: Re: 665-tone equal temperament

>
> From: Mark Rankin <markrankin95511@yahoo.com>
>
>
> I first heard of 665-ET in 1975 from Jacques Dudon who
> had worked it out by himself and used it for
> measurement. He showed it to Alain Danielou, who
> proceeded to publish it without even mentioning
> Jacques' name. I later found it in Barbour.
>
>
> As mentioned by others, 665-ET is from 3-limit
> Pythagorean, an approximation of a massive circle of
> 665 3/2 Fifths. 612-ET is an approximation of 5-limit
> J.I. (3/2, 5/4, and 6/5), and was found by Isaac
> Newton in the 1660's (see Penelope Gouk's article in
> the book called 'Let Newton Be!').
>
> -- Mark Rankin

>
> Message: 9
> Date: Sat, 02 Aug 2003 23:00:03 -0000
> From: "Carl Lumma" <ekin@lumma.org>
> Subject: Re: 665-tone equal temperament?
>
> Kraig wrote...
> >I am appauled by this group attitude against first hand
> >knowledge with real sound in real time.
>
> Kraig dude, don't make a simple misunderstanding into a
> holy war! Do you, or do you not claim:
>
> () To be able to hear the difference between a 665-et
> 3:2 and a pure 3:2?

i did not claim this if you read my very first sentance. such an exercise is rediculous and tells us nothing about real music

>
>
> () To have an instrument that would even allow you to
> test such a thing?

I do not have access to 665 not does anyone else here

>
>
>
> Kraig, what on Earth are you talking about? All of us
> involved in this thread have experience listening to and
> work with various tunings and mistunings on JI on highly
> accurate equipment.

that would be easy to believe , burt obviously there has been no acknowledgement of the problem inherent in 768 which leads to the only conclusion i can
make. no one is listening to it as it compares to JI or even other resolutions.

>
>
> From: "Justin Weaver" <improvist@usa.net>
>
>
> I actually agree with this-- I think you can feel any change in air movement, of which
> sound is a measurement, no matter how small. Plus, if you tune A one millionth of a
> cent above A440, it might not rain in China six years later! -Justin

what is important is that peopple actually listen to such things

>
>
> -
>
> Message: 16
> Date: Sun, 03 Aug 2003 00:32:56 -0000
> From: "Justin Weaver" <improvist@usa.net>
> Subject: Re: 665-tone equal temperament?
>
> Are the harmonics on Secor's organ perfectly harmonic? -Justin

i do not know

>
>
>
> Message: 18
> Date: Sun, 03 Aug 2003 01:15:18 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> Subject: Re: 665-tone equal temperament?
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > I think you and Gene are talking about different things. You are
> > talking about the diff. between 1200-et and 665-et. Gene's talking
> > about something else (the diff. between 665-et and JI?).
>
> I'm talking about the difference between 3-limit JI and 3-limit 665.
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 19
> Date: Sun, 03 Aug 2003 01:17:38 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> Subject: Re: 665-tone equal temperament?
>
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > > --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...>
> wrote:
> > >
> > > > 665 is predicted by viggo brun's algorithm see
> > > > http://www.anaphoria.com/viggo.PDF
> > > > page 4
> > >
> > > Eh? 665 is a denominator for a convergent of l2(3/2); as such it
> > > would not surprise me to hear that someone like Huygens, Newton
> or
> > > Mersenne knew about it. Certainly Barbour did.
> >
> > ok, so what does "eh?" mean here??
>
> "Eh" means why drag Brun's algorithm into it.

since one of the other great useless activites of this list is its reinvention of the wheel, iIt seem useful to show 665 in the whole chain that Brun
predicts . i know this cuts into the possibilities of those putting their graffiti ( there own name) next to it. Message: 20

> Date: Sun, 03 Aug 2003 01:35:37 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> Subject: Re: 665-tone equal temperament?
>
> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> > >
> >
> > Hello Gene!
> > I would suggest that you calulator knows nothing of such
> subltleties.
>
> If you have no clue what you are talking about, the best course is
> silence.

I am not interested in taking the very course practiced here in evauluating tunings. I have yet to witness any recognition of the blantant problems of 768
by the power circle elite. get out your calulators , i am sure you can find them

>
>

>
>
>
> Subject: Re: Re: 665-tone equal temperament?
> Both Erv and
> Kraig (!) complained of the scalatron's tuning resolution
> in 1998, which IIRC was 1024-et.

the tuning, as i just said in my last post is a subharmonic series from 1024-2048. it has been one of my main sources of being able to hear pure JI and hence
is extremely valuable. What is everyone's else's source from which they can comment at all as to what can be heard or not.

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>> () To be able to hear the difference between a 665-et
>> 3:2 and a pure 3:2?
>
>i did not claim this if you read my very first sentance.

Well, that's why it's important to realize there was a
misunderstanding between you and Gene.

>such an exercise is rediculous and tells us nothing about
>real music

The assumption is that if you can't hear it in a single
fifth (under optimal conditions), then you won't be able
to hear it in music.

>> () To have an instrument that would even allow you to
>> test such a thing?
>
> I do not have access to 665 not does anyone else here

Certainly some here do. MAX, CSound, Matlab, Maple, and
others have arbitrary precision.

>> Kraig, what on Earth are you talking about? All of us
>> involved in this thread have experience listening to and
>> work with various tunings and mistunings on JI on highly
>> accurate equipment.
>
>that would be easy to believe , burt obviously there has been
>no acknowledgement of the problem inherent in 768 which leads
>to the only conclusion i can make.

Monz has written about this problem but not every synth has
it!

>> "Eh" means why drag Brun's algorithm into it.
>
> since one of the other great useless activites of this list
>is its reinvention of the wheel,

You mean bringing other people on board? Everyone must
create understanding for themselves. Papert may have even
called it reinvention.

-Carl

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> > Kraig wrote...
> > >I am appauled by this group attitude against first hand
> > >knowledge with real sound in real time.
> >
> > Kraig dude, don't make a simple misunderstanding into a
> > holy war! Do you, or do you not claim:
> >
> > () To be able to hear the difference between a 665-et
> > 3:2 and a pure 3:2?
>
> i did not claim this if you read my very first sentance. such an
>exercise is rediculous and tells us nothing about real music

but that's the only claim that was made by the other side, kraig!
there's no attitude against your experiences or anything like that,
let's take a deep breath and start again, ok?

> >
> >
> > () To have an instrument that would even allow you to
> > test such a thing?
>
> I do not have access to 665 not does anyone else here

i could easily create a .wav file of a pythagorean piece left alone
and then in 665-equal. care to submit a score, anyone?

>
> >
> >
> >
> > Kraig, what on Earth are you talking about? All of us
> > involved in this thread have experience listening to and
> > work with various tunings and mistunings on JI on highly
> > accurate equipment.
>
> that would be easy to believe , burt obviously there has been no
>acknowledgement of the problem inherent in 768 which leads to the
>only conclusion i can
> make. no one is listening to it as it compares to JI or even other
>resolutions.

first of all, i don't know how 768 got dragged into this thread about
665. secondly, i've been reporting for years that monz's "ji" midi
files exhibit noticeable beating, due of course to the resolution of
the rendering equipment. since that was likely 768, your claim is
likely wrong.

> > > > Eh? 665 is a denominator for a convergent of l2(3/2); as such
it
> > > > would not surprise me to hear that someone like Huygens,
Newton
> > or
> > > > Mersenne knew about it. Certainly Barbour did.
> > >
> > > ok, so what does "eh?" mean here??
> >
> > "Eh" means why drag Brun's algorithm into it.
>
> since one of the other great useless activites of this list is its
>reinvention of the wheel, iIt seem useful to show 665 in the whole
>chain that Brun
> predicts . i know this cuts into the possibilities of those putting
>their graffiti ( there own name) next to it.

kraig, is barbour still alive? huygens, newton, and mersenne aren't.
so what is this graffiti nonsense? please, take a look at what people
are actually saying.
>
> I am not interested in taking the very course practiced here in
>evauluating tunings. I have yet to witness any recognition of the
>blantant problems of 768
> by the power circle elite.

i brought up quite a few when monz brought up 768-equal recently. of
course, i'm not a member of the power circle elite, which i suppose
would include the manufacturers of our sound cards and synthesizers.

