50
minutes at 1 kHz.

Regards,
-- Dave Keenan
http://dkeenan.com

David C Keenan wrote,

> Yes. I believe a ring modulator (approximately) multiplies the
instantaneous values of the waves fed into its two inputs. This is a
very nonlinear thing to do.

Following in the peculiar footsteps of Qubais Reed Ghazala, there's
actually a whole school of "circuit-benders" out there regularly
engaged in all sorts of nonlinear hotwiring, i.e., "an electronic art
which implements creative audio short-circuiting". For anyone who's
interested, here's the definitive site:

<http://www.anti-theory.com/soundart/>

> As people have tried to explain many times before, this definition
is so simple it is useless. It includes all possible tuning systems,
including 12-tET. It is unfortunate that some people get very
emotional about this, when it is simply an observable fact of maths
and physics.

Dave, I think just intonation is a term much like "jazz" in a way...
and it's obvious that someone saying, 'well that ain't jazz because of
x and y reasons', is apt to incite the ire of some!

In any event, it's an umbrella term, and I think a precise definition
of the sort you seem interested in might just be a case of one barking
up the wrong tree... this is not to say that what your trying to
actually say (explain, delimit, demystify, etc.) is not without
obvious merit, 'cause I think it is, but changing the term "just
intonation" accordingly is (to my mind) a miscalculation.

> one can always find large enough whole numbers to make a ratio that
is for all practical purposes indistinguishable from any irrational
number.

Speaking of gray areas... <!>, in the recent U/O morphing series posts
I did, walking one series into the other involves passing through
ultra gray "over-equal series" and "under-equal series".

--Dan Stearns

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/15598

> I don't think I suggested otherwise. I've never heard it, by the
way. I wish I could. I was just saying that I imagine it has to be
loud to get the important (nonlinear) effects. Joe Pehrson confirmed
that it _is_ loud.
>

mighty loud...

> There are two different ways of looking at a wave. Time domain
(waveform) and frequency domain (spectrum). Multiplication in the
time
domain looks like addition and subtraction in the frequency domain.
>

But, if you multiply a waveform by 2X in the time domain, doesn't it
double the frequency by 2X?? So that's multiplication, right?? So
where does the addition and subtraction come in?? Dunno...

>
> Is the following a JI tuning system?
>
> 12
> !
> 1059463/1000000
> 561231/500000
> 1189207/1000000
> 1259921/1000000
> 33371/25000
> 707107/500000
> 1498307/1000000
> 1587401/1000000
> 1681793/1000000
> 1781797/1000000
> 1887749/1000000
> 2/1
>
> It deviates from 12-tET by less than 0.0006 of a cent, or one beat
every 50 minutes at 1 kHz.
>

This is hilarious. (Well, it was to ME anyway) Thanks for it!

________ ___ __ _
Joe Pehrson

David C Keenan wrote:

> David Beardsley:
>
> >Just Intonation is a tuning system where intervals are tuned by whole
> >number ratios.
> >The definition is that simple and covers all of the above. A simple
> >concept that you're
> >trying to make more complicated.
>
> As people have tried to explain many times before, this definition is so
> simple it is useless. It includes all possible tuning systems, including
> 12-tET. It is unfortunate that some people get very emotional about this,
> when it is simply an observable fact of maths and physics. The real world
> (even electronic instruments) has limited precision. As long as the
> temperature is above absolute zero there will always be noise to interfere.
> And of course the precision of the human ear-brain system is way more
> limited than that. Therefore one can always find large enough whole numbers
> to make a ratio that is for all practical purposes indistinguishable from
> any irrational number.
>
> Is the following a JI tuning system?
>
<pile of numbers snipped>
>
> It deviates from 12-tET by less than 0.0006 of a cent, or one beat every 50
> minutes at 1 kHz.

By my defenition, yes. Maybe I should include the
term "low interger ratios" in there. For some, low
interger means 5 or less. For me, less than a few
hundred, not THOUSANDS.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

Joseph Pehrson wrote:
>
> --- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
>
> http://www.egroups.com/message/tuning/15598
>
> > I don't think I suggested otherwise. I've never heard it, by the
> way. I wish I could. I was just saying that I imagine it has to be
> loud to get the important (nonlinear) effects. Joe Pehrson confirmed
> that it _is_ loud.
> >
>
> mighty loud...

I've never check with a db meter but Dream House
is probably not as loud as a symphony at full blast.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:

http://www.egroups.com/message/tuning/15611

> I've never check with a db meter but Dream House
> is probably not as loud as a symphony at full blast.
>

Oh David... it certainly is! Particularly in the "main room."
Unless you were standing, perhaps, right in front of the brass
section (!!) That's not meant as a criticism.

With my limited knowledge of acoustics (which I'm trying to gradually
improve by being on this list along with other ignorances) I had NO
IDEA WHY it had to be that loud. La Monte Young obviously does.

I guess I'll have to go in there again and "turn my head back and
forth" to see if I can catch more of the "combination tones." I
think that's what I'm getting now when I do that.

How those differ from "subtones" is anybody's guess (well, at least
MY guess). It seems, according to Paul Erlich, that the "combination
tones" would be somewhat in the vicinity of the harmonic series
tones, since the "sums and differences" really wouldn't take them too
farout of the same otonal range... or at least that's what I'm
getting
(??)

Then there is always the little room in the back where one can
"retire" momentarily, where the sound is not as loud.

The carpet isn't quite so nice in that room, though... but maybe this
has been improved.

__________ ___ __ __
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:

http://www.egroups.com/message/tuning/15613

Oh... I should hasten to add that the volume in the Dream House is
not at all unpleasant. In fact, it is very clever that La Monte Young
has managed to make absolutely the loudest TOLERABLE sound possible.

OK... I had to "retire" to the back room from time to time, but it's
not loud like the "classical" rock band. It IS more in the upper
volume limit range of traditional "classical" music... just, however
the VERY UPPER limit...

I didn't know why that was... but now I have a better understanding
of what Young was trying to accomplish... so I'll have to go back
there again.

In fact, I would urge everyone to go back there again:

http://www.virtulink.com/mela/S&LPRESS.HTM

_________ ___ ___ _
Joseph Pehrson

Joseph Pehrson wrote:

> --- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
>
> http://www.egroups.com/message/tuning/15611
>
> > I've never check with a db meter but Dream House
> > is probably not as loud as a symphony at full blast.
> >
>
> Oh David... it certainly is! Particularly in the "main room."
> Unless you were standing, perhaps, right in front of the brass
> section (!!)

That's what I was thinking. There is a second louder setting on themixer that is "approved" by
La Monte, but usually it's at one pre-set volume.
I find the 2nd setting a bit too much.

> That's not meant as a criticism.

None taken.

> With my limited knowledge of acoustics (which I'm trying to gradually
> improve by being on this list along with other ignorances) I had NO
> IDEA WHY it had to be that loud. La Monte Young obviously does.

My guess is to bring up the sum and difference tones to a levelwhere they can be heard. There's
probablly more reasons, I'll
have to check my archives when I'm home this weekend.

> I guess I'll have to go in there again and "turn my head back and
> forth" to see if I can catch more of the "combination tones." I
> think that's what I'm getting now when I do that.

When you're in the DH, you're always hearing parts ofthe chord with it's sum and difference
tones. Most of the
35 sine tones are clustered inbetween the 9th and
7th harmonics.

I know that if I drone a C below middle C as a sine tone,
theres a spot about 12 ft. from the speakers where there
is a dead spot where I can't hear the tone. With 35 pitches,
the DH is full of these dead spots (nodes?).

> How those differ from "subtones" is anybody's guess (well, at least
> MY guess). It seems, according to Paul Erlich, that the "combination
> tones" would be somewhat in the vicinity of the harmonic series
> tones, since the "sums and differences" really wouldn't take them too
> farout of the same otonal range... or at least that's what I'm
> getting
> (??)
>
> Then there is always the little room in the back where one can
> "retire" momentarily, where the sound is not as loud.
>
> The carpet isn't quite so nice in that room, though... but maybe this
> has been improved.

Before DH was reopened in Sept., it got new paint, new carpet and a new lease.I think the
carpet in the sculpture room was changed as well.

--
* D a v i d B e a r d s l e y
* http://www.virtulink.com/immp/lookhere.htm

>I've never check with a db meter but Dream House
>is probably not as loud as a symphony at full blast.

Definitely not.

-Carl

{David B. wrote...}
>I've never check with a db meter but Dream House
>is probably not as loud as a symphony at full blast.

Well, someday take a check for grins -- I've never done a db reading out front in the house, but back in the percussion section, behind the damn French horns, we've used a db meter and clocked those guys pumping out 114-116 decibels (Prokofiev "Romeo and Juliet"). Those guys and gals blow their brains out on a regular basis...

Cheers,
Jon

`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`
Real Life: Orchestral Percussionist
Web Life: "Corporeal Meadows" - about Harry Partch
http://www.corporeal.com/

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:

> How those differ from "subtones" is anybody's guess (well, at least
> MY guess).

What do you mean by "subtones"?

> It seems, according to Paul Erlich, that the "combination
> tones" would be somewhat in the vicinity of the harmonic series
> tones,

They would be _exactly_ harmonic series tones.

> since the "sums and differences" really wouldn't take them too
> farout of the same otonal range...

The sums and differences would correspond _exactly_ to identities in the same otonality.

I wrote:

>> Is the following a JI tuning system?
>>
>> <ratios of 5, 6 and 7 digit integers snipped>
>>
>> It deviates from 12-tET by less than 0.0006 of a cent, or one beat every 50
>> minutes at 1 kHz.

David Beardsley wrote:

>By my defenition, yes. Maybe I should include the
>term "low interger ratios" in there. For some, low
>interger means 5 or less. For me, less than a few
>hundred, not THOUSANDS.

Yes! I consider this a major breakthrough in our shared understanding of
what a useful definition of JI might look like. The important point is that
there _is_ a limit to how big those numbers can be and still be JI. It
seems we both (all on this list?) agree that ratios with greater than 3
digit integers are not JI.

I'm certainly glad you didn't begin proclaiming me as a new JI genius who
had pushed his way further up the harmonic series than anyone else. ;-)

Perhaps we can even agree that JI music using ratios of integers greater
than say 50 is a very different kind of JI to that using ratios of integers
less than say 20 (as consonances).

Regards,
-- Dave Keenan
http://dkeenan.com

>>I've never check with a db meter but Dream House
>>is probably not as loud as a symphony at full blast.
>
>
>Oh David... it certainly is! Particularly in the "main room."
>Unless you were standing, perhaps, right in front of the brass
>section (!!) That's not meant as a criticism.

Symphonies can play at different volumes, and are usually heard
in spaces much larger than the dream house. But if you filled
the house with an orchestra and asked for forte, you'd be blown
out.

-Carl

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

> Perhaps we can even agree that JI music using ratios of integers
greater
> than say 50 is a very different kind of JI to that using ratios of
integers
> less than say 20 (as consonances).

