RE: [tuning] planets, gravity, and Darren's one dimensional tunin g space

Darren McDougall wrote,

>I consider the realm of tuning as a one dimensional space where distance is
>measured in cents, and the heavenly bodies are the rational intervals
situated
>at points along on a straight line. I regard the mass of each body as
being
>the number of common harmonics shared between the pitches forming the
rational
>interval, which is inversely proportional to the numerator;

Can you explain how you obtained this result? You must be making some hidden
assumption that explains the asymmetry.

>a perfect fourth
>(4/3) has half the number of common harmonics as the interval of an octave
>(2/1). (Others on this list have stated a preference for the geometric
mean of
>the numerator and denominator as a measure of an intervals consonance.)
>Instead of measuring a force of gravitational attraction (in Newtons), I
have a
>unit that I have been calling _uncertainty_ which is equal to the numerator
>when the interval is just, and gets higher according to the amount of pain.
>Any arbitrary point along the line representing the space will have
rational
>intervals nearby which all engage in a gravity-like tug-of-war over that
point.
>The one having the lowest uncertainty wins, and that value becomes the
measure
>of uncertainty of that point. I have been scaling pain so that the point
at
>49.9 cents is claimed by unity but the point at 50.1 cents is pulled away
by
>another. I chose a switchover point of 50 cents because I _assume_ my
ears'
>pitch selectivity/fussiness/personal-taste/whatever-it's-called has been
formed
>by 12tET. Someone who is more fussy/selective would choose to have pain
affect
>uncertainty more strongly. I realise a carefully executed experiment would
>provide me with a more honest pain scaling factor, but for now I am not
willing
>to sacrifice music/art/play time for psychology/science/research time.
Maybe I
>will later.

A similar, but more sophisticated (if I may say so) approach, is "harmonic
entropy", a concept I've been developing for the last few years. If you wish
to learn more, join the list harmonic_entropy@egroups.com, or to see a
compilation of my older (a bit outdated) ideas on the subject, see
http://www.ixpres.com/interval/td/entropy.htm. In this model, _every_ ratio
pulls on you no matter where you are on the number line, though much like
gravity the "pull" decreases the farther you are from the ratio . . . like
you, I measure "uncertainty", but I use the standard information-theoretic
measure of uncertainty, called entropy (which satisfies some basic
additivity and subsetting properties), which measures how "confused" you are
as to which ratio you're actually hearing. There are some graphs in the
"files" section of this list, and of the harmonic entropy list as well.

>I am now contemplating using the series of parabolas as a means of
selecting
>rational intervals to form scales. If you picture them as buckets catching
>objects dropped from above, the objects will settle at rational interval
>points. Imagine 13 equally spaced objects (representing 12 steps) falling
into
>an octave wide series of parabolic buckets, you would be provided with 12
>pitches to tune a synth to.

This reminds me of a concept put forward by Hajdu. The problem with this is,
though you'll be ensuring that each note is consonant with 1/1, you won't be
giving any consideration to whether the non-1/1 pitches are consonant with
one another. However, I used this same "drop into buckets idea" using an N-1
dimensional total dyadic harmonic entropy function for N notes to derive
some "optimal" N-tone scales some time ago on this list. I'd be happy to
review this if you're interested -- it takes _all_ the intervals into
account.

JdL:
> A question about your model: do you intend it to be used to create a
> static scale, or to be used in dynamic (adaptive) tuning as a piece
> modulates on the fly?

I would use it to make a set of static scales. I would select the appropriate
scale on the fly.

BTW, thanks for the suggestion to check out harmonic entropy. I will.

DARREN McDOUGALL
Australia

> Darren McDougall wrote,
>
> >I consider the realm of tuning as a one dimensional space
> where distance is
> >measured in cents, and the heavenly bodies are the rational intervals
> situated
> >at points along on a straight line. I regard the mass of
> each body as
> being
> >the number of common harmonics shared between the pitches forming the
> rational
> >interval, which is inversely proportional to the numerator;
>
> Can you explain how you obtained this result? You must be
> making some hidden
> assumption that explains the asymmetry.

Result?? Do you mean the 'inversly proportional to numerator' part?
I'm also not sure what you mean by 'the asymmetry'.

> A similar, but more sophisticated (if I may say so) approach,
> is "harmonic
> entropy", a concept I've been developing for the last few
> years. If you wish
> to learn more, join the list harmonic_entropy@egroups.com, or to see a
> compilation of my older (a bit outdated) ideas on the subject, see
> http://www.ixpres.com/interval/td/entropy.htm. In this model,
> _every_ ratio
> pulls on you no matter where you are on the number line,
> though much like
> gravity the "pull" decreases the farther you are from the
> ratio . . . like
> you, I measure "uncertainty", but I use the standard
> information-theoretic
> measure of uncertainty, called entropy (which satisfies some basic
> additivity and subsetting properties), which measures how
> "confused" you are
> as to which ratio you're actually hearing. There are some
> graphs in the
> "files" section of this list, and of the harmonic entropy
> list as well.

Thanks, I'll have a squiz.