> get out your calulators , i am sure you can find them

whom are you speaking to now?

> > Subject: Re: Re: 665-tone equal temperament?
> > Both Erv and
> > Kraig (!) complained of the scalatron's tuning resolution
> > in 1998, which IIRC was 1024-et.
>
> the tuning, as i just said in my last post is a subharmonic series
>from 1024-2048. it has been one of my main sources of being able to
>hear pure JI and hence
> is extremely valuable.

yes. nb carl.

>>the tuning, as i just said in my last post is a subharmonic series
>>from 1024-2048. it has been one of my main sources of being able to
>>hear pure JI and hence is extremely valuable.
>
>yes. nb carl.

Yeah, that rocks. Analog electronics rock! Both Kraig and Erv
complained about it, though. Maybe Kraig remembers why.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>the tuning, as i just said in my last post is a subharmonic series
> >>from 1024-2048. it has been one of my main sources of being able to
> >>hear pure JI and hence is extremely valuable.
> >
> >yes. nb carl.
>
> Yeah, that rocks. Analog electronics rock! Both Kraig and Erv
> complained about it, though. Maybe Kraig remembers why.
>
> -Carl

Carl,

Frequency dividers are digital, not analog.

Why does a technology have to be superceded before it can be romanticised?
-- Dave Keenan

Dave,

>Frequency dividers are digital, not analog.

There are analog freq. dividers too, as in a Baldwin
organ, and possibly the Scalatron.

>Why does a technology have to be superceded before
>it can be romanticised?

If you've ever checked out Mark Tilden's robots,
you'd know there's more to analog than romanticism.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Dave,
>
> >Frequency dividers are digital, not analog.
>
> There are analog freq. dividers too, as in a Baldwin
> organ, and possibly the Scalatron.

You're right, there are electronic circuits that go by the name of
"analog frequency divider", however I understand they are only used at
very high radio frequencies (where counter-based dividers are
impractical or where some modulation of the original waveform must be
preserved). There may have been some aspects of the Baldwin organs
that were analog, but I severely doubt that the frequency dividers
were. Please provide more information if you can.

I am quite confident that the Motorola Scalatrons use electronic
binary counters for frequency division. "Counting" is of course an
inherently digital activity if ever there was one.

>> >Frequency dividers are digital, not analog.
>>
>> There are analog freq. dividers too, as in a Baldwin
>> organ, and possibly the Scalatron.
>
>You're right, there are electronic circuits that go by the name of
>"analog frequency divider", however I understand they are only used
>at very high radio frequencies (where counter-based dividers are
>impractical or where some modulation of the original waveform must
>be preserved). There may have been some aspects of the Baldwin
>organs that were analog, but I severely doubt that the frequency
>dividers were. Please provide more information if you can.

The Baldwin organ predates anything digital in the consumer space,
for sure.

Analog synths had tunable oscillators, so it doesn't seem hard to
imagine them having frequency dividers. In fact I'm pretty sure
they did.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >Frequency dividers are digital, not analog.
> >>
> >> There are analog freq. dividers too, as in a Baldwin
> >> organ, and possibly the Scalatron.
> >
> >You're right, there are electronic circuits that go by the name of
> >"analog frequency divider", however I understand they are only used
> >at very high radio frequencies (where counter-based dividers are
> >impractical or where some modulation of the original waveform must
> >be preserved). There may have been some aspects of the Baldwin
> >organs that were analog, but I severely doubt that the frequency
> >dividers were. Please provide more information if you can.
>
> The Baldwin organ predates anything digital in the consumer space,
> for sure.

I think you are using the current "popular" meaning of the word
"digital" as applied to consumer goods. I'm using the technical
meaning as it has always been applied to electronic circuits. You
might consider the fact that I have made my living for some decades
now by doing electronic design, both digital and analog (among other
things).

> Analog synths had tunable oscillators, so it doesn't seem hard to
> imagine them having frequency dividers. In fact I'm pretty sure
> they did.

At around the time when they were popular, I designed and built, for
my own amusement, both a monophonic analog synthesizer and a
polyphonic top-octave-divider organ. I can assure you there is no
reason why the presence of tunable oscillators would make analog
frequency dividers more likely, and I would be very surprised to learn
that an analog frequency divider had _ever_ been used in an electronic
musical instrument.

I would also be surprised to learn that any (predominantly) analog
synthesizer ever contained _any_ kind of frequency divider for the
purpose of generating the pitches of the keys. In an analog
synthesizer the pressing of each key causes a different voltage (or
current or other continuous electrical quantity) to be sent to the
oscillator to tell it what frequency is required. They are generally a
one-key-at-a-time instrument, although there were split keyboard
models. The degree of polyphony added in the later models was
generally in direct proportion to the degree of digital (and in
particular microprocessor) electronics that they contained.

>I think you are using the current "popular" meaning of the word
>"digital" as applied to consumer goods. I'm using the technical
>meaning as it has always been applied to electronic circuits.

All circuits are analog. Some are additionally digital, if you
choose to interpret the signals as having only two states. I'm
not sure what you mean by "popular".

>You might consider the fact that I have made my living for some
>decades now by doing electronic design, both digital and analog
>(among other things).

I did consider that, and I concluded that we have a terminology
mismatch.

>> Analog synths had tunable oscillators, so it doesn't seem hard to
>> imagine them having frequency dividers. In fact I'm pretty sure
>> they did.
>
>At around the time when they were popular, I designed and built, for
>my own amusement, both a monophonic analog synthesizer and a
>polyphonic top-octave-divider organ.

That's what the Baldwin organs are. So what do you call tha circuit?

>I can assure you there is no reason why the presence of tunable
>oscillators would make analog frequency dividers more likely,

Not the presence, the existence, was what I was trying to say.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I think you are using the current "popular" meaning of the word
> >"digital" as applied to consumer goods. I'm using the technical
> >meaning as it has always been applied to electronic circuits.
>
> All circuits are analog. Some are additionally digital,

It is sometimes useful to adopt that stance, but only rarely. The
usual stance, even by electronics designers, is that if it's digital
then it's not analog. But a device may of course contain both digital
and analog circuits and circuits for converting between them.

> if you
> choose to interpret the signals as having only two states.

It's not only the choosing to interpret but also the fact that the
signals spend no significant amount of time in any intermediate state.

> I'm not sure what you mean by "popular".

Consumer goods are often considered digital or analog based only on
the method used in the storage medium or the final output stage or
whatever, irrespective of what goes on internally. Or something may
only be considered digital if it uses a microprocessor internally as
opposed to discrete logic or ICs as opposed to individual transistors.
The popular concept of "digital" versus "analog" is somewhat fuzzy.

> I did consider that, and I concluded that we have a terminology
> mismatch.

OK.

> >> Analog synths had tunable oscillators, so it doesn't seem hard to
> >> imagine them having frequency dividers. In fact I'm pretty sure
> >> they did.
> >
> >At around the time when they were popular, I designed and built, for
> >my own amusement, both a monophonic analog synthesizer and a
> >polyphonic top-octave-divider organ.
>
> That's what the Baldwin organs are. So what do you call that
> circuit?

The divider part is definitely digital, consisting of binary counters
made of flip-flops made of NAND or NOR gates made of transistors...
but the important thing is that it's square waves in and square waves
out and anywhere you poked a probe inside of it you'd see square (or
rectangular) waves, i.e. signals spending most of their time in one of
two states.

After the divisions are all done and you have a square wave for every
note on the keyboard and possibly for some additional harmonics of the
highest notes, then the circuitry becomes analog -- mixing and
filtering or distorting the square waves to approximate the desired
waveshapes or spectra for the different stops.

> >I can assure you there is no reason why the presence of tunable
> >oscillators would make analog frequency dividers more likely,
>
> Not the presence, the existence, was what I was trying to say.