Below is a 12 pitch (subset) spectrum scale of a metal drum, which I
have used as the basis for a 7 minute composition. Is it JI? If not,
what is it? Very interesting that many of the intervals are close to
lower number ratios.

0: 1/1 0.000
1: 244043/229688 104.952
2: 193799/172266 203.908
3: 136377/114844 297.510
4: 107666/86133 386.310
5: 301465/229688 470.778
6: 473731/344532 551.316
7: 516797/344532 701.952
8: 186621/114844 840.523
9: 1162793/689064 905.861
10: 301465/172266 968.823
11: 53833/28711 1088.265
12: 2/1 1200.000

Spectrum Scale for Methophone, 11/12/00

Jacky Ligon

Jacky Ligon wrote,

> Very interesting that many of the intervals are close to lower
number ratios.

Looks like an 8 - 16 harmonic series hunk with a handful of holes in
it -- 23, 25, 29, and 31.

good to see you back,

--Dan Stearns

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...>
wrote:

> > There are two different ways of looking at a wave. Time domain
> (waveform) and frequency domain (spectrum). Multiplication in the
> time
> domain looks like addition and subtraction in the frequency domain.
> >
>
> But, if you multiply a waveform by 2X in the time domain, doesn't
it
> double the frequency by 2X??

No, it just gets louder.

> So
> where does the addition and subtraction come in?? Dunno...

It comes from some trigonometric formulae you may have learned in
high school:

cos A cos B = 1/2 (cos (A - B) + cos (A + B))
sin A sin B = 1/2 (cos (A - B) - cos (A + B))

In other words, multiplying two sine waves is the same as adding the
difference tone and sum tone. And since our ears decompose what we
hear into a _sum_ of sine waves, we hear the sum tone and difference
tones; we can't hear the original two tones that we multiplied.

David C Keenan wrote:
>
> I wrote:
>
> >> Is the following a JI tuning system?
> >>
> >> <ratios of 5, 6 and 7 digit integers snipped>
> >>
> >> It deviates from 12-tET by less than 0.0006 of a cent, or one beat every 50
> >> minutes at 1 kHz.
>
> David Beardsley wrote:
>
> >By my defenition, yes. Maybe I should include the
> >term "low interger ratios" in there. For some, low
> >interger means 5 or less. For me, less than a few
> >hundred, not THOUSANDS.
>
> Yes! I consider this a major breakthrough in our shared understanding of
> what a useful definition of JI might look like. The important point is that
> there _is_ a limit to how big those numbers can be and still be JI. It
> seems we both (all on this list?) agree that ratios with greater than 3
> digit integers are not JI.

Probably not, but I can agree with that.

> I'm certainly glad you didn't begin proclaiming me as a new JI genius who
> had pushed his way further up the harmonic series than anyone else. ;-)

I'd call you obnoxious before I'd call you a new JI genius. Besides, I'd
need some musical proof!

> Perhaps we can even agree that JI music using ratios of integers greater
> than say 50 is a very different kind of JI to that using ratios of integers
> less than say 20 (as consonances).

Is 13/8 a consonance? It's all JI man. Each harmonic has a differing
amount
of consonance.

db

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

> http://www.egroups.com/message/tuning/15598
>
> David Beardsley:
>
> > Just Intonation is a tuning system where intervals are tuned
> > by whole number ratios.
> >
> > The definition is that simple and covers all of the above.
> > A simple concept that you're trying to make more complicated.
>
>
> As people have tried to explain many times before, this
> definition is so simple it is useless. It includes all possible
> tuning systems, including 12-tET. It is unfortunate that some
> people get very emotional about this, when it is simply an
> observable fact of maths and physics. The real world (even
> electronic instruments) has limited precision. As long as the
> temperature is above absolute zero there will always be noise
> to interfere.
>
> And of course the precision of the human ear-brain system is
> way more limited than that. Therefore one can always find large
> enough whole numbers to make a ratio that is for all practical
> purposes indistinguishable from any irrational number.
>
> Is the following a JI tuning system?
>
> 12
> !
> 1059463/1000000
> 561231/500000
> 1189207/1000000
> 1259921/1000000
> 33371/25000
> 707107/500000
> 1498307/1000000
> 1587401/1000000
> 1681793/1000000
> 1781797/1000000
> 1887749/1000000
> 2/1
>
> It deviates from 12-tET by less than 0.0006 of a cent, or one
> beat every 50 minutes at 1 kHz.

Thanks, Dave. I think this pretty much sums up the argument I've
made that one should make a distinction between 'JI' and 'rational'
tuning systems. While this particular tuning is 100% 'rational',
by my reasoning one could never call it 'JI'.

At the same time, however, one should be careful not to say that
Dave's definition 'includes all possible tuning systems, including
12-tET'. I understand that your point is that the above tuning
is aurally indistinguishable from 12-tET, but mathematically,
it is *not* the same as 12-tET.

For the sake of keeping the argument clear, you should have said
something like 'includes scales that are aurally indistinguishable
from all possible tunings, including 12-tET'.

I guess the bottom line is that some folks (like Dave) don't care
about distinctions that can't be heard, which I suppose is a
point that can be reasonably defended as long as the tuning is
under consideration specifically as it applies to *music*.

But some of us (including me) are interested as well in other
aspects of tuning that relate to domains other than music, in
which case tiny distinctions like this can become important.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

Monz wrote:

> Thanks, Dave.

Dave who?

> At the same time, however, one should be careful not to say that
> Dave's definition 'includes all possible tuning systems, including
> 12-tET'. I understand that your point is that the above tuning
> is aurally indistinguishable from 12-tET, but mathematically,
> it is *not* the same as 12-tET.

Dave who?

>
> For the sake of keeping the argument clear, you should have said
> something like 'includes scales that are aurally indistinguishable
> from all possible tunings, including 12-tET'.
>
> I guess the bottom line is that some folks (like Dave) don't care
> about distinctions that can't be heard,

Dave who?

At least use initials Joe. DB, DK.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
>
>
> David C Keenan wrote:

> > >By my defenition, yes. Maybe I should include the
> > >term "low interger ratios" in there. For some, low
> > >interger means 5 or less. For me, less than a few
> > >hundred, not THOUSANDS.
> >
> > Yes! I consider this a major breakthrough in our shared understanding of
> > what a useful definition of JI might look like. The important point is that
> > there _is_ a limit to how big those numbers can be and still be JI. It
> > seems we both (all on this list?) agree that ratios with greater than 3
> > digit integers are not JI.
>
> Probably not, but I can agree with that.

Guys, the Hammond organ is, strictly speaking, a JI instrument, with all ratios containing
only 2-digit integers. Yet it's designed to be a 12-tET instrument -- it's within 0.7 cents of
mathematically exact 12-tET. Is it JI or 12-tET? Clearly contextual considerations, not
simply the size of the integers, would have to come into a useful definition of whether or
not a given piece of music is in JI.
>
> > Perhaps we can even agree that JI music using ratios of integers greater
> > than say 50 is a very different kind of JI to that using ratios of integers
> > less than say 20 (as consonances).
>
> Is 13/8 a consonance? It's all JI man. Each harmonic has a differing
> amount
> of consonance.

Well, of course I don't think a harmonic by itself can have any amount of consonance --
it's intervals and chords which do. No need to hammer at this point -- I know D.B. is
familiar with the one-footed bride. But anyway, D.K. is trying to make an important
distinction here.

JI music using ratios of integers greater than 50 is most likely going to have to be rather
loud and rely on combinational tones for its effect, rather on than coinciding partials
(since you can't tell if, say, the 50th partial of one tone coincides with, say, the 49th
partial of another tone). Hence one can dispense with the partials altogether, and use
sine waves. The useful chords in this music are going to be otonal ones. It might be fair to
say that the Dream House installation depends on the physiological phenomenon of
combination tones insofar as JI is responsible for its musical effect.

This is very different from a lot of JI music out there that uses timbres with harmonic
partials, whether it's voices, strings, horns, or computer sounds. If we're talking about 5-
limit JI, utonal and symmetrical chords will be about as common as otonal ones; as long
as the volume isn't too loud, the consonance of these chords will depend largely on how
well the harmonics of each tone line up with the harmonics of the other tones. It might
be fair to say that a Renaissance vocal piece sung in adaptive JI depends mainly on the
physiological phenomenon of roughness (through its avoidence) insofar as JI is
responsible for its musical effect.

Another phenomenon, the virtual pitch phenomenon, seems most important in an
intermediate range of integer sizes (roughly speaking). Already in the second case
above, some musicians feel that the 5-limit minor triad is better retuned as 16:19:24. This
likely is due to the ear's attempt to fit all chords or subsets of chords into a single
harmonic series, and the musician's desire to have the fundamental implied so strongly
by the root and fifth octave-equivalent to the fundamental of the whole chord. It is this
phenomemon which governs (along with the other phenomena) the intermediate-limit,
otonal-leaning JI of Stamm, Gann, and Alves, for example . . .

One implication of the differing phenomena responsible for the musical usefulness of JI
in these different circumstances is that a different tolerance for mistuning will apply in
these situations . . .

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

> But some of us (including me) are interested as well in other
> aspects of tuning that relate to domains other than music, in
> which case tiny distinctions like this can become important.

Would you mind giving an example or two of a domain other than music to which
"tuning", as we discuss it on this list, relates? I was under the impression that this list is
concerned with tuning as it relates to _music_ -- correct me if I'm wrong.

> One implication of the differing phenomena responsible for the musical usefulness of JI
> in these different circumstances is that a different tolerance for mistuning will apply in
> these situations . . .

. . . and another implication is that there is a strong correlation between the musical
media you are using and the kind of JI tuning system that is likely to be useful. If you're
using loud sine waves, you're not likely to find utonal JI chords useful; while if you're
using quiet bassoons, you're not likely to find chords from high up in the overtone series
useful. "Useful" here implies more than just "conducive to music" -- any arbitrary tuning
system could be used to good effect by a good enough composer -- in the context of this
argument, by "useful" I really mean "the music would suffer if JI were not strictly
adhered to".

Paul Erlich wrote:

> Guys, the Hammond organ is, strictly speaking, a JI instrument,
> with all ratios containing only 2-digit integers. Yet it's designed
> to be a 12-tET instrument -- it's within 0.7 cents of
> mathematically exact 12-tET. Is it JI or 12-tET? Clearly contextual
> considerations, not simply the size of the integers, would have
> to come into a useful definition of whether or
> not a given piece of music is in JI.

And all this time I thought it was called additive synthesis!

> > > Perhaps we can even agree that JI music using ratios of integers greater
> > > than say 50 is a very different kind of JI to that using ratios of integers
> > > less than say 20 (as consonances).
> >
> > Is 13/8 a consonance? It's all JI man. Each harmonic has a differing
> > amount
> > of consonance.
>
> Well, of course I don't think a harmonic by itself can have any amount of consonance --
> it's intervals and chords which do. No need to hammer at this point -- I know D.B. is
> familiar with the one-footed bride. But anyway, D.K. is trying to make an important
> distinction here.