> >I am now contemplating using the series of parabolas as a means of
> selecting
> >rational intervals to form scales. If you picture them as
> buckets catching
> >objects dropped from above, the objects will settle at
> rational interval
> >points. Imagine 13 equally spaced objects (representing 12
> steps) falling
> into
> >an octave wide series of parabolic buckets, you would be
> provided with 12
> >pitches to tune a synth to.
>
> This reminds me of a concept put forward by Hajdu. The
> problem with this is,
> though you'll be ensuring that each note is consonant with
> 1/1, you won't be
> giving any consideration to whether the non-1/1 pitches are
> consonant with
> one another.

That is true, but a lot of them will be.

> However, I used this same "drop into buckets
> idea" using an N-1
> dimensional total dyadic harmonic entropy function for N
> notes to derive
> some "optimal" N-tone scales some time ago on this list. I'd
> be happy to
> review this if you're interested -- it takes _all_ the intervals into
> account.

That would be appreciated, but will I understand it easily or should I first
have a look into harmonic entropy in general?

DARREN McDOUGALL
Australia

Darren McDougall wrote,

>Result?? Do you mean the 'inversly proportional to numerator' part?

Yes.

>I'm also not sure what you mean by 'the asymmetry'.

Well, you can ignore that for now, but I meant asymmetry between numerator
and denominator.

>> However, I used this same "drop into buckets
>> idea" using an N-1
>> dimensional total dyadic harmonic entropy function for N
>> notes to derive
>> some "optimal" N-tone scales some time ago on this list. I'd
>> be happy to
>> review this if you're interested -- it takes _all_ the intervals into
>> account.

>That would be appreciated, but will I understand it easily or should I
first
>have a look into harmonic entropy in general?

As long as you understand that harmonic entropy is a smooth
uncertainty/dissonance function, that should be enough. Of course, I'm sure
you'd enjoy looking into the details of how harmonic entropy is derived.

Also please note that so far both of our models are limited to evaluating
dyads and not larger chords (except by treating them one dyad at a time,
hence ignoring otonal/utonal distinctions) -- though I've described a plan
to extend harmonic entropy to triads, the computational requirements will be
enormous . . .

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

/tuning/topicId_17877.html#17877

> A similar, but more sophisticated (if I may say so) approach,
> is "harmonic entropy", a concept I've been developing for the
> last few years. If you wish to learn more, join the list
> harmonic_entropy@egroups.com, or to see a compilation of my
> older (a bit outdated) ideas on the subject, see
> http://www.ixpres.com/interval/td/entropy.htm.

Hi Paul (or whomever else is able to do this),

As I just wrote in my post responding to Jacky Ligon, I'm way
to busy with other things in my life right now to do a whole lot
of work on my webpages. But I would very much like to bring
the Harmonic Entropy pages more up-to-date. If possible,
please send or post something that updates your thoughts on
it, or at least a list of Harmonic Entropy List links that
revise what's on the pages on my site. Thanks.

-monz
http://www.monz.org
"All roads lead to n^0"

> Darren McDougall wrote,
>
> >I consider the realm of tuning as a one dimensional space
> where distance is
> >measured in cents, and the heavenly bodies are the rational intervals
> situated
> >at points along on a straight line. I regard the mass of
> each body as
> being
> >the number of common harmonics shared between the pitches forming the
> rational
> >interval, which is inversely proportional to the numerator;

Paul E. asked:
> Can you explain how you obtained this result?

Sorry about the delay in responding (Australian public holiday on Friday).
I hope this example illustrates the relationship between shared harmonics and
the numerator. If you consider the harmonics of 300 Hz and 400 Hz:
300 400
600 800
900 1200
1200 1600
1500 2000
1800 2400
2100 2800
2400 3200
2700 3600
3000 .
3300 .
3600
. you will notice that 1200, 2400 and 3600 Hz are common.
. This is the overtone series for 1200 Hz. The lowest
common multiple (LCM) of 300 and 400 is also 1200.
In my opinion, the more shared harmonics two notes have, the more *substance*
that interval has. And the lower the LCM, the more shared harmonics there will
be.

If you calculate the LCM of 'X' and all the integers between 'X' and '2X'
(unison to an octave) then divide all the LCMs by 'X', you end up with:
unison = 1
octave = 2
fifth = 3
fourth = 4
maj3rd = 5
maj6th = 5
min3rd = 6 etc.
These numbers happen to be the numerators of the ratios that describe the
intervals. Unison is the 'heaviest', an octave is then half as heavy, the
fourth half as heavy again, and so on.

I suspect that there is at least one other factor affecting the importance of
each rational interval, and that is the number of harmonics that are NOT in
common. Note that the maj3rd and maj6th both have 1/5 the number of shared
harmonics as unison, but the 6th has one extra surplus harmonic per shared
harmonic than the 5th (the number of surplus per shared is n+d -2). But to my
ears, the effect of the surplus harmonics is way less than the common.

> However, I used this same "drop into buckets
> idea" using an N-1
> dimensional total dyadic harmonic entropy function for N
> notes to derive
> some "optimal" N-tone scales some time ago on this list. I'd
> be happy to
> review this if you're interested -- it takes _all_ the intervals into
> account.

Over the weekend I read a bit of harmonic entropy stuff that I printed, I think
I am ready for the optimal N-tone review. E-mail me privately if it is
lengthy. Thanks Paul.

DARREN McDOUGALL
Australia

--- In tuning@