That's what I meant too.

By the way, I have no idea how an analog frequency divider works.

>It's not only the choosing to interpret but also the fact that the
>signals spend no significant amount of time in any intermediate
>state.

What sort of device can perform such a feat?

>> That's what the Baldwin organs are. So what do you call that
>> circuit?
>
>The divider part is definitely digital, consisting of binary counters
>made of flip-flops made of NAND or NOR gates made of transistors...

I stand corrected.

>By the way, I have no idea how an analog frequency divider works.

Nor do I (pun intended). I remember learning the circuit diagram
for a VCO, and seem to remember some bit about a loop consisting
of a capacitor that discharges-after a certain period... could
perhaps be used to divide a frequency... too fuzzy now, I'm afraid.

-Carl

on 8/4/03 9:02 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> That's what the Baldwin organs are. So what do you call that
>>> circuit?
>>
>> The divider part is definitely digital, consisting of binary counters
>> made of flip-flops made of NAND or NOR gates made of transistors...
>
> I stand corrected.

Stand uncorrected. I have an old Baldwin tube circuit diagram in front of
me and it operates in an analog mode. There is no flip flop. Each
successive divider stage is tuned by using progressively larger capacitors
for progressively lower frequencies. In the old days digital mode would
have been very expensive compared to analog. The tube or transistor count
would be much higher, and that would determine the price. For tubes,
unnecessary digital-mode operation would have been prohibitive. And analog
division by 2 would probably have been quite reliable, even with
not-that-accurate components, because components would have to be off by a
good bit for the circuit to go into a divide-by-3 mode.

>> By the way, I have no idea how an analog frequency divider works.

"Digitally", by which I mean in a mode involving states that are reasonably
modelled as "finite" in number, whereas analog approaches infinite, but not
digitally in the sense that the progression toward the state of "rolling
over" occurs in a digital mode, i.e. a mode in which counting is explicit,
but rather in a mode wherein counting is done by analog accumulation, such
as the charging of a capacitor.

An example of an analog frequency divider might be an old analog
oscilloscope (predating "triggered" scopes, or perhaps cheaper) with a
synchronization input. The sync signal of an old TV works similarly. There
is a free-running oscillator whose frequency is influenced by a sync signal
being mixed into one of the oscillator signal paths. This might officially
be a chaotic system, but when sync is achieved, you have the sync signal
being frequency divided. So this is not done by counting and in that sense
is not really "digital". The frequency division ratio is determined by the
relationship between the free-running frequency of the sweep oscillator,
which you can tune using a knob. If the sweep oscillators free-runing
frequencye is a little more than 3 times the frequency of the sync signal,
then the sweep oscillator will probably increase slightly to fall into sync,
and a frequency division of 3 will probably result.

I think there have been analog synths that take advantage of such a mode of
operation explicitly. However, any analog synth that allows arbitrary
patching could achieve such a thing easily by mixing a sync signal in with
the VCO input. This essentially follows the form of FM synthesis, however,
and you may not get locking unless you also mix the output of the VCO back
to the input. So the VCO input might be a linear mixture of a pitch control
signal, a sync signal, and the output fed back. Nonlinear combination of
the sync and feedback is also possible.

A lot of bird song is probably involves analog frequency division. (I'm not
restricting analog to be electronic. Nature is analog, and differs from
electronic analog by having more inherent dimensions. A signal is usually
essentially one-dimensional.)

Whenever a vibrating thing comes into contact with another vibrating thing,
such that they hit each other when the vibrate, you have the potential for
frequency division. If you place a speaker cone producing a fixed frequency
so that it is facing upwards, and lay a small light object on it, maybe a
piece of paper, chances are you can create frequency division by creating
the situation wherein the object "bounces" against the cone, hitting it
perhaps on every 3rd vibration of the speaker.

> Nor do I (pun intended). I remember learning the circuit diagram
> for a VCO, and seem to remember some bit about a loop consisting
> of a capacitor that discharges-after a certain period... could
> perhaps be used to divide a frequency... too fuzzy now, I'm afraid.

So does what I said above ring a bell? There are no doubt other ways to
achieve it. I don't think you need to add a capacitor, but who knows what
someone might have done.

Analog frequency division is a VERY general phenomenon, and I suspect it is
a special-case of many chaotic systems. I think neurons are very inclined
toward firing behavior which can in simple situations approximate analog
frequency division. Probably there are special "circuits" which are
intended to do just that for things like heart-beat, etc.

I wonder whether subharmonic chanting actually involves parts of the throat
flapping at subharmonics of the frequency of the vocal chords, or whether it
is the vocal chords themselves that are looser and vibrating at a
subharmonics of the primary resonance.

-Kurt

>
> -Carl

on 8/3/03 7:23 PM, Dave Keenan <D.KEENAN@UQ.NET.AU> wrote:

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>>> the tuning, as i just said in my last post is a subharmonic series
>>>> from 1024-2048. it has been one of my main sources of being able to
>>>> hear pure JI and hence is extremely valuable.
>>>
>>> yes. nb carl.
>>
>> Yeah, that rocks. Analog electronics rock! Both Kraig and Erv
>> complained about it, though. Maybe Kraig remembers why.
>>
>> -Carl
>
> Carl,
>
> Frequency dividers are digital, not analog.
>
> Why does a technology have to be superceded before it can be romanticised?
> -- Dave Keenan

For many people, analog electronics "rocked" before it was superceded, if
indeed it has been.

-Kurt Bigler

on 8/4/03 11:00 PM, Kurt Bigler <kkb@breathsense.com> wrote:

> Whenever a vibrating thing comes into contact with another vibrating thing,
> such that they hit each other when the vibrate, you have the potential for
> frequency division. If you place a speaker cone producing a fixed frequency
> so that it is facing upwards, and lay a small light object on it, maybe a
> piece of paper, chances are you can create frequency division by creating
> the situation wherein the object "bounces" against the cone, hitting it
> perhaps on every 3rd vibration of the speaker.

And I should have mentioned that term "phase-locked" (which already came up)
is a key here. Phase-locking can be achieved by digital frequency division,
but to me the term usually applies to process with an analog aspect, wherein
the locking is less mechanical and more subject to conditions being correct.
A phase-locked-loop (PLL) tuner does not lock to a signal unless it is
already tuned somewhere nearby.

The most interesting (to me) kinds of phase-locking are the physical
"analog" ones. Of course I misuse the word analog in applying to a physical
process, since analog came into being originaly in how circuits could be
created that perform in ways that are analog-ous to physical process.

Another relevant kind of phase-locking is what happens in organ pipes, and
is the reason why the harmonics are exactly harmonic. However this is true
only when steady-state is achieved. In a malfunctioning pipe steady-state
may not be achieved. The phase locking is due to something non-linear
occuring in the midst of relatively linear processes in which resonance is a
primary aspect. The resonances would create partials that are not
necessarily harmonic, and the phase-locking locks them into exact multiples
of the fundamental. Even functioning pipes are also not phase-locked during
the attack phase, until a steady state is reached.

-Kurt

Thanks for writing in, Kurt.

>The most interesting (to me) kinds of phase-locking are the physical
>"analog" ones.

You'd probably really like Mark Tilden's work.

>Analog frequency division is a VERY general phenomenon, and I
>suspect it is a special-case of many chaotic systems.

There have been discussions here about the period-doubling route
to chaos. Here's part of a thread from 1999...

/tuning/topicId_3631.html#3631

...messages that far back aren't threaded, I don't think, making
it hard to find one's way around. There was a bigger discussion on
the subject back in the Mills days, but those messages aren'
archived on the yahoo site at all.

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 8/4/03 9:02 PM, Carl Lumma <ekin@l...> wrote:
>
> >>> That's what the Baldwin organs are. So what do you call that
> >>> circuit?
> >>
> >> The divider part is definitely digital, consisting of binary counters
> >> made of flip-flops made of NAND or NOR gates made of transistors...
> >
> > I stand corrected.
>
> Stand uncorrected.

Yes, please do, Carl. I was only telling you what the divider was in
the organ I built. As I think I said earlier, I know nothing about
Baldwin organs.