It wouldn't be a 13/8 without a 1/1.

> Another phenomenon, the virtual pitch phenomenon, seems most
> important in an intermediate range of integer sizes (roughly speaking).
> Already in the second case above, some musicians feel that the
> 5-limit minor triad is better retuned as 16:19:24. This
> likely is due to the ear's attempt to fit all chords or subsets
> of chords into a single harmonic series, and the musician's desire
> to have the fundamental implied so strongly by the root and fifth
> octave-equivalent to the fundamental of the whole chord. It is this
> phenomemon which governs (along with the other phenomena) the
> intermediate-limit, otonal-leaning JI of Stamm, Gann, and Alves, for example . . .

16:19:24 is so close to 12tet, of course some will like it the best.
Maybe cultural programing?

Who's Stamm?

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

Paul Erlich wrote,

> "Useful" here implies more than just "conducive to music" -- any
arbitrary tuning system could be used to good effect by a good enough
composer -- in the context of this argument, by "useful" I really mean
"the music would suffer if JI were not strictly adhered to".

While these are all good, sound practical points, I think Kraig
Grady's 'we should listen to what the composer is saying' point makes
just as good practical (if not always so sound) sense as well.

I mean where do you draw the line? A lot of folks, even experienced
musicians, simply cannot tell the difference between a piece of actual
music in tuning A and tuning B... do we take their "word" for what
point the music would suffer if such and such were not strictly
adhered to... or do we take say Dave Keenan's well argued
non-composerly conservative word for what point the music would suffer
if this or that were not strictly adhered to...

How about common sense, a body of sound aesthetic and technical
arguments, *and* some respect for what the composer says/is saying?

A good case in point would be Ives. His music does not tick like a
clock. And often times the music is so dense that an entire part could
be misplayed or omitted and there's a good chance very few would
notice. However, the composer put those details there for a reason
which may or may not hinge on a strict fidelity to each and every
detail... The point is, I think, that it's important to at least
attempt to emphasize with the composers point of view/intentions. This
is really what the music is all about... A detailed
inventory/understanding of every moving part is all well and fine but
it ain't music (though it may be quite artful and self-sustaining in
its own right).

Oh well, just some thoughts as they were passing by...

--Dan stearns

Jacky Ligon [hooray!!!] wrote:

> Below is a 12 pitch (subset) spectrum scale of a metal drum

Jacky, how arbitrary were your selections from the overtone series
of the drum? Did you pick and choose those that were close to low
prime JI ratios or were those the lowest 12 overtones?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:

> http://www.egroups.com/message/tuning/15653
>
> Dave who?
> ...
> At least use initials Joe. DB, DK.

I know...sorry. It wasn't until I looked at that post later that
I realized I should have specified which of you two Daves I was
referring to. Here's a repost of that message which uses includes
last-name initials and some other editorial comments to make things
clearer:

> [revised version of http://www.egroups.com/message/tuning/15652:]
>
>
> Thanks, Dave K. I think this pretty much sums up the argument I've
> made that one should make a distinction between 'JI' and 'rational'
> tuning systems. While this particular tuning is 100% 'rational',
> by my reasoning one could never call it 'JI'.
>
>
> At the same time, however, one should be careful not to say
> that Dave B.'s definition 'includes all possible tuning systems,
> including 12-tET'. I understand that your point is that the
> above tuning is aurally indistinguishable from 12-tET, but
> mathematically, it is *not* [exactly] the same as 12-tET.
>
> For the sake of keeping the argument clear, you should have said
> something like 'includes scales that are aurally indistinguishable
> from all possible tunings, including 12-tET'.
>
>
> I guess the bottom line is that some [actually, most] folks
> (like Dave B.) don't care about distinctions that can't be heard,
> which I suppose is a point that can be reasonably defended as
> long as the tuning is under consideration specifically as it
> applies to *music*.
>
> But some of us (including me) are interested as well in other
> aspects of tuning that relate to domains other than music, in
> which case tiny distinctions like this can become important.

To which Paul Erlich responded:

> http://www.egroups.com/message/tuning/15655
>
> Would you mind giving an example or two of a domain other than
> music to which "tuning", as we discuss it on this list,
> relates? I was under the impression that this list is concerned
> with tuning as it relates to _music_ -- correct me if I'm wrong.

Of course I wouldn't say that you're 'wrong', Paul. But to me
this list isn't *exclusively* about music.

The example that comes immediately to mind is the ancient idea
of the 'Music of the Spheres': the idea that relationships of
orbital periods, distances, etc., between various heavenly bodies
are in 'harmonic' proportions.

Even tho I consider myself to be foremost a musician, and I know
that almost everyone else on this list is interested in tuning
primarily because of its application to music, I and some others
are also interested in studying tunings because of their application
to concepts like 'Music of the Spheres'. My own particular
interest in this is mainly historical; others may be interested
in it because they're astronomers, etc.

My own feeling is that this list is here so that anyone with *any*
interest in tuning can discuss and debate interesting points.
It doesn't *always* have to be relevant *only* to music. And
even in the cases where the research does pertain to non-musical
domains, it usually ultimately has some connection with music
anyway, as in my 'Solar System' compositional project begun a
few months ago.

Just because music is perceived most obviously thru the ear/brain
system doesn't necessarily mean that it doesn't affect us in
other ways. For example, I'm intrigued by the idea (developed
most by W. A. Mathieu) that JI may be more 'pleasing' to us
(in whatever sense one wishes to make of that term) because
our body cavities have resonance frequencies which reverberate
'in tune' with low-integer frequency ratios (i.e., JI music).

As a further example, I'm also interested very much in visual
representations of tunings, not only because they 'explain'
the mathematics so easily to my eyes, but also because they
have an aesthetic beauty of their own as visual art... which
is part of the reason why the organization I'm associated with
is called 'Sonic Arts'. While my main purpose is to use these
diagrams to illustrate *music*, I also get much enjoyment from
simply looking at them *as* visual art. What I'm really reaching
for is a complete synthesis of the aural and the visual.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:
> Jacky Ligon [hooray!!!] wrote:
>
> > Below is a 12 pitch (subset) spectrum scale of a metal drum
>
> Jacky, how arbitrary were your selections from the overtone series
> of the drum? Did you pick and choose those that were close to low
> prime JI ratios or were those the lowest 12 overtones?
>

Hello David,

I use these kinds of 12 tone scales for some of my synths which can
only accommodate an "octave tuning". The synths that will allow full
keyboard retuning, I'll use more of the ratios. For that scale, the
choices were mostly by taste. Actually, when I start out I have a
list of about 500+ frequencies from the FFT. This list I sort by
order of amplitude, then I generally choose the first 25-50 (although
I have tried mapping the midi range with different ratios - giving
non-octave scales usually) pitches from which to create the scales.
This set of frequencies I will convert to cents and reduce within
2/1, 4/1 or 8/1. It is always surprising to see the resulting ratios
of an inharmonic sample - just a small slice of the chaos that's
happening on the surface of an instrument vibrating from many
directions. What you see there doesn't seem to obey the orderly logic
of musical instruments with a *more* linear harmonic structure. Just
for fun, I thought I'd paste in the raw ratios from which the scale I
posted was derived. You can see that there are some duplicate
intervals and some that are so close to one another as to be
indistinguishable. With so many close pitches, one must do a bit of
sorting out - I guess that's when personal preference and taste steps
in. My plan for the composition included a 24 part male choir, which
my friend Richard Hunt impeccably performed. So I wanted to use this
12 pitch scale as the "singable" portion (the choir part actually
uses an octatonic scale with "whole step-half step" pattern). It
wasn't easy to sing at first, but after we rehearsed it a few times,
the tuning sank in (5 to 7 limit is easy to sing, but this was a
challenge!). The metal drum plays an dominant rhythmic role, and
provides a sort of inharmonic timbre chordal drone, and the choir is
in it's tuning - it is a wonderful sound. I've always fancied that
the chosen tuning system, can or should somewhat dictate the
idiomatic/stylistic approach of the composition. Working in this way,
timbres become chords (or behave like chords) in the overall sound of
the piece. So naturally one is compelled do some creative things with
these rhythmic chordal effects. Nothing new really - most of the
world does it by instinct. It is fascinating though to put
timbres "under the microscope", and mine out the scales there. It is
a process that basically blew the lid off the Limits for me, as I
used to have it easy in my nice cozy 31 Prime Limit world - now I'm
getting vertigo from these high number ratios - I need a parachute up
here!

All the best,

Jacky Ligon

P.S. Please note that I did not impose the 2/1 octave boundary on the
below scale. 2/1 was in the spectrum.

0: 1/1 0.000
1: 244043/229688 104.952
2: 193799/172266 203.908
3: 136377/114844 297.510
4: 107666/86133 386.310
5: 107666/86133 386.310
6: 107666/86133 386.310
7: 301465/229688 470.778
8: 947461/689064 551.314
9: 473731/344532 551.316
10: 990527/689064 628.270
11: 129199/86133 701.948
12: 129199/86133 701.948
13: 516797/344532 701.952
14: 516797/344532 701.952
15: 269165/172266 772.623
16: 186621/114844 840.523
17: 1119727/689064 840.525
18: 1162793/689064 905.861
19: 401953/229688 968.822
20: 301465/172266 968.823
21: 301465/172266 968.823
22: 53833/28711 1088.265
23: 1335059/689064 1145.032
24: 344531/172266 1199.995
25: 459375/229688 1199.996
26: 689063/344532 1199.997
27: 2/1 1200.000

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15647

> >
> > But, if you multiply a waveform by 2X in the time domain, doesn't
> it double the frequency by 2X??
>
> No, it just gets louder.
>

I should be embarassed that I'm not getting this... but I think you
know by now that I'm 'way past embarassment...

If the frequencies are twice as "frequent" why wouldn't they be
double in pitch?? Or am I getting the "time domain" thing wrong??

So on an oscilloscope wouldn't the sine waves would be more
"scrunched up??" Doesn't that mean the pitch is higher?? I thought
when on a graphic oscilloscope the waveforms got "bigger" in a
vertical sense they were louder (??) That CAN'T be the "time domain"
is it??

(Sorry to bug you about these rudiments but, actually, they're pretty
important and I should know them!)

> > So
> > where does the addition and subtraction come in?? Dunno...
>
> It comes from some trigonometric formulae you may have learned in
> high school:
>
> cos A cos B = 1/2 (cos (A - B) + cos (A + B))
> sin A sin B = 1/2 (cos (A - B) - cos (A + B))
>

Gee... it's hard to remember back 10 years, but it *IS* vaguely
coming back...

> In other words, multiplying two sine waves is the same as adding
the
> difference tone and sum tone. And since our ears decompose what we
> hear into a _sum_ of sine waves, we hear the sum tone and
difference
> tones; we can't hear the original two tones that we multiplied.

So are you saying that in the Dream house there are a lot of harmonic
series pitches that are obliterated by "difference tones??" Or
doesn't that have anything to do with this...

Thanks, Paul!

_________ ___ __ _
Joseph

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15654

I found this to be an EXCEPTIONALLY interesting post...
I "saved" it...
___________ ___ __
Joseph Pehrson

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

http://www.egroups.com/message/tuning/15659

> attempt to emphasize with the composers point of view/intentions.
This is really what the music is all about... A detailed
> inventory/understanding of every moving part is all well and fine
but
> it ain't music (though it may be quite artful and self-sustaining in
> its own right).
>
> Oh well, just some thoughts as they were passing by...
>
> --Dan stearns

You know, Dan, it's just amazing to have an "alternately tuned"
contemporary-composer version of Charles Ives around (!!)

That's just my own thought as it was passing by!

_________ ___ _ _
Joseph Pehrson

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/15663

>
> My own feeling is that this list is here so that anyone with *any*
> interest in tuning can discuss and debate interesting points.
> It doesn't *always* have to be relevant *only* to music.

Actually, I would also "vote" (we're obviously in a voting phase) for
a fairly open list, with ideas of aesthetics and other arts included
from time to time in with the musical tuning "core." People can
always "scroll down." I realize there may be others with more
specifically focussed interests...

__________ ___ __
Joseph Pehrson

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
>
>
> Paul Erlich wrote:
>
> > Guys, the Hammond organ is, strictly speaking, a JI instrument,
> > with all ratios containing only 2-digit integers. Yet it's designed
> > to be a 12-tET instrument -- it's within 0.7 cents of
> > mathematically exact 12-tET. Is it JI or 12-tET? Clearly contextual
> > considerations, not simply the size of the integers, would have
> > to come into a useful definition of whether or
> > not a given piece of music is in JI.
>
> And all this time I thought it was called additive synthesis!

Huh? Timbres on the Hammond are created through a process similar to additive synthesis, but
this is an issue completely apart from how the Hammond is tuned. Again, the Hammond is
tuned in exact 2-digit ratios that are within 0.7 cents of 12-tET.

> 16:19:24 is so close to 12tet, of course some will like it the best.
> Maybe cultural programing?

Probably, though Xavier posted recenlty that he finds other minor triads unacceptable on his
violin because the combination tones only agree with the root when it's tuned 16:19:24. It's
been argued that the use of the Picardy third at the end of pieces in minor disappeared around
the time 12-tET-like tunings began to displace meantone, which would make sense given the
greater "rootedness" of 16:19:24 vs. 10:12:15.
>
> Who's Stamm?

Hans-Andre Stamm, the guy with the big microtonal organ.

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

> While these are all good, sound practical points, I think Kraig
> Grady's 'we should listen to what the composer is saying' point makes
> just as good practical (if not always so sound) sense as well.
>
> How about common sense, a body of sound aesthetic and technical
> arguments, *and* some respect for what the composer says/is saying?

I don't see what I said that could be construed as disrespect for what any composer said, is
saying, or may say in the future about his or her own music.

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15675

It's
> been argued that the use of the Picardy third at the end of pieces
in minor disappeared around the time 12-tET-like tunings began to
displace meantone, which would make sense given the
> greater "rootedness" of 16:19:24 vs. 10:12:15.

Hang on a minute... Are you saying that the Picardy third had
something to do with the fact that the minor triad wasn't really "in
tune" in meantone and they ended in major because of this. If so,
this is CERTAINLY something that is rarely taught in music schools...

________ ___ __ _
Joseph Pehrson

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> Just because music is perceived most obviously thru the ear/brain
> system doesn't necessarily mean that it doesn't affect us in
> other ways. For example, I'm intrigued by the idea (developed
> most by W. A. Mathieu) that JI may be more 'pleasing' to us
> (in whatever sense one wishes to make of that term) because
> our body cavities have resonance frequencies which reverberate
> 'in tune' with low-integer frequency ratios (i.e., JI music).

Even assuming that our body cavities are tuned in JI relative to one another (which is of course
ridiculous), our music would have to be tuned to exactly the right absolute frequency --
otherwise the music, even if in JI, wouldn't excite these resonances. Of course, different people
have different-sized body cavities; the sizes of these cavities will vary over time, etc . . . now
overtone singing is an example of where the singer varies the sizes of various resonating areas
in the vocal tract in order to make certain overtones stick out . . . if unaccompanied, the singer
might choose a fundamental to correspond with the resonance of some other body cativity . . .
but that seems as far as we can take this.

But in any case, Mathieu is still talking about _music_, isn't he?

--- In tuning@egroups.com, ligonj@n... wrote:

> This set of frequencies I will convert to cents and reduce within
> 2/1, 4/1 or 8/1.

How can you justify this reduction if the timbre's partials don't repeat themselves at these octave
ratios? It seems that the Setharization you're attempting won't work otherwise.

> It is always surprising to see the resulting ratios
> of an inharmonic sample - just a small slice of the chaos that's
> happening on the surface of an instrument vibrating from many
> directions. What you see there doesn't seem to obey the orderly logic
> of musical instruments with a *more* linear harmonic structure.

Well, it's not usually chaos, it's just more complicated to calculate -- but the underlying logic is just
as orderly. One usually sees the derivation of the partials of a round drumhead as the simplest
example of an inharmonic spectrum.

> Just
> for fun, I thought I'd paste in the raw ratios from which the scale I
> posted was derived. You can see that there are some duplicate
> intervals and some that are so close to one another as to be
> indistinguishable.

Where did these come from? Were they originally separated by octaves? And what purpose
do the 6-digit ratios serve?

Your description of the music is fascinating. I hope we get to hear it.

> I
> used to have it easy in my nice cozy 31 Prime Limit world - now I'm
> getting vertigo from these high number ratios - I need a parachute up
> here!

The ratios aren't doing you any good as far as I can tell. You can tune a Hammond organ in
12-tET with 2-digit ratios -- I'm sure these 6-digit ratios are far too accurate to allow for the natural
changes in your metal drum with temperature, etc.

>
> P.S. Please note that I did not impose the 2/1 octave boundary on the
> below scale. 2/1 was in the spectrum.
>
Oh!
>
> 0: 1/1 0.000
> 1: 244043/229688 104.952
> 2: 193799/172266 203.908
> 3: 136377/114844 297.510
> 4: 107666/86133 386.310
> 5: 107666/86133 386.310
> 6: 107666/86133 386.310
> 7: 301465/229688 470.778
> 8: 947461/689064 551.314
> 9: 473731/344532 551.316
> 10: 990527/689064 628.270
> 11: 129199/86133 701.948
> 12: 129199/86133 701.948
> 13: 516797/344532 701.952
> 14: 516797/344532 701.952
> 15: 269165/172266 772.623
> 16: 186621/114844 840.523
> 17: 1119727/689064 840.525
> 18: 1162793/689064 905.861
> 19: 401953/229688 968.822
> 20: 301465/172266 968.823
> 21: 301465/172266 968.823
> 22: 53833/28711 1088.265
> 23: 1335059/689064 1145.032
> 24: 344531/172266 1199.995
> 25: 459375/229688 1199.996
> 26: 689063/344532 1199.997
> 27: 2/1 1200.000

Jacky, may I humbly suggest that this is, for all intents and purposes, and to the probable
accuracy of your FFT, harmonics 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, and 32!
In other words, it's not an inharmonic timbre at all -- it's just the plain old harmonic series! How you
got this out of a metal drum, I don't know -- perhaps you did something wrong -- could some
distortion have been introduced into the signal? (Distorting a sine wave will give you integer
harmonics.)

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...> wrote:
> --- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
>
> http://www.egroups.com/message/tuning/15647
>
> > >
> > > But, if you multiply a waveform by 2X in the time domain, doesn't
> > it double the frequency by 2X??
> >
> > No, it just gets louder.
> >
>
> I should be embarassed that I'm not getting this... but I think you
> know by now that I'm 'way past embarassment...
>
> If the frequencies are twice as "frequent" why wouldn't they be
> double in pitch??

Yes, but they're not more frequent.

>Or am I getting the "time domain" thing wrong??

I guess so. Doubling in the time domain just means, at each point in time, multiply the amplitude
of the wave by 2.
>
> > In other words, multiplying two sine waves is the same as adding
> the
> > difference tone and sum tone. And since our ears decompose what we
> > hear into a _sum_ of sine waves, we hear the sum tone and
> difference
> > tones; we can't hear the original two tones that we multiplied.
>
> So are you saying that in the Dream house there are a lot of harmonic
> series pitches that are obliterated by "difference tones??" Or
> doesn't that have anything to do with this...

Our ears don't act as a ring modulator -- but when the sound is loud enough they do act
nonlinearly. Let's say this nonlinearity is a small quadratic one. Then, if you play two sine waves,
sin A and sin B, what you'll hear will be sin A, sin B, a small amount of (sin A)^2, a small amount
of (sin B)^2, and a small amount of sin A sin B. The last term, as we now know, contributes the
sum and difference tones; the first two terms contribute the second harmonics of the two sine
waves. Cubic and higher nonlinearities will contribute higher-order combination tones. This is
explained best in the _Feynman Lectures on Physics_.

:y?duning@egroups.com, "Joseph Pehrson" <josephpehrson@c...> wrote:
> --- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
>
> http://www.egroups.com/message/tuning/15675
>
> It's
> > been argued that the use of the Picardy third at the end of pieces
> in minor disappeared around the time 12-tET-like tunings began to
> displace meantone, which would make sense given the
> > greater "rootedness" of 16:19:24 vs. 10:12:15.
>
> Hang on a minute... Are you saying that the Picardy third had
> something to do with the fact that the minor triad wasn't really "in
> tune" in meantone and they ended in major because of this. If so,
> this is CERTAINLY something that is rarely taught in music schools...

Well, Daniel Wolf has questioned this theory (van Eck's) on a historical basis . . . but I can say
this:

The minor triad is very much in tune in meantone -- it's almost exactly 10:12:15, or 1/6:1/5:1/4,
and that's very consonant. This is certainly the right tuning for the minor triad in Renaissance
music, before tonality developed. However, Xavier and others have a very negative reaction to
pieces of _tonal_ music in a minor key ending on a 10:12:15 triad. This is because the implied
fundamental is the submediant rather than the tonic. So a piece in A minor ending on a meantone
or Just minor triad would sound sort of like it was tonicizing F, especially if the tuning was really
accurate and the chord was loud enough for combination tones to be produced. Helmholtz
shared these perceptions. A 16:19:24 minor triad, though, has all combination tones supporting
the sensation of the root as fundamental -- in this case, they're all overtones of A -- so it makes
for a more "stable" tonicization of the root. So, if the final chord of a piece in minor is to be played
loud, with the tonic in the bass, you'll get a much more "supportive" tuning from 16:19:24 than
from 10:12:15.