> I have an old Baldwin tube circuit diagram in front of
> me and it operates in an analog mode. There is no flip flop. Each
> successive divider stage is tuned by using progressively larger
capacitors
> for progressively lower frequencies. In the old days digital mode would
> have been very expensive compared to analog. The tube or transistor
count
> would be much higher, and that would determine the price. For tubes,
> unnecessary digital-mode operation would have been prohibitive. And
analog
> division by 2 would probably have been quite reliable, even with
> not-that-accurate components, because components would have to be
off by a
> good bit for the circuit to go into a divide-by-3 mode.

This sounds a bit like the timebase circuit in a CRO [I see that you
confirm this below], and would presumably have been used to produce
all the other octaves given the top octave. But we were actually
talking about producing the top octave by dividing a single very high
master frequency by various large numbers, where this sort of divider
would clearly be unreliable. Did the Baldwin have 12 independent
oscillators for its top octave?

> >> By the way, I have no idea how an analog frequency divider works.
>
> "Digitally",

Hee hee!. I love it.

> by which I mean in a mode involving states that are reasonably
> modelled as "finite" in number, whereas analog approaches infinite,
but not
> digitally in the sense that the progression toward the state of "rolling
> over" occurs in a digital mode, i.e. a mode in which counting is
explicit,
> but rather in a mode wherein counting is done by analog
accumulation, such
> as the charging of a capacitor.

OK. So although it's digital in and digital out, there's actually a
sawtooth wave there in the middle. I'll agree this can be called analog.

> A lot of bird song is probably involves analog frequency division.
(I'm not
> restricting analog to be electronic. Nature is analog,

I expect evolution has discovered digital too.

> and differs from
> electronic analog by having more inherent dimensions. A signal is
usually
> essentially one-dimensional.)
>
> Whenever a vibrating thing comes into contact with another vibrating
thing,
> such that they hit each other when the vibrate, you have the
potential for
> frequency division. If you place a speaker cone producing a fixed
frequency
> so that it is facing upwards, and lay a small light object on it,
maybe a
> piece of paper, chances are you can create frequency division by
creating
> the situation wherein the object "bounces" against the cone, hitting it
> perhaps on every 3rd vibration of the speaker.

Right. But it isn't division by a fixed number, the divisor depends on
the input frequency, such that the output frequency stays within some
range.

>Did the Baldwin have 12 independent
>oscillators for its top octave?

It had 12 screws. I suppose they all could have been
divided from an unseen master source, but I doubt it.

-C.

on 8/5/03 12:48 AM, Dave Keenan <D.KEENAN@UQ.NET.AU> wrote:

>> I have an old Baldwin tube circuit diagram in front of
>> me and it operates in an analog mode. There is no flip flop. Each
>> successive divider stage is tuned by using progressively larger
> capacitors
>> for progressively lower frequencies. In the old days digital mode would
>> have been very expensive compared to analog. The tube or transistor
> count
>> would be much higher, and that would determine the price. For tubes,
>> unnecessary digital-mode operation would have been prohibitive. And
> analog
>> division by 2 would probably have been quite reliable, even with
>> not-that-accurate components, because components would have to be
> off by a
>> good bit for the circuit to go into a divide-by-3 mode.
>
> This sounds a bit like the timebase circuit in a CRO [I see that you
> confirm this below]

I don't know what a CRO is, but I assume cardio something oscillator. I was
merely speculating about actual neural circuitry. I take it a CRO might be
something electronic, i.e. like a pacemaker?

> , and would presumably have been used to produce
> all the other octaves given the top octave.

That's right.

> But we were actually
> talking about producing the top octave by dividing a single very high
> master frequency by various large numbers, where this sort of divider
> would clearly be unreliable. Did the Baldwin have 12 independent
> oscillators for its top octave?

Yes, it did, but I remember there being organs that did derive all 12
top-octave notes from a single high-frequency oscillator. I seem to recall
such a "how to" project being covered in Radio Electronics about 30 years
ago.

>>>> By the way, I have no idea how an analog frequency divider works.
>>
>> "Digitally",
>
> Hee hee!. I love it.
>
>> by which I mean in a mode involving states that are reasonably
>> modelled as "finite" in number, whereas analog approaches infinite,
> but not
>> digitally in the sense that the progression toward the state of "rolling
>> over" occurs in a digital mode, i.e. a mode in which counting is
> explicit,
>> but rather in a mode wherein counting is done by analog
> accumulation, such
>> as the charging of a capacitor.
>
> OK. So although it's digital in and digital out, there's actually a
> sawtooth wave there in the middle. I'll agree this can be called analog.

Hmm. Well I'd say that any frequency divider in an organ is "analog out".
In this case it doesn't have to be considered digital anywhere - there is
nothing even detecting an edge, although I suppose it is possible that
something depends on the input being amplified to saturation for stable
functioning. It is going into a waveshaping filters from there, so
immediately the important aspect is the analog aspect. I can't model the
circuit in my head so well that I can tell whether the signal out is closer
to sawtooth or square wave. But in either case that is not what you need to
imitate a physical instrument. Meanwhile square wave has only the odd
harmonics. This might help or might hurt depending on what you are trying
to create. Often the waveshaping circuitry based on square-wave source
combines square waves an octave apart (or even 3 different octaves) to flesh
out the harmonic spectrum before filtering, but it can also apply different
filters to the different octaves before combining. In the latter case
square wave offers an advantage, but I don't know whether this was actually
used by anyone.

>> A lot of bird song is probably involves analog frequency division.
> (I'm not
>> restricting analog to be electronic. Nature is analog,
>
> I expect evolution has discovered digital too.

Yes, but I'd expect frequency-independent digital division (as you refer to
below) is less far less common. Maintaining a "state" in a highly dynamic
system is really a special-case of chaos, and not something you can count on
for functioning, I would suppose. Not that it wouldn't be developed if it
were really useful for survival.

>> and differs from
>> electronic analog by having more inherent dimensions. A signal is
> usually
>> essentially one-dimensional.)
>>
>> Whenever a vibrating thing comes into contact with another vibrating
> thing,
>> such that they hit each other when the vibrate, you have the
> potential for
>> frequency division. If you place a speaker cone producing a fixed
> frequency
>> so that it is facing upwards, and lay a small light object on it,
> maybe a
>> piece of paper, chances are you can create frequency division by
> creating
>> the situation wherein the object "bounces" against the cone, hitting it
>> perhaps on every 3rd vibration of the speaker.
>
> Right. But it isn't division by a fixed number, the divisor depends on
> the input frequency, such that the output frequency stays within some
> range.

Right. I would guess that analog frequency division never is. The baldwin
circuits I am referring to probably can not work with the same component
values for the top and bottom notes in an octave, though they might get away
with 2 or 3 different sets of components.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> ...messages that far back aren't threaded, I don't think, making
> it hard to find one's way around. There was a bigger discussion on
> the subject back in the Mills days, but those messages aren'
> archived on the yahoo site at all.

the mills messages have been collected by robert walker and the posts
by the six or so people who gave robert explicit approval to
publically archive them can be found here:

http://members.tripod.com/~tuning_archive/Mills/html/

unfortunately i can't figure out how to do a search over these
archives. i was looking for detailed information about andrzej
gawel's 36-equal proposal, in particular a date.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Yes, it did, but I remember there being organs that did derive all
12
> top-octave notes from a single high-frequency oscillator.

well, that's what secor's organ in question was doing, in fact he
just posted the actual Hz values (~3,000,000 Hz, IIRC) he used for
the single high-frequency oscillator here, yesterday i think.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 8/5/03 12:48 AM, Dave Keenan <D.KEENAN@U...> wrote:
> > This sounds a bit like the timebase circuit in a CRO [I see that you
> > confirm this below]
>
> I don't know what a CRO is, but I assume cardio something
oscillator. I was
> merely speculating about actual neural circuitry. I take it a CRO
might be
> something electronic, i.e. like a pacemaker?

Cathode Ray Oscilloscope. Just a plain old oscilloscope to some.
Electronic yes.

> > , and would presumably have been used to produce
> > all the other octaves given the top octave.
>
> That's right.
>
> > But we were actually
> > talking about producing the top octave by dividing a single very high
> > master frequency by various large numbers, where this sort of divider
> > would clearly be unreliable. Did the Baldwin have 12 independent
> > oscillators for its top octave?
>
> Yes, it did, but I remember there being organs that did derive all 12
> top-octave notes from a single high-frequency oscillator. I seem to
recall
> such a "how to" project being covered in Radio Electronics about 30
years
> ago.

Well yes. That's what we were originally talking about. These always
use fully-digital frequency dividers as far as I know.

But I stand corrected on my supposition that analog dividers were
never used in musical instruments. I think there must be more than one
circuit that goes by the name of analog frequency divider. These
circuits with digital in and digital out but analog in the middle, are
barely deserving of the name. I imagined something with analog in and
analog out, e.g sine wave in, sine wave out.

> Hmm. Well I'd say that any frequency divider in an organ is "analog
out".
> In this case it doesn't have to be considered digital anywhere -
there is
> nothing even detecting an edge, although I suppose it is possible that
> something depends on the input being amplified to saturation for stable
> functioning. It is going into a waveshaping filters from there, so
> immediately the important aspect is the analog aspect.

You're forgetting that, except for the bottom octave, each output has
to drive the input of the divider for the next lower octave. So
"digital out" is definitely important.

>http://members.tripod.com/~tuning_archive/Mills/html/
>
>unfortunately i can't figure out how to do a search over these
>archives. i was looking for detailed information about andrzej
>gawel's 36-equal proposal, in particular a date.

Huh; google's domain-restricted search is letting me down here.

I have all of Mills back to 9/21/97 (digest 1185), but the only
instance of *gawel* in that bit is by you, when you and monz
were establishing the et advocates list. However, from the
onelist days, there's this:

>One thing your example reminds me of is Andrzej Gawel's
>19-of-36-tET scale. Gawel ingeniously took the 7-of-12-tET
>diatonic scale and divided each of the six instances of the
>generator, 7/12 oct. = 19/12 oct., into a chain of three
>sub-generators, 19/36 oct., allowing all six of the ordinary
>diatonic triads to be completed as 7-limit tetrads, and in
>fact the scale has 14 7-limit tetrads.

As for period-doubling posts from Mills, here are four...

>Date: Tue, 5 May 1998 10:58:06 -0400
>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>To: "'tuning@eartha.mills.edu'" <tuning@eartha.mills.edu>
>Subject: Chaos, octave equivalence, and subharmonics
>
>One of the major results in chaos theory is the universality of
>certain features of non-linear dynamical systems. In virtually
>all dynamical systems which can behave chaotically, there is a
>phenomenon called period-doubling which describes the transition
>from stable, periodic behavior to chaotic behavior. As a relevant
>parameter of the system is increased from a stable to a chaotic
>value, the system repeats itself every 2, 4, 8, 16, . . . periods
>of the stable period. The transitions from one power of two to
>the next occur closer and closer together; the changes in
>parameter values required to produce each successive period
>doubling approach a decreasing geometric sequence with scaling
>parameter equal to the Feigenbaum constant (4.6692016091029 . . .
>). This means that at a finite value of the parameter, the period
>will be infinite, i.e., we have chaos.
>
>There may be many stages in the hearing process in which non-
>linear dynamics come into play. It would probably be
>counterproductive to allow this non-linearity to be enough to
>lead to chaos, while a parallel structure of processors with
>different, lesser degrees of non-linearity might actually aid in
>the recognition of pitch. It is known that when a (not too high)
>pitch is heard, there are neurons that fire at the same rate as
>the vibration rate of the pitch itself. Other neurons in the
>brain are known to have a non-linear response to their input from
>other neurons. Since a response non-linear enough to lead to
>chaos would essentially be destroying all frequency information,
>most of the neurons would oscillate at the input frequency or at
>octave equivalents below that frequency. Perhaps a certain, low
>octave range is where pitch judgments are actually made. Notice
>how very high tones seem ambiguous in pitch.
>
>Whether this or the winding of the cochlea explains octave
>equivalence, there may have been evolutionary advantages
>conferred by the ability to reduce unimportant information and
>potential confusion from overtones by compressing pitch
>information to within one octave, which led to the brain or ear
>being designed the way they are.
>
>As for the apparantly irregular "subharmonic" which Gary observed
>in the bassoon waveform, this can easily be explained by assuming
>some parameter of non-linearity (perhaps lip pressure) was
>hovering around a value at which an initial period doubling
>occurs. So the amplitude of this period-2 subharmonic could have
>been changing, and it could cease to exist for a while, returning
>again just as easily after either an odd or even number of
>period-1 oscillations.
>
>Here's an observation about instrument or vocal "subharmonics":
>Beyond the onset of chaos, chaotic regions alternate (in a
>fractal pattern) with regions whose periods are non-power-of-two
>multiples of the stable period. The last of these subharmonic
>periods to occur, but the broadest in allowed parameter values,
>is period 3. So within a wide enough range of highly chaotic
>parameter values, one is likely to stumble upon period-3
>behavior. Increasing the parameter value further leads to the
>period doublings, which in this case means period-6, period-12,
>period-24, . . . with the same Feigenbaum constant, and back to
>chaos. But decreasing the parameter leads directly back to chaos,
>via a phenomemon known as intermittency, where very nearly
>period-3 behavior persists for stretches of time, unpredictably
>alternating with stretches of chaotic behavior. (The same thing
>is true for every odd number above 3, although the smallest
>parameter value needed to achieve a given odd subharmonic, and
>the range of parameter values in which it persists, are
>decreasing functions of that odd number). Therefore, assuming the
>parameter value varies smoothly with time, and at some times
>takes on values corresponding to simple period-1 vibration, the
>only subharmonics which can exist without chaos ever occuring are
>the subharmonics corresponding to powers of 2. Period-3
>oscillation (or, to a lesser extent, periods of higher odd
>numbers) can be relatively common but cannot smoothly connect
>with simpler behavior.

>Date: Fri, 8 May 1998 17:34:51 -0400
>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>To: "'tuning@eartha.mills.edu'" <tuning@eartha.mills.edu>
>Subject: Replies to Graham Breed
>
>>>I maintain that any systematic inharmonicity in brass
>>>instruments, which is either nonexistent or extremely small,
>>>has little or nothing to do with the failure of the resonant
>>>modes of the instrument to form an exact harmonic series.
>>
>>Now you're getting suitably equivocal. Before you were saying
>>that no inharmonicity could possibly occur, which places too
>>much faith in the simplified models. Ideally, we should develop
>>more complicated models to give a quantitative estimate of the
>>(upper limit of the) inharmonicity.
>
>I'm not getting equivocal at all. The "simplified models", as
>Gary observed, seem to handle the vast majority of cases. Dave
>Hill did find specific values for the inharmonicities, which I
>would ascribe to methodological errors, etc. But even if you
>don't believe me, and take Dave's values for the inharmonicities,
>they are clearly of a different order of magnitude (typically < 1
>cent) than the departures of the series of resonant frequencies
>from a harmonic series (typically dozens of cents). That's what
>I've been trying to say.
>
>>Any noise in the input will cause peaks at the resonant modes.
>
>They are not really "peaks" so much as "hills" since there are no
>constructively interfering standing waves to sharpen them. While
>the noise energy will be spread throughout the spectrum, the
>energy from the driver (reed, lips, bow) will manifest in Dirac-
>delta-function-like peaks at harmonic overtones in the spectrum.
>
>>Inharmonicity will cause problems with JI whether or not it's
>>systematic. Any noise in the input will cause peaks at the
>>resonant modes. The result will be inharmonicity.
>
>Listen, Gary and I have been defining systematic inharmonicity as
>cases where, after the noise is removed, the partials deviate
>from a harmonic series. Since noise doesn't exhibit interference
>effects, the only inharmonicity relevant to JI is systematic
>inharmonicity.
>
>>If the overtones are not centered at the resonant peaks, the
>>amplitude of each overtone on output will depend on it's pitch.
>
>Good point. I don't know about the rest, but you said it isn't
>important, so . . .
>
>>Examples of recorded resonances were given for a range of wind
>>instruments, and they're all significantly off from the integer
>>ratios.
>
>Yes indeed!
>
>>The resonances of flutes were also given relative to 12-equal.
>>They were up to 20 cents out! It's only the flautist's skill
>>that adjusts them to the desired scale.
>
>The standard fingerings take this into account, in case you
>didn't know that already. So, if knowing the standard fingerings
>is considered part of "skill", then you are right.
>
>>Yes, chaos would be counterproductive. There are plenty of
>>nonlinear equations that produce stable subharmonics without
>>being chaotic, though. These would probably be more efficient
>>than one that could suddenly lapse into chaotic behaviour.
>
>>Jordan & Smith give the following example of a stable order 1/3
>>subharmonic:
>>
>>x'' + x - x^3 / 6 = 1.5*cos(2.85*t)
>>
>>Where the '' means differentiate twice with respect to t.
>
>While most physical nonlinearities are characterized by a
>quadratic fixed point, this appears to be one with a zero-slope
>cubic nonlinearity (inflection point). Instead of a period-
>doubling cascade, changes in the parameter value for this type of
>system lead to a Fibonacci sequence of period lengths. So
>changing the parameter in this case could decrease the period to
>2 or increase it to 5. I don't see how this is more "stable" that
>a typical order 1/n^2 subharmonic for a quadratic-fixed-point
>nonlinearity.
>
>>>It is known that when a (not too high) pitch is heard,
>>>there are neurons that fire at the same rate as the vibration
>>>rate of the pitch itself.
>>
>>Your theory suggests that neurons should fire at a 1/2^n
>>subharmonic of higher pitches. Has this been observed?
>
>I don't know, but this would be at a higher processing level than
>the neurons forming the auditory nerve, which is what is
>typically probed. In any case, Gary was asking for a mechanism by
>which powers of two would be distinguished from powers of any
>other number. Chaos theory presents a very clear reason to think
>that such distinctions are often made in nature.
>
>>> Other neurons in the brain are known to have a non-linear
>>>response to their input from other neurons. Since a response
>>>non-linear enough to lead to chaos would essentially be
>>>destroying all frequency information, most of the neurons
>>>would oscillate at the input frequency or at octave equivalents
>>>below that frequency. Perhaps a certain, low octave range is
>>>where pitch judgments are actually made. Notice how very high
>>>tones seem ambiguous in pitch.
>>
>>I would question whether neurons can react at such high
>>frequencies at all.
>
>What are "such high frequencies"? I did say "not too high" above!
>
>>I don't have experimental data to hand, but I remember evidence
>>that low frequencies are perceived in a different way to high
>>ones, probably because of this.
>
>The actual frequency at which periodicity pitch gives up is about
>5-6 kHz. Really high!
>
>>The latest book's I've read -->which, as they came from
>>libraries, are a decade or three old -- say that the original
>>experiments showing octave invariance were performed on
>>musically literate subjects.
>
>Are white rats musically literate?
>
>>More specifically, literate in Western Classical Music, where
>>the notation is octave based.
>
>Many other cultures use octave-based notation.
>
>>>As for the apparantly irregular "subharmonic" which Gary
>>>observed in the bassoon waveform, this can easily be explained
>>>by assuming some parameter of non-linearity (perhaps lip
>>>pressure) was hovering around a value at which an initial
>>>period doubling occurs. So the amplitude of this period-2
>>>subharmonic could have been changing, and it could cease to
>>>exist for a while, returning again just as easily after either
>>>an odd or even number of period-1 oscillations.
>>
>>This assumes the system is teetering in a bifurcation point,
>>which is actually very unlikely.
>
>Not unlikely at all! Look again at a bifurcation diagram -- there
>is quite a large range of values where the period-2 behavior has
>a relatively constant amplitude, but at one end of this range the
>amplitude suddenly decreases and period-1 behavior ensues. If the
>bassoonist is trying to keep within this range, say because he is
>trying to play as loud as he can and octave subharmonics don't
>bother him but chaos does, the parameter will likely take on just
>such a range of values.
>
>>I'd guess the amplitude of the subharmonic is chaotic. Or, the
>>subharmonic is entirely chaotic but appears to be order 1/2 at
>>certain times
>
>>A period doubling cascade can be linked to amplitude, though.
>
>These don't sound like examples of real-world dynamics.
>
>>It's unlikely that a period 3 cascade could be picked out from a
>>chaotic region with one parameter.
>
>Not unlikely at all, I've done it with my vocal cords.
>
>>In the Mandelbrot set, though, there is a fairly large period 3
>>region. Similar things might occur with differential equations
>>of 2 parameters for all I'd know.
>
>Actually, a differential equation needs to have at least 3
>parameters for chaos to occur. However, the picture along
>whichever of the parameters is responsible for chaos will be the
>good old bifurcation diagram. The Mandelbrot set represents a
>discrete (not continuous) dynamical system, which only needs one
>parameter to exhibit chaos. If you restrict yourself to the real
>line, you see the usual bifurcation diagram again (proceeding
>from right to left).
>
>>I think there might even be a period 3, 9, 27, ... cascade in
>>the Mandelbrot set.
>
>Naah, the Mandelbrot set is clearly all about period doubling;
>the iteration of a point on the set's boundary doubles the angle
>the point makes on the unit circle (to which the boundary can be
>conformally mapped). Although you could probably find some
>contorted path in the complex plane to support your guess.

>Date: Sat, 9 May 1998 14:56 +0100 (BST)
>From: gbreed@cix.compulink.co.uk (Graham Breed)
>To: tuning@eartha.mills.edu
>Subject: More on instrument aperiodicity
>
>Paul Erlich wrote:
>
>>I'm not getting equivocal at all. The "simplified models", as Gary
>>observed, seem to handle the vast majority of cases. Dave Hill did
>>find specific values for the inharmonicities, which I would ascribe to
>>methodological errors, etc. But even if you don't believe me, and take
>>Dave's values for the inharmonicities, they are clearly of a different
>>order of magnitude (typically < 1 cent) than the departures of the
>>series of resonant frequencies from a harmonic series (typically
>>dozens of cents). That's what I've been trying to say.
>
>Hang on. In TD 1392, Gary Morrison started the discussion with:
>
>>Another consideration that deserves some thought along these lines
>>is that very few acoustic instruments have their overtones within 1
>>cent of exact harmonics.
>
>Then you said
>
>>Last time we had this discussion, we agreed (I thought) that bowed
>>strings, winds, and brass instruments had exact integer harmonics for
>>as long as they sustain a consistent tone. That is a physical fact,
>>true of any system with a 1-component driver, which oscillates
>>periodically and
>
>Firstly, I took "exact" to mean an integer to an infinite number of
>decimal places. "How can anyone assert such accuracy?" I thought.
>Also, the implication is obviously of far _higher_ accuracy than the <1
>cent previously suggested. I also queried how you could be so sure that
>a reed or whatever is exactly a 1-component driver, but Gary made the
>point for me.
>
>You replied with a model for the operation of reeds/lips that assumed
>laminar airflow, and concluded with:
>
>>Hence the amplitude of the partials can vary, sometimes wildly,
>>rendering the waveform aperiodic. However, for the time domains
>>relevant for the ear's analysis, this does not lead to any relevant
>>alterations in the frequencies of the partials.
>
>So, you're presumably still with exact integer ratios. Maybe, though,
>you mean that pitch fluctuations place a lower limit on the pitch
>precision that is meaningful, and that inharmonicity is below this
>threshold. It isn't clear.
>
>I presumed, with all the uncertainties in the construction of the reed,
>that resonance with the air column was the main factor governing
>inharmonicity, and was corrected on that point.
>
>If you're now saying that inharmonicities of a fraction of a cent are
>not physically impossible, I'm happy to concur with that. My guess
>would be less than 0.1 cents, although I have no proof of that, and
>would like some. Hopefully, we can stop theorising and wait for some
>more data.
>
>This is interesting, though:
>
>>>Any noise in the input will cause peaks at the resonant modes.
>>
>>They are not really "peaks" so much as "hills" since there are no
>>constructively interfering standing waves to sharpen them. While the
>>noise energy will be spread throughout the spectrum, the energy from
>>the driver (reed, lips, bow) will manifest in
>>Dirac-delta-function-like peaks at harmonic overtones in the
>>spectrum.
>
>Why couldn't they be reinforced by standing waves? If the noise
>provides a constant impulse at all frequencies, that should lead to
>peaks at the resonances like with a flute, maybe still corrected by an
>order of magnitude towards harmonicity. Whether this means two peaks
>occur close together, or the original peak is shifted slightly, I don't
>know. Definitely smaller than the 0.1 cents for most instruments,
>though, because of the low noise level. Maybe not even the only source
>of inharmonicity, but the easiest one to quantify.
>
>I'm not deliberately causing trouble here, only I usually agree with you
>and it distresses me when I don't. Hopefully, this was just a problem
>of expression, and we can put it all behind us, marching defiantly
>towards a better future.
>
>>Listen, Gary and I have been defining systematic inharmonicity as
>>cases where, after the noise is removed, the partials deviate from a
>>harmonic series. Since noise doesn't exhibit interference effects, the
>>only inharmonicity relevant to JI is systematic inharmonicity.
>
>I can't find this definition anywhere. I was assuming "systematic"
>meant the deviation from integers had to be constant from one cycle to
>the next.
>
>>>The resonances of flutes were also given relative to 12-equal. They
>>>were up to 20 cents out! It's only the flautist's skill that adjusts
>>>them to the desired scale.
>>
>>The standard fingerings take this into account, in case you didn't know
>>that already. So, if knowing the standard fingerings is considered part
>>of "skill", then you are right.
>
>I can't go back and check now, but I think both cases were covered.
>Harmonics played by the performer were out by less than however else
>they were measured. There is still a deviation with the usual
>fingerings, though, of quite a few cents. Enough to scupper the
>difference between 12-equal and meantone. Instrument manufacture may
>have improved since then, of course.
>
>The inharmonicity in the resonances is caused by the holes being there,
>and there's some complex relationship between this and the holes
>producing slightly the wrong pitches. I didn't read it for long enough
>to be sure. I usually ignore acoustic instruments, but thought I'd
>better clue up for this discussion.

>Date: Sat, 9 May 1998 14:56 +0100 (BST)
>From: gbreed@cix.compulink.co.uk (Graham Breed)
>To: tuning@eartha.mills.edu
>Subject: More chaotic stuff
>
>Ah yes, more on chaos theory shoehorned into a musical discussion.
>Musicians can still be reassured that they don't need to know this
>stuff.
>
>Paul Erlich wrote on the version of Duffing's equation I posted:
>
>>While most physical nonlinearities are characterized by a quadratic
>>fixed point, this appears to be one with a zero-slope cubic
>>nonlinearity (inflection point).
>
>"Most physical nonlinearities" is something of a generalisation. I
>thought Duffing's equation is pretty much a classic for forced
>oscillations, but maybe only in pure mathematics. I never did any chaos
>theory in physics. Not important here, as I assume we're still thinking
>of an artificial process in the brain.
>
>>Instead of a period-doubling cascade, changes in the
>>parameter value for this type of system lead to a Fibonacci sequence of
>>period lengths. So changing the parameter in this case could decrease
>>the period to 2 or increase it to 5. I don't see how this is more
>>"stable" that a typical order 1/n^2 subharmonic for a
>>quadratic-fixed-point nonlinearity.
>
>I meant "stable" in the usual sense that, of you perturb the system
>slightly, it will not tend towards an entirely different behaviour.
>This is the minimum criterion for a limit cycle to be at all useful.
>There are, in fact, four centres and three (unstable) saddles. Centres
>are stable, but not asymptotically stable, and this is usually how limit
>cycles arise, although I don't know if that's relevant in this case.
>Slight damping means the centres become stable spirals, and their basins
>of attraction are much reduced.
>
>Here's some text:
>
>"For the linear equation this[not the system above] periodic motion
>appears to be merely an anomalous case of the usual almost-periodic
>motion, depending on a precise relation between the forcing and natural
>frequencies. Also, any damping will cause its disappearance. When the
>equation is nonlinear, however, the generation of alien harmonics by the
>nonlinear terms may cause a stable subharmonic to appear for a range of
>the parameters, and in particular for a range of applied frequencies.
>Also, the forcing amplitude plays a part in generating and sustaining
>the subharmonic even in the presence of damping. Thus there will exist
>the tolerance of slightly varying conditions necessary for the
>consistent appearance of a subharmonic and its use in physical systems."
>
>D.W.Jordan & P. Smith "Nonlinear Ordinary Differential Equations" 2nd
>edition, Oxford, 1987, p.196.
>
>The following general case (Duffing's equation without damping) is then
>covered:
>
> x'' + a*x + b*x^3 = G*cos(w*t)
> w^2*x'' + a*x + b*x^3 = G*cos(tau)
>
>In the first equation, differentiation is with respect to t, in the
>second wrt to tau. The two are equivalent, and the second is used as
>the frequency is not known before the parameters are chosen. There's a
>lengthy calculation, that I haven't followed, the result being (p.200):
>
> a =~ w^2/9
>
>As "the calculations are valid without the restriction on b." I think b
>should still be small, though. It looks like this is a good way of
>generating a subharmonic of a particular value, if that's what you want
>to do. Not all of these subharmonics are stable, though, so I quoted
>one that is proved such as an example in the next section.
>
>Whatever mechanism, some kind of nonlinear equation seems to be the way
>to perform frequency reduction. The most useful subharmonic will still
>be of order 1/2, but that is not inevitable. According to the text,
>this sort of thing goes on in quartz clocks.
>
>>The actual frequency at which periodicity pitch gives up is
>>about 5-6 kHz. Really high!
>
>Yes, this is higher than I thought. I was expecting a few hundred Hz.
>
>>>The latest book's I've read -- which, as they came from libraries,
>>>are a decade or three old -- say that the original experiments showing
>>>octave invariance were performed on musically literate subjects.
>>
>>Are white rats musically literate?
>
>No, obviously not. I'd still prefer humans, but please outline any
>experiments you know of. Not having access to a university library,
>I can't get hold of academic papers very easily.
>
>>>More specifically, literate in Western
>>>Classical Music, where the notation is octave based.
>>
>>Many other cultures use octave-based notation.
>
>Whoa! Hold on there! When did I say anything else?
>
>>>>As for the apparantly irregular "subharmonic" which Gary observed
>>>>in the bassoon waveform, this can easily be explained by assuming
>>>>some parameter of non-linearity (perhaps lip pressure) was hovering
>>>>around a value at which an initial period doubling occurs. So the
>>>>amplitude of this period-2 subharmonic could have been changing,
>>>>and it could cease to exist for a while, returning again just as
>>>>easily after either an odd or even number of period-1 oscillations.
>>>
>>>This assumes the system is teetering in a bifurcation point, which is
>>>actually very unlikely.
>>
>>Not unlikely at all! Look again at a bifurcation diagram -- there is
>>quite a large range of values where the period-2 behavior has a
>>relatively constant amplitude, but at one end of this range the
>>amplitude suddenly decreases and period-1 behavior ensues. If the
>>bassoonist is trying to keep within this range, say because he is
>>trying to play as loud as he can and octave subharmonics don't bother
>>him but chaos does, the parameter will likely take on just such a
>>range of values.
>
>If there's some reason to move toward a bifurcation, that would make a
>difference. If "the period-2 behaviour has a relatively constant
>amplitude" why not play right in the middle of the period 2 region,
>instead of veering toward the bifurcation?
>
>>>I'd guess the amplitude of the subharmonic is chaotic. Or, the
>>>subharmonic is entirely chaotic but appears to be order 1/2 at
>>>certain times
>>
>>>A period doubling cascade can be
>>>linked to amplitude, though.
>>
>>These don't sound like examples of real-world dynamics.
>
>On thinking about it, the chaotic subharmonic is unlikely.
>
>It's obvious I'm relying on one book, but anyway ... The example is for
>Duffing's equation again:
>
> x'' + k*x' + x^3 = G*cos(w*t)
>
>The bifurcation parameter is G (really Gamma) which is the forcing
>amplitude. A period doubling cascade is demonstrated (pp336-345).
>
>>>It's unlikely that a period 3 cascade could be picked out from a
>>>chaotic region with one parameter.
>>
>>Not unlikely at all, I've done it with my vocal cords.
>
>Well done! How did you perform this experiment?
>
>>>In the Mandelbrot set, though, there is a fairly large period 3
>>>region. Similar things might occur with differential equations
>>>of 2 parameters for all I'd know.
>>
>>Actually, a differential equation needs to have at least 3 parameters
>>for chaos to occur. However, the picture along whichever of the
>>parameters is responsible for chaos will be the good old bifurcation
>>diagram. The Mandelbrot set represents a discrete (not continuous)
>>dynamical system, which only needs one parameter to exhibit chaos. If
>>you restrict yourself to the real line, you see the usual bifurcation
>>diagram again (proceeding from right to left).
>
>I meant that the bifurcation parameter becomes a plane, like the
>Mandelbrot set, rather than that the system has two variables. I can
>see that isn't clear.
>
>Minor correction: that should be "an autonomous differential
>equation..."
>
>>>I think there might even be a period 3, 9, 27, ... cascade in the
>>>Mandelbrot set.
>>
>>Naah, the Mandelbrot set is clearly all about period doubling; the
>>iteration of a point on the set's boundary doubles the angle the point
>>makes on the unit circle (to which the boundary can be conformally
>>mapped). Although you could probably find some contorted path in the
>>complex plane to support your guess.
>
>You bet! The way I would implement twelfth-reduction, though, is to
>use a 1/3 order subharmonic generator, and repeat the process until
>the frequency became suitably low.
>
>Incidentally, Benoit Mandelbrot is a Polish mathematician, with some
>links with France, working in the USA. I have no idea how he pronounces
>his name, but everyone I know, including one Italian who's met him,
>sounds the last 't' in 'Mandelbrot'.

-Carl

>i don't think secor's subharmonic series organ, which was of course
>polyphonic, followed the larger trends of electronic keyboard
>manufacturing in any way. but i should review his post 3xxxx to see
>if he gave any details on this . . .

Well, it is Motorola, and all of a sudden I seem to remember the
image of an IC in the old Xenharmonikon ad.

-C.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >http://members.tripod.com/~tuning_archive/Mills/html/
> >
> >unfortunately i can't figure out how to do a search over these
> >archives. i was looking for detailed information about andrzej
> >gawel's 36-equal proposal, in particular a date.
>
> Huh; google's domain-restricted search is letting me down here.
>
> I have all of Mills back to 9/21/97 (digest 1185), but the only
> instance of *gawel* in that bit is by you, when you and monz
> were establishing the et advocates list.

then how come he *still* isn't on there? (monz?)

on 8/5/03 3:41 PM, Carl Lumma <ekin@lumma.org> wrote:

> As for period-doubling posts from Mills, here are four...
>
>> Date: Tue, 5 May 1998 10:58:06 -0400
>> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>> To: "'tuning@eartha.mills.edu'" <tuning@eartha.mills.edu>
>> Subject: Chaos, octave equivalence, and subharmonics
>>
>>> I think there might even be a period 3, 9, 27, ... cascade in
>>> the Mandelbrot set.
>>
>> Naah, the Mandelbrot set is clearly all about period doubling;
>> the iteration of a point on the set's boundary doubles the angle
>> the point makes on the unit circle (to which the boundary can be
>> conformally mapped). Although you could probably find some
>> contorted path in the complex plane to support your guess.

I guess Graham Breed already replied to this years ago, but I'll add my
experience here. I wrote a Mandelbrot program that plots the
converging/diverging series in another window as you move the mouse through
the Mandelbrot set. This plot can be either discrete points, or points
connected by straight lines, to help make the exact pattern of the
periodicity visible.

And yes, I just checked, you can get any periodicity number you like, plus
nested periodicity with any number you like embedded within another
periodicity of any other number you like (thus a net period which is the
product of the two embedded periods), and so on. The set is a wealth of
numerical meaning. And actually the Julia set defined by a given point in
the Mandelbrot set also gives this numerical signature, though I find it a
little harder to interpret. It appears the periodicity is only exact at the
set boundary where there is neither convergence nor divergence. Points on
the boundary of the main set (main bubble) have simple periodicity. Points
on sets branching off the main set have duplex nested periodicity. Points
where sets branch off are points where the periodicity becomes an integer
(or you could say, becomes a periodicity). Each sub-bubble off the main set
has an integer associated with it. Each sub-sub-bubble has a nested integer
associated with it, so at a point where a bubble branches off a bubble the
first bubble has for example number 5, the second, nested number 2, with a
resulting periodicity of 10 at the point where the sub-sub-bubble branches
off. This apparently goes on to any desired level of nesting.

If someone wants to see this, and has a Mac, let me know. The program by
default displays a Mandelbrot and Julia window pair, and a command-click in
either window establishes the reference point for the other. Usually the
alternate Mandelbrot sets are not discussed - they look like the original
except for an aspect of partial disintegration. As I see it there is really
just one Mandelbrot/Julia set in a 4-dimensional space. The M set is a
particular 2-space cross-section, and the J sets are normally considered any
2-space cross-section orthogonal to the M set. This program can display
other cross-sections based chosing any 2 axes, which are perhaps less
beautiful than the M or J but also other-worldly, looking like something you
would expect to see at an event horizon.

-Kurt

on 8/7/03 12:03 PM, Kurt Bigler <kkb@breathsense.com> wrote:

> on 8/5/03 3:41 PM, Carl Lumma <ekin@lumma.org> wrote:
>
>> As for period-doubling posts from Mills, here are four...
>>
>>> Date: Tue, 5 May 1998 10:58:06 -0400
>>> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>>> To: "'tuning@eartha.mills.edu'" <tuning@eartha.mills.edu>
>>> Subject: Chaos, octave equivalence, and subharmonics
>>>
>>>> I think there might even be a period 3, 9, 27, ... cascade in
>>>> the Mandelbrot set.
>>>
>>> Naah, the Mandelbrot set is clearly all about period doubling;
>>> the iteration of a point on the set's boundary doubles the angle
>>> the point makes on the unit circle (to which the boundary can be
>>> conformally mapped). Although you could probably find some
>>> contorted path in the complex plane to support your guess.
>
> And yes, I just checked, you can get any periodicity number you like, plus
> nested periodicity with any number you like embedded within another
> periodicity of any other number you like (thus a net period which is the
> product of the two embedded periods), and so on. The set is a wealth of
> numerical meaning. And actually the Julia set defined by a given point in the
> Mandelbrot set also gives this numerical signature, though I find it a little
> harder to interpret. It appears the periodicity is only exact at the set
> boundary where there is neither convergence nor divergence.

Correction: the periodicity is exact at what appears to be the center of a
bubble. Level 1 bubbles off the main (level 0) set have simple periodicity
at their centers. Sub-bubbles (level 2) have duplex nested periodicity at
their centers, etc.

> Points on the
> boundary of the main set (main bubble) have simple periodicity. Points on
> sets branching off the main set have duplex nested periodicity. Points where
> sets branch off are points where the periodicity becomes an integer (or you
> could say, becomes a periodicity). Each sub-bubble off the main set has an
> integer associated with it. Each sub-sub-bubble has a nested integer
> associated with it, so at a point where a bubble branches off a bubble the
> first bubble has for example number 5, the second, nested number 2, with a
> resulting periodicity of 10 at the point where the sub-sub-bubble branches
> off. This apparently goes on to any desired level of nesting.