Note that, from a harmonic entropy standpoint, the voicing matters a great deal -- so a given
voicing of a minor triad might be optimally tuned 1:4:12:19, rather than 5:20:60:96; while a
different voicing might be better tuned 5:6:15 rather than 16:19:48 -- in both cases because the
first voicing represents a much lower segment of the harmonic series and is thus much more
recognizable as such.

P.S. A little while ago (was it last year?) I posed a bunch of minor triads for people to listen to
(blindly) and evaluate. Gerry Eskelin, who was "sure" that the right minor triad would turn out to
be 10:12:15, actually chose one very close to 16:19:24, but on the _other side_ of 10:12:15.
In the triad that was actually 10:12:15, he found the third "uncomfortably sharp". He was
surprised to find out the actual ratios of the triads he had reacted to, but I wasn't -- since we've
been so acculturated to 12-tET, and the acoustical effects of the combination tones of a
near-16:19:24 are so familiar, I had confidently predicted that Gerry and others would prefer the
chords in the vicinity of 16:19:24 to those near 10:12:15.

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> --- In tuning@egroups.com, ligonj@n... wrote:
>
> > This set of frequencies I will convert to cents and reduce within
> > 2/1, 4/1 or 8/1.
>
> How can you justify this reduction if the timbre's partials don't
repeat themselves at these octave
> ratios? It seems that the Setharization you're attempting won't
work otherwise.

Paul,

Should have been more clear on that. When 2/1 octaves are found in
the spectrum.

>
> Well, it's not usually chaos, it's just more complicated to
calculate -- but the underlying logic is just
> as orderly. One usually sees the derivation of the partials of a
round drumhead as the simplest
> example of an inharmonic spectrum.

I was being metaphorical. Unpredictable pitch behavior might have
been more accurate.

I should have noted that a huge part of what I've been trying to
achieve with this it to create sampled instruments of metal and
membranophone percussion which can play melodic/harmonic music. Just
for example (one out of many) - when I made recordings of the frame
drum "one-shots", I recorded 6 strikes each, of 5 different sounds:
Dum, Tak, Snap, 1/2 Finger Mute and 1/4 Finger Mute. When I look at
the fundamental pitches of all these raw sounds, each and every one
is at a different fundamental pitch - or had a different dominant
pitch. Part of creating the final "instrument" multi-sample, is to
tune all of these to the same or complementary pitches.

>
> > Just
> > for fun, I thought I'd paste in the raw ratios from which the
scale I
> > posted was derived. You can see that there are some duplicate
> > intervals and some that are so close to one another as to be
> > indistinguishable.
>
> Where did these come from? Were they originally separated by
octaves? And what purpose
> do the 6-digit ratios serve?

The ratios come from converting the timbre frequencies from the FFT
into cents values. Every time the results are very different. Some
timbres would have octaves while others would have some interval near
an octave. The 6 digit ratios are uncompromisingly showing the
frequency relationships found in the timbre. Sure, you could reduce
them to lower numbers, but something close, isn't what's there.

>
> Your description of the music is fascinating. I hope we get to hear
it.
>

I'm better than half way through with the CD - I do plan on sharing
the music with friends. Probably, I'll cycle out my Tuning Punk
pieces for this new crop.

>
> The ratios aren't doing you any good as far as I can tell. You can
tune a Hammond organ in
> 12-tET with 2-digit ratios -- I'm sure these 6-digit ratios are far
too accurate to allow for the natural
> changes in your metal drum with temperature, etc.

Yes, as above, I've been creating multi-sampled and tuned
instruments, so as to control some of the idiosyncratic behavior of
these kinds of instruments, thus making them much more friendly to
compose tonal music. But I will frequently combine acoustic
performance, with these special sampled instruments (e.g live singing
is a favorite, also every piece on the new CD has live drumming on
it.).

The ratios are good in that they just show the raw translation from
the fft of the timbre.

>
> >
> > P.S. Please note that I did not impose the 2/1 octave boundary on
the
> > below scale. 2/1 was in the spectrum.
> >
> Oh!
> >
>
> Jacky, may I humbly suggest that this is, for all intents and
purposes, and to the probable
> accuracy of your FFT, harmonics 16, 17, 18, 19, 20, 21, 22, 23, 24,
26, 27, 28, 30, 31, and 32!
> In other words, it's not an inharmonic timbre at all -- it's just
the plain old harmonic series! How you
> got this out of a metal drum, I don't know -- perhaps you did
something wrong -- could some
> distortion have been introduced into the signal? (Distorting a sine
wave will give you integer
> harmonics.)

Actually, this was one of the more "tame" scales. You are correct
that it's intervals are close to the harmonic series. When I was
listening to my percussionist collaborator playing this drum, I could
plainly hear the major third harmonic in the sound.
I try to take every precaution possible to make sure that distortion
and noise aren't introduced into the signal when I record through
microphones. Also the sample that was used for the fft, I applied no
compression or limiting, although I did normalize the wave.

Have a great day,

Jacky

>The ratios come from converting the timbre frequencies from the FFT
>into cents values. Every time the results are very different.

Seems like an argument for less precision.

>Some
>timbres would have octaves while others would have some interval near
>an octave. The 6 digit ratios are uncompromisingly showing the
>frequency relationships found in the timbre. Sure, you could reduce
>them to lower numbers, but something close, isn't what's there.

I'm not sure . . .

>The ratios are good in that they just show the raw translation from
>the fft of the timbre.

Why wouldn't the cents values be sufficient for that?

>> Jacky, may I humbly suggest that this is, for all intents and
purposes, and to the probable
>> accuracy of your FFT, harmonics 16, 17, 18, 19, 20, 21, 22, 23, 24,
26, 27, 28, 30, 31, and 32!
>> In other words, it's not an inharmonic timbre at all -- it's just
the plain old harmonic series! How you
>> got this out of a metal drum, I don't know -- perhaps you did
something wrong -- could some
>> distortion have been introduced into the signal? (Distorting a sine
wave will give you integer
>> harmonics.)

>Actually, this was one of the more "tame" scales. You are correct
>that it's intervals are close to the harmonic series.

They're _too_ close. You'd be lucky to get harmonics this accurate from an
FFT of a computer-generated sawtooth wave, let alone a supposedly
_inharmonic_ instrument. Would you mind presenting the results for a less
"tame" case, using the same FFT procedure?

>When I was
>listening to my percussionist collaborator playing this drum, I could
>plainly hear the major third harmonic in the sound.
>I try to take every precaution possible to make sure that distortion
>and noise aren't introduced into the signal when I record through
>microphones. Also the sample that was used for the fft, I applied no
>compression or limiting, although I did normalize the wave.

Can you explain your process in more detail? How long was the sample, and
what was the normalization process you used?

P.S. Your comments on this scale vs. 31-limit JI make no sense to me, since
this scale is aurally indistinguishable from a 31-limit otonality. What am I
missing?

-Paul

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> >The ratios come from converting the timbre frequencies from the
FFT
> >into cents values. Every time the results are very different.
>
> Seems like an argument for less precision.

Please reveal how I might improve the accuracy.

>
> >The ratios are good in that they just show the raw translation
from
> >the fft of the timbre.
>
> Why wouldn't the cents values be sufficient for that?

Sorry - I didn't understand the nature of the original question. The
ratios are secondary to the cents values. But they are interesting
all the same.

>
> They're _too_ close. You'd be lucky to get harmonics this accurate
from an
> FFT of a computer-generated sawtooth wave, let alone a supposedly
> _inharmonic_ instrument. Would you mind presenting the results for
a less
> "tame" case, using the same FFT procedure?

Perhaps it was that the particular sample that I used for the
analysis was a soft stroke, which would made the timbre less complex.
I think it would be interesting to show some other values. Allow me
to produce something. Perhaps I could upload a timbre to check from
our different locations, to see the difference.

What's the best FFT size to use Paul?

Please note that as I wrote to David, I begin with the frequency list
from the fft, which is about 500+ frequencies with amplitudes. I sort
the list by amplitude, and derive the scale from the first 25-50 (or
depending on how much of the midi range I may like to map even
higher) frequencies - with the loudest partial being the 1/1.

>
> Can you explain your process in more detail? How long was the
sample, and
> what was the normalization process you used?

This sample was short, but some are longer - up to about 8 seconds,
and in the case of some of the gongs, well they sustain so long that
it becomes necessary to only use a portion of the total wave. The
normalization I used just maximizes the level of the wave file up to
0 dBs (if needed).

>
> P.S. Your comments on this scale vs. 31-limit JI make no sense to
me, since
> this scale is aurally indistinguishable from a 31-limit otonality.
What am I
> missing?

What is missing is that we are only discussing an isolated case. In
this one you are correct. The most significantly different thing is
the way the scale is generated from the timbre, rather than by some
other process of scale construction. In other words, I might discover
an appropriate scale for a given timbre in it's spectrum, whereas I
might not deliberately construct such a scale with a more subjective
method of choosing intervals by taste, or a mathematical method of
selection. That it is near a certain limit is something I only take
note of afterwards.

Thanks,

Jacky

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15682

> Our ears don't act as a ring modulator -- but when the sound is
loud enough they do act nonlinearly. Let's say this nonlinearity is a
small quadratic one. Then, if you play two sine waves,
> sin A and sin B, what you'll hear will be sin A, sin B, a small
amount of (sin A)^2, a small amount of (sin B)^2, and a small amount
of sin A sin B. The last term, as we now know, contributes the
> sum and difference tones; the first two terms contribute the second
harmonics of the two sine waves. Cubic and higher nonlinearities
will contribute higher-order combination tones. This is explained
best in the _Feynman Lectures on Physics_.

Thanks, Paul... This is really interesting, particularly when
thinking of sound installations a la Dream House... But, where can
theFeynman Lectures on Physics be found... Who is the publisher??
_________ ___ __ _ _
Joseph Pehrson

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15683

> Well, Daniel Wolf has questioned this theory (van Eck's) on a
historical basis . . .

Oh! So somebody did actually propose that the minor triad was "out
of tune" in meantone... van Eck??

Who was van Eck?? Not the painter, I assume (!!)

but I can say this:
>
> The minor triad is very much in tune in meantone -- it's almost
exactly 10:12:15, or 1/6:1/5:1/4, and that's very consonant.

OK, so if I am understanding this, you are deriving all your minor
triads from the UTONAL series, correct?? In this case the "lowest
common denominator" is 60, right??

This is certainly the right tuning for
the minor triad in Renaissance music, before tonality developed.
However, Xavier and others have a very negative reaction to
> pieces of _tonal_ music in a minor key ending on a 10:12:15 triad.
This is because the implied fundamental is the submediant rather
than the tonic. So a piece in A minor ending on a meantone
> or Just minor triad would sound sort of like it was tonicizing F,
especially if the tuning was really accurate and the chord was loud
enough for combination tones to be produced. Helmholtz shared these
perceptions. A 16:19:24 minor triad,
though, has all combination tones supporting the sensation of the
root as fundamental -- in this case, they're all overtones of A -- so
it makes for a more "stable" tonicization of the root. So, if the
final chord of a piece in minor is to be played loud, with the tonic
in the bass, you'll get a much more "supportive" tuning from 16:19:24
than from 10:12:15.
>
Ok, I think I'm getting this... but what are the "fractions" for
16:19:24 again?? I should be able to do this quickly, but it's "not
happening..."

> Note that, from a harmonic entropy standpoint, the voicing matters
a great deal

Oh sure... we went over this in our harmonic entropy studies so far...

-- so a given
> voicing of a minor triad might be optimally tuned 1:4:12:19, rather
than 5:20:60:96; while a different voicing might be better tuned
5:6:15 rather than 16:19:48 in both cases because the
> first voicing represents a much lower segment of the harmonic
series and is thus much more recognizable as such.

Very interesting Paul... but I assume we should always derive our
minor chords from the utonal series (??) What happens to the simple
otonal 6:7:9 (??)

_________ ___ __ _
Joseph

Joseph Pehrson wrote:

> Ok, I think I'm getting this... but what are the "fractions" for
> 16:19:24 again?? I should be able to do this quickly, but it's "not
> happening..."

16 = 1/1
19 = 19/16
24 = 3/2

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

[Paul E:]
>One implication of the differing phenomena responsible for the musical
>usefulness of JI in these different circumstances is that a different
>tolerance for mistuning will apply in these situations . . .

Could you amplify on this? For example, of the tunings 10:12:15 vs.
16:19:24, which has greater tolerance for mistuning, and why?

JdL

John
i would say the 16:19:24 which has the prime would more easily be tuned
incorrectly.

Pat Pagano, Director
South East Just Intonation Society
http://indians.australians.com/meherbaba/
http://www.screwmusicforever.com/SHREESWIFT/
----- Original Message -----
From: John A. deLaubenfels <jdl@adaptune.com>
To: <tuning@egroups.com>
Sent: Tuesday, November 21, 2000 10:01 AM
Subject: [tuning] Re: Graylessness and limit (was: notation systems)

> [Paul E:]
> >One implication of the differing phenomena responsible for the musical
> >usefulness of JI in these different circumstances is that a different
> >tolerance for mistuning will apply in these situations . . .
>
> Could you amplify on this? For example, of the tunings 10:12:15 vs.
> 16:19:24, which has greater tolerance for mistuning, and why?
>
> JdL
>
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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emails.
>
>
>

Jacky Ligon wrote,

>Perhaps it was that the particular sample that I used for the
>analysis was a soft stroke, which would made the timbre less complex.

It wouldn't change the inharmonicity, which should be the same regardless.

>I think it would be interesting to show some other values. Allow me
>to produce something. Perhaps I could upload a timbre to check from
>our different locations, to see the difference.

If you give me a .wav file, I can do an FFT on it.

>What's the best FFT size to use Paul?

As long as possible a sample, in which the amplitude stays relatively
constant.

>What are ways to insure accuracy with FFT? My FFT size can be set to
>a maximum of 65536.

I may know what your problem is. What is the fundamental frequency,
according to your FFT? And according to your ear? Do they agree? What I'm
guessing it that you're using a fundamental frequency corresponding to the
period of your entire sample, which would of course be a subsonic frequency,
and all the other frequencies are necessarily (by the design of FFT) integer
multiples of that frequency. Could this be your error?

Joseph Pehrson wrote,

>But, where can
>theFeynman Lectures on Physics be found... Who is the publisher??

Addison-Wesley. See
http://www.amazon.com/exec/obidos/ASIN/020102117X/ref=ed_oe_p/107-5192794-32
41338.

>Who was van Eck?? Not the painter, I assume (!!)

He wrote the book _J.S. Bach's Critique of Pure Music_, in which he attempts
to decipher Bach's tuning intentions, and in the appendix he came up with
the model that harmonic entropy is based on.

> The minor triad is very much in tune in meantone -- it's almost
>exactly 10:12:15, or 1/6:1/5:1/4, and that's very consonant.

>OK, so if I am understanding this, you are deriving all your minor
>triads from the UTONAL series, correct?? In this case the "lowest
>common denominator" is 60, right??

Well, that's not too relevant here -- what's relevant is that this triad is
composed of three very consonant, 5-limit just intervals.

>Ok, I think I'm getting this... but what are the "fractions" for
>16:19:24 again?? I should be able to do this quickly, but it's "not
>happening..."

You mean 1/1 19/16 3/2?

>Very interesting Paul... but I assume we should always derive our
>minor chords from the utonal series (??)

Why would you assume that? 16:19:24 is most definitely an otonal triad, yet
it's the minor triad most modern musicians prefer.

John deLaubenfels wrote,

>Could you amplify on this? For example, of the tunings 10:12:15 vs.
>16:19:24, which has greater tolerance for mistuning, and why?

That's a difficult question to answer precisely, but if you has asked

>of the tunings 1/6:1/5:1/4 vs.
>16:19:24, which has greater tolerance for mistuning, and why?

I would have said that 1/6:1/5:1/4 has much greater tolerance for mistuning
-- so much so that 16:19:24 actually falls into this range. In other words,
if you think of the minor chord as purely utonal, then only the individual
intervals matter, and as long as each interval is recognizable as the
appropriate 5-limit consonance, the chord is fine. Since 19:16 is "heard" as
6:5, and 19:24 is "heard" as 5:4, 16:19:24 is within this range. However, if
you're actually after 16:19:24, you're interested in the proportions that
will imply these particular harmonics over a fundamental. In that framework,
the tolerance for mistuning would be similar to (slightly smaller than) that
of 10:12:15 (which implies a submediant fundamental), but this time the two
ranges would be mutually exclusive possibilities.

Hope that makes some sense -- the answer to your question depends on which
psychoacoustic phenomena you're trying to exploit by using simple-integer
chords, and so depends on timbre, register, duration, etc.

>John
>i would say the 16:19:24 which has the prime would more easily be tuned
>incorrectly.

I take that to mean that a smaller mistuning would make the chord sound
incorrect relative to the JI rendition, meaning it has a lower tolerance for
mistuning. Yes?

[Paul E:]
>Hope that makes some sense -- the answer to your question depends on
>which psychoacoustic phenomena you're trying to exploit by using
>simple-integer chords, and so depends on timbre, register, duration,
>etc.

Yes, I think it does. My ears perked up at your original statement,
because the next big level of refinement in my adaptive techniques
should attempt to address this very issue: which intervals (especially
in the chordal sense, beyond collections of dyads) have little tuning
tolerance and which have more? I already have a mechanism to express
the answer (springs with different constants), but I don't HAVE the
answer to express yet!

This whole minor question is interesting. I missed your sample files,
but can generate some via midi. 6:7:9 is of course another option,
with a wonderfully dark minor sound, but terrible problems doing the
ol' A,C,E -> A,C,E,G -> C,E,G transition without ending on a horrible
"car horn" major triad (14:18:21) or forcing painful retuning motion.

JdL

>This whole minor question is interesting. I missed your sample files,
>but can generate some via midi. 6:7:9 is of course another option,
>with a wonderfully dark minor sound, but terrible problems doing the
>ol' A,C,E -> A,C,E,G -> C,E,G transition without ending on a horrible
>"car horn" major triad (14:18:21) or forcing painful retuning motion.

As you know, I wouldn't use this tuning of the minor triad in a piece of
classical music, but since you're already tuning dominant ninths to
4:5:6:7:9 in your 7-limit renditions, this tuning of the minor triad may at
least eliminate retuning motion in some circumstances like D,F,A ->
G,B,D,F,A.

[Paul E:]
>As you know, I wouldn't use this tuning of the minor triad in a piece
>of classical music, but since you're already tuning dominant ninths to
>4:5:6:7:9 in your 7-limit renditions, this tuning of the minor triad
>may at least eliminate retuning motion in some circumstances like
>D,F,A -> G,B,D,F,A.

Yes, and my program in its existing state has the potential of picking
that up through the process of condensing tunings across time if doing
so reduces horizontal motion sufficiently to offset additional vertical
pain (my existing tuning files favor 10:12:15 minor triads, and have
no provision at all for 16:19:24). And/or, I could introduce other
tuning files to favor 6:7:9 or 16:19:24 directly.

JdL

Yes
i think the higher the prime the more difficult to tune
the 19 being rather tricky for me at least.
and i do assume you mean tuning by ear No?

Pat Pagano, Director
South East Just Intonation Society
http://indians.australians.com/meherbaba/
http://www.screwmusicforever.com/SHREESWIFT/
----- Original Message -----
From: Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>
To: <tuning@egroups.com>
Sent: Tuesday, November 21, 2000 1:58 PM
Subject: RE: [tuning] Re: Graylessness and limit (was: notation systems)

> >John
> >i would say the 16:19:24 which has the prime would more easily be tuned
> >incorrectly.
>
> I take that to mean that a smaller mistuning would make the chord sound
> incorrect relative to the JI rendition, meaning it has a lower tolerance
for
> mistuning. Yes?
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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emails.
>
>
>

>and i do assume you mean tuning by ear No?

No -- what John was actually asking was in the context of a computer-tuned
rendition, in which the pain of deviating from JI is traded off against the
pain of shifting pitches, whether the mistuning allowable for a 16:19:24
would be larger or smaller than the mistuning allowable for a 10:12:15.

Well with the 19 you are gonna get a "thirdness" regardless
and what do you consider "allowable" ? one hz?

Pat Pagano, Director
South East Just Intonation Society
http://indians.australians.com/meherbaba/
http://www.screwmusicforever.com/SHREESWIFT/
----- Original Message -----
From: Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>
To: <tuning@egroups.com>
Sent: Tuesday, November 21, 2000 4:00 PM
Subject: RE: [tuning] Re: Graylessness and limit (was: notation systems)

> >and i do assume you mean tuning by ear No?
>
> No -- what John was actually asking was in the context of a computer-tuned
> rendition, in which the pain of deviating from JI is traded off against
the
> pain of shifting pitches, whether the mistuning allowable for a 16:19:24
> would be larger or smaller than the mistuning allowable for a 10:12:15.
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@egroups.com - join the tuning group.
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>
>
>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Jacky Ligon wrote,
>
> >Perhaps it was that the particular sample that I used for the
> >analysis was a soft stroke, which would made the timbre less
complex.
>
> It wouldn't change the inharmonicity, which should be the same
regardless.
>
>
> I may know what your problem is. What is the fundamental frequency,
> according to your FFT? And according to your ear? Do they agree?
What I'm
> guessing it that you're using a fundamental frequency corresponding
to the
> period of your entire sample, which would of course be a subsonic
frequency,
> and all the other frequencies are necessarily (by the design of
FFT) integer
> multiples of that frequency. Could this be your error?

Paul,

Hello.

I'm not convinced that it is in error. The reason being is that the
frequencies that I used to construct the scale were from 27-30 of the
highest amplitude - and correct me if I'm wrong, these certainly
*could* have been the found in such a sample as the prominent
frequencies. It was the other 500 that I didn't use (of lesser
amplitude), that we haven't (yet) considered.

All the same these investigations are great fun and hugely
interesting to me.

One other thing that makes me believe that it's either correct, or
else, very close, is that the tuning *sounds* correct with the music.
As I said in a previous post, the drum plays a prominent role - a
sort of rhythmic chordal drone, over which a number of instruments
and singing are heard. All of these are in this tuning, and seem to
belong there. I'll let you be the judge of that when I wrap up this
CD.

Thanks,

Jacky

>Why would you assume that? 16:19:24 is most definitely an otonal triad,
>yet it's the minor triad most modern musicians prefer.

Eeep! Would you care to revise and/or clarify that statement?

-Carl

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15728

So am I getting that the 16:19:24 is actually close to 12-tET, moreso
than the 10:12:15?? What's the arithmetic behind that??

_______ ___ __ _
Joseph Pehrson

In
http://www.egroups.com/message/tuning/15654
and
http://www.egroups.com/message/tuning/15656
Paul Erlich beautifully described and expanded on the distinction I was
trying to make:

approx 50 to 1000

>JI music using ratios of integers greater than 50 is most likely going to
>have to be rather loud and rely on combinational tones for its effect,
>rather on than coinciding partials (since you can't tell if, say, the 50th
>partial of one tone coincides with, say, the 49th partial of another
>tone). Hence one can dispense with the partials altogether, and use sine
>waves. The useful chords in this music are going to be otonal ones.

approx 1 to 20

>This is very different from a lot of JI music out there that uses timbres
>with harmonic partials, whether it's voices, strings, horns, or
>computer sounds. If we're talking about 5- limit JI, utonal and symmetrical
>chords will be about as common as otonal ones; as long as the
>volume isn't too loud, the consonance of these chords will depend largely on
>how well the harmonics of each tone line up with the harmonics
>of the other tones.

approx 20 to 50

>Another phenomenon, the virtual pitch phenomenon, seems most important in
an
>intermediate range of integer sizes (roughly speaking).
>Already in the second case above, some musicians feel that the 5-limit minor
>triad is better retuned as 16:19:24. This likely is due to the
>ear's attempt to fit all chords or subsets of chords into a single
>harmonic series, and the musician's desire to have the fundamental implied so
>strongly by the root and fifth octave-equivalent to the fundamental of the
>whole chord. It is this phenomemon which governs (along with
>the other phenomena) the intermediate-limit, otonal-leaning JI of Stamm,
>Gann, and Alves, for example . . .

>One implication of the differing phenomena responsible for the musical
>usefulness of JI in these different circumstances is that a
>different tolerance for mistuning will apply in these situations
>and another implication is that there is a strong correlation between the
>musical media you are using and the kind of JI tuning system
>that is likely to be useful. If you're using loud sine waves, you're not
>likely to find utonal JI chords useful; while if you're using quiet
>bassoons, you're not likely to find chords from high up in the overtone
>series useful. "Useful" here implies more than just "conducive to
>music" -- any arbitrary tuning system could be used to good effect by a
>good enough composer -- in the context of this argument, by
>"useful" I really mean "the music would suffer if JI were not strictly
>adhered to".

Paul also wrote:

>Guys, the Hammond organ is, strictly speaking, a JI instrument, with all
>ratios containing only 2-digit integers. Yet it's designed to be a
>12-tET instrument -- it's within 0.7 cents of mathematically exact 12-tET.
>Is it JI or 12-tET? Clearly contextual considerations, not simply
>the size of the integers, would have to come into a useful definition of
>whether or not a given piece of music is in JI.

Ah yes. Tone wheel organs. I would have said instead that, despite having
all ratios containing only 2-digit integers, it is clearly a 12-tET
instrument and therefore not JI. How do you feel about this one Dave
Beardsley?

It doesn't seem likely that one could get within 0.7 cents with only 2
digit integers. Paul, can you tell us what the numbers are (or work out a
suitable set yourself)? I tried to find them on the web with no luck. When
the Hommond-Leslie FAQ doesn't have it, things are looking bad.
http://www.theatreorgans.com/hammond/faq/

This also brings to mind the old top-octave divider ICs. I have one in an
electronic organ I built several decades ago. I'm pretty sure they had 3
digit divisors (between 256 and 512?).

Jacky Ligon made me realise an important point. A scale can be described in
(irreducible) ratios of 4 digit integers and still be 5-limit JI (just as
easily as it can be 12-tET). (i.e. indistingushable from it by listening).

I have to disagree with both Monz and Margo Schulter in this thread. More
later.

Regards,
-- Dave Keenan
http://dkeenan.com

[Joseph Pehrson:]
>So am I getting that the 16:19:24 is actually close to 12-tET, moreso
>than the 10:12:15?? What's the arithmetic behind that??

Joe! Didn't we cover this in the early posts on the H.E. list? Let's
run thru the math again - get your calculator handy.

19/16 = 1.1875 (frequency ratio)

To convert to cents,

cents = 1200.0 * log(freq_ratio)/log(2) = 1200.0 * .24793 = 297.51

(just 2.5 cents from 12-tET minor third, 300.0 cents)

24/19 = 1.26316 (frequency ratio)

cents = 1200.0 * log(freq_ratio)/log(2) = 1200.0 * .33703 = 404.44

(about 4.4 cents from 12-tET major third, 400.0 cents)

You know the numbers for 10:12:15, or can now calculate them, yes?

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

http://www.egroups.com/message/tuning/15752

>
> Joe! Didn't we cover this in the early posts on the H.E. list?
Let's run thru the math again - get your calculator handy.
>

Oh sure.... I can do this! So the 12/10 is approx. 315 cents, quite
a bit larger, and the 15/12 is 386 cents, quite a bit smaller than it
"should" be for 12-tET. And, of course, those are the "just"
deviations... obviously.

Curious I didn't realize immediately how to do it...

Thanks again!

________ ___ __ _ _
JP

Pat wrote,

>and what do you consider "allowable" ? one hz?

Well, if beat rate is the criterion, then the higher-numbered ratios will
indeed require more accuracy than lower-numbered ratios. However, the 19th
harmonic is normally too weak to result in audible beating, so other
phenomena, such as virtual pitch and combination tones, are the relevant
ones for 16:19:24.

>>Why would you assume that? 16:19:24 is most definitely an otonal triad,
>>yet it's the minor triad most modern musicians prefer.

>Eeep! Would you care to revise and/or clarify that statement?

In my experience with testing musician friends, according to both the "rich"
and the "pure" listeners in the famous experiment that is brought up so
often, and according to Xavier, 16:19:24 is usually preferred over 10:12:15
as the "correct" tuning of the minor triad. A capella maestro Gerry Eskelin,
you may recall, thought that 10:12:15 was the correct tuning of the minor
triad, but in a blind test he preferred one very close to 16:19:24 (actually
further from 10:12:15 than 16:19:24), and thought that the third of 10:12:15
was "uncomfortably sharp". Now it only takes a little conditioning to
reverse these preferences, such as a bit of time playing Renaissance music,
but the point is that 16:19:24 has a very distinct, audibly "rooted" quality
that 10:12:15 doesn't, and the closeness of the former to 12-tET has made
this quality one that most modern musicians expect to hear.

Joseph Pehrson wrote,

>So am I getting that the 16:19:24 is actually close to 12-tET, moreso
>than the 10:12:15?? What's the arithmetic behind that??

16:19:24 = 0¢, 297.5¢, 702¢

10:12:15 = 0¢, 315.6¢, 702¢

12-tET m = 0¢, 300.0¢, 700¢

I wrote,

>>Guys, the Hammond organ is, strictly speaking, a JI instrument, with all
>>ratios containing only 2-digit integers. Yet it's designed to be a
>>12-tET instrument -- it's within 0.7 cents of mathematically exact 12-tET.

>>Is it JI or 12-tET? Clearly contextual considerations, not simply
>>the size of the integers, would have to come into a useful definition of
>>whether or not a given piece of music is in JI.

Dave Keenan wrote,

>Ah yes. Tone wheel organs. I would have said instead that, despite having
>all ratios containing only 2-digit integers, it is clearly a 12-tET
>instrument and therefore not JI. How do you feel about this one Dave
>Beardsley?

>It doesn't seem likely that one could get within 0.7 cents with only 2
>digit integers. Paul, can you tell us what the numbers are

It's in Barbour -- anyone have a copy on hand?

>(or work out a
>suitable set yourself)?

I can't remember whether the ratios relative to a "fundamental" were 2-digit
integers, or if the successive ratios were 2-digit integers -- the point,
though, is that none of the tone-gears had more than 100 teeth on either
side, and the resulting pitches were within 0.7 cents of 12-tET.

I remember something from one of the electronic hobby magazines called a
DigiSynTone. This was a monophonic digital synthesizer which used frequency
dividers and some "magic ratio" which approximated the 12th root of two.
I've got the article buried somewhere, but I did some calculations with
Derive and the best approximation to the 12th root of two using three-digit
integers, which also conveniently uses numbers less than 255 (an eight-bit
register sufficeth :-) is 196/185. If you are limited to 2 decimal digits,
89/84 is almost as good.

Having done that, I decided to go ahead and search for eight-bit
approximations for all the equal tempered intervals from 200 to 1100 cents.
The Derive calculations are in the attached (small) PDF file; if anyone
needs the URL for the Acrobat Reader, let me know and I'll post it.
Converting the resulting ratios to cents is left as an exercise for the
student :-).
--
M. Edward Borasky
mailto:znmeb@teleport.com
http://www.borasky-research.com

Cold leftover pizza: it's not just for breakfast any more!

M. Edward Borasky wrote,

> I've got the article buried somewhere, but I did some calculations
with Derive and the best approximation to the 12th root of two using
three-digit integers, which also conveniently uses numbers less than
255 (an eight-bit register sufficeth :-) is 196/185.

While a bit larger, the 3 digit integer ratio of 873/824 is also a bit
closer to the 12th root of 2 than the 196/185 is.

--Dan Stearns

David C Keenan wrote:

> >Guys, the Hammond organ is, strictly speaking, a JI instrument, with all
> >ratios containing only 2-digit integers. Yet it's designed to be a
> >12-tET instrument -- it's within 0.7 cents of mathematically exact 12-tET.
> >Is it JI or 12-tET? Clearly contextual considerations, not simply
> >the size of the integers, would have to come into a useful definition of
> >whether or not a given piece of music is in JI.
>
> Ah yes. Tone wheel organs. I would have said instead that, despite having
> all ratios containing only 2-digit integers, it is clearly a 12-tET
> instrument and therefore not JI. How do you feel about this one Dave
> Beardsley?

The Hammond organ is obvioiusly a 12tet instrument.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

http://www.egroups.com/message/tuning/15752

> [Joseph Pehrson:]
> >So am I getting that the 16:19:24 is actually close to 12-tET,
moreso than the 10:12:15??
>

Ok... Now going back to this (I'm a little behind...)

Where do we actually FIND the ratios 16:19:24 AND 10:12:15??

Do they exist "in reality" anyplace??

Thanks!

_________ ___ __ __
Joseph Pehrson

Joseph Pehrson wrote:

> --- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:
>
> http://www.egroups.com/message/tuning/15752
>
> > [Joseph Pehrson:]
> > >So am I getting that the 16:19:24 is actually close to 12-tET,
> moreso than the 10:12:15??
> >
>
> Ok... Now going back to this (I'm a little behind...)
>
> Where do we actually FIND the ratios 16:19:24 AND 10:12:15??
>
> Do they exist "in reality" anyplace??

In the harmonic series.

--
* D a v i d B e a r d s l e y
* xouoxno@virtulink.com
*
* J u x t a p o s i t i o n E z i n e
* M E L A v i r t u a l d r e a m house monitor
*
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:

http://www.egroups.com/message/tuning/15928

> >
> > Where do we actually FIND the ratios 16:19:24 AND 10:12:15??
> >
> > Do they exist "in reality" anyplace??
>
> In the harmonic series.
>
>
Hmmmm... Thanks David... I guess that would make sense(!!) It's
still a little ironic, though, isn't it, that 16:19:24 would end up
being close to 12-tET.... well, at least it seems that way to ME...
_______ ____ __ _
Joseph Pehrson

Joseph Pehrson wrote:

> --- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
>
> http://www.egroups.com/message/tuning/15928
>
> > >
> > > Where do we actually FIND the ratios 16:19:24 AND 10:12:15??
> > >
> > > Do they exist "in reality" anyplace??
> >
> > In the harmonic series.
> >
> >
> Hmmmm... Thanks David... I guess that would make sense(!!) It's
> still a little ironic, though, isn't it, that 16:19:24 would end up
> being close to 12-tET.... well, at least it seems that way to ME...

Some of the harmonics are close to 12tet: 9, 17, 19
Some are not: 5, 7, 13, 11

I'm not familiar enough with the relationship of harmonics higher
than 19 and 12tet. I know I made a chart of the series up to 127
with cents values and all the octave repitions once. I'll look for it
when I get home.

--
* D a v i d B e a r d s l e y
* xouoxno@virtulink.com
*
* J u x t a p o s i t i o n E z i n e
* M E L A v i r t u a l d r e a m house monitor
*
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:

> http://www.egroups.com/message/tuning/15933
>
> --- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
>
> http://www.egroups.com/message/tuning/15928
>
> > >
> > > Where do we actually FIND the ratios 16:19:24 AND 10:12:15??
> > >
> > > Do they exist "in reality" anyplace??
> >
> > In the harmonic series.
> >
> >
> Hmmmm... Thanks David... I guess that would make sense(!!)
> It's still a little ironic, though, isn't it, that 16:19:24
> would end up being close to 12-tET.... well, at least it seems
> that way to ME...

It's not at all ironic, Joe. As one 'travels' up the harmonic
series, because it's an arithmetic progression, the intervals
between successive harmonics get closer and closer together.
These between-degree intervals therefore cover the entire
range of possibility, limited only by how far one decides
to observe the progression. It's only natural that a certain
few of them will be close to 12-tET, or to any other possible
tuning. This is the whole point behind the current debate
we're having about JI and integer-ratios vs. ETs.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

Joseph Pehrson wrote,

> >So am I getting that the 16:19:24 is actually close to 12-tET,
moreso than the 10:12:15??
>

>Ok... Now going back to this (I'm a little behind...)

>Where do we actually FIND the ratios 16:19:24 AND 10:12:15??

>Do they exist "in reality" anyplace??

What on earth do you mean? Are we getting into an argument of Platonic
philosophy? (I hope not!)

Joseph Pehrson wrote,

> > > Where do we actually FIND the ratios 16:19:24 AND 10:12:15??
> > > Do they exist "in reality" anyplace??

David Beardsley wrote,

> > In the harmonic series.

>Hmmmm... Thanks David... I guess that would make sense(!!) It's
>still a little ironic, though, isn't it, that 16:19:24 would end up
>being close to 12-tET.... well, at least it seems that way to ME...

The higher you go in the harmonic series, the more accurate an approximation
you can find of _any_ interval or chord. That's because of the property that
the rational numbers are "dense" in the real number line. In other words,
there are no "holes" in the real number line where there are no rationals --
between any two real numbers there is at least one, and in fact an infinite
number, of rationals.

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:

http://www.egroups.com/message/tuning/15935

>
> Some of the harmonics are close to 12tet: 9, 17, 19
> Some are not: 5, 7, 13, 11
>
> I'm not familiar enough with the relationship of harmonics higher
> than 19 and 12tet. I know I made a chart of the series up to 127
> with cents values and all the octave repitions once. I'll look for
it when I get home.
>
Hi David...

I wish this could be posted to the list, or placed in the "files"
section...
________ _____ ___ __ _
Joseph Pehrson

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/15938

>
> It's not at all ironic, Joe. As one 'travels' up the harmonic
> series, because it's an arithmetic progression, the intervals
> between successive harmonics get closer and closer together.
> These between-degree intervals therefore cover the entire
> range of possibility, limited only by how far one decides
> to observe the progression. It's only natural that a certain
> few of them will be close to 12-tET, or to any other possible
> tuning. This is the whole point behind the current debate
> we're having about JI and integer-ratios vs. ETs.
>

This is fascinating, Monz! So then why don't we just call EVERYTHING
Just Intonation and be DONE with it! :)
______ ___ __ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/15943

> you can find of _any_ interval or chord. That's because of the
property that the rational numbers are "dense" in the real number
line. In other words, there are no "holes" in the real number line
where there are no rationals -- between any two real numbers there is
at least one, and in fact an infinite number, of rationals.

Well, that certainly is interesting to think about.. and I guess I
haven't for a while. Of course, that makes sense, since there is
always a fraction between two other fractions... only the terms get
larger... We have infinity right at our doorstep, and we don't even
have to go very far looking for it!!
_______ ___ __ _
Joseph Pehrson

Joseph Pehrson wrote:
>
> --- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
>
> http://www.egroups.com/message/tuning/15935
>
> >
> > Some of the harmonics are close to 12tet: 9, 17, 19
> > Some are not: 5, 7, 13, 11
> >
> > I'm not familiar enough with the relationship of harmonics higher
> > than 19 and 12tet. I know I made a chart of the series up to 127
> > with cents values and all the octave repitions once. I'll look for
> it when I get home.
> >
> Hi David...
>
> I wish this could be posted to the list, or placed in the "files"
> section...

It needs cents values, I'll finish it up later this week.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:

> --- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> http://www.egroups.com/message/tuning/15943
>
> > you can find of _any_ interval or chord. That's because of
> > the property that the rational numbers are "dense" in the
> > real number line. In other words, there are no "holes" in
> > the real number line where there are no rationals -- between
> > any two real numbers there is at least one, and in fact an
> > infinite number, of rationals.
>
> Well, that certainly is interesting to think about.. and I guess
> I haven't for a while. Of course, that makes sense, since there
> is always a fraction between two other fractions... only the
> terms get larger... We have infinity right at our doorstep, and
> we don't even have to go very far looking for it!!

Joe, Paul is saying essentially the same thing I said in answer
to this question you posed (he's probably saying it better...).

Below is a table giving the first 256 harmonics with their '8ve'-
reduced cents values (that is, cents mod 1200), so you can see
which ones approximate 12-tET. To make it easier to visualize,
I've posted a graph of this at:

http://www.egroups.com/files/tuning/monz/harmonics.gif

It's very easy to see on the graph how the harmonics get closer
and closer together the higher up the series you go.

(And again, this info - at least for the first 128 harmonics -
is in my book, with a similar graph. Look in the appendices at
the back.)

harmonic cents

1 0
2 0
3 702
4 0
5 386
6 702
7 969
8 0
9 204
10 386
11 551
12 702
13 841
14 969
15 1088
16 0
17 105
18 204
19 298
20 386
21 471
22 551
23 628
24 702
25 773
26 841
27 906
28 969
29 1030
30 1088
31 1145
32 0
33 53
34 105
35 155
36 204
37 251
38 298
39 342
40 386
41 429
42 471
43 512
44 551
45 590
46 628
47 666
48 702
49 738
50 773
51 807
52 841
53 874
54 906
55 938
56 969
57 999
58 1030
59 1059
60 1088
61 1117
62 1145
63 1173
64 0
65 27
66 53
67 79
68 105
69 130
70 155
71 180
72 204
73 228
74 251
75 275
76 298
77 320
78 342
79 365
80 386
81 408
82 429
83 450
84 471
85 491
86 512
87 532
88 551
89 571
90 590
91 609
92 628
93 647
94 666
95 684
96 702
97 720
98 738
99 755
100 773
101 790
102 807
103 824
104 841
105 857
106 874
107 890
108 906
109 922
110 938
111 953
112 969
113 984
114 999
115 1015
116 1030
117 1044
118 1059
119 1074
120 1088
121 1103
122 1117
123 1131
124 1145
125 1159
126 1173
127 1186
128 0
129 13
130 27
131 40
132 53
133 66
134 79
135 92
136 105
137 118
138 130
139 143
140 155
141 167
142 180
143 192
144 204
145 216
146 228
147 240
148 251
149 263
150 275
151 286
152 298
153 309
154 320
155 331
156 342
157 354
158 365
159 375
160 386
161 397
162 408
163 418
164 429
165 440
166 450
167 460
168 471
169 481
170 491
171 501
172 512
173 522
174 532
175 541
176 551
177 561
178 571
179 581
180 590
181 600
182 609
183 619
184 628
185 638
186 647
187 656
188 666
189 675
190 684
191 693
192 702
193 711
194 720
195 729
196 738
197 746
198 755
199 764
200 773
201 781
202 790
203 798
204 807
205 815
206 824
207 832
208 841
209 849
210 857
211 865
212 874
213 882
214 890
215 898
216 906
217 914
218 922
219 930
220 938
221 945
222 953
223 961
224 969
225 977
226 984
227 992
228 999
229 1007
230 1015
231 1022
232 1030
233 1037
234 1044
235 1052
236 1059
237 1066
238 1074
239 1081
240 1088
241 1095
242 1103
243 1110
244 1117
245 1124
246 1131
247 1138
248 1145
249 1152
250 1159
251 1166
252 1173
253 1180
254 1186
255 1193
256 0

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/15989

>
> Joe, Paul is saying essentially the same thing I said in answer
> to this question you posed (he's probably saying it better...).
>
>
> Below is a table giving the first 256 harmonics with their '8ve'-
> reduced cents values (that is, cents mod 1200), so you can see
> which ones approximate 12-tET. To make it easier to visualize,
> I've posted a graph of this at:
>
> http://www.egroups.com/files/tuning/monz/harmonics.gif
>

Wow. It's a RAINBOW! Why that pattern??

__________ ___ __ _
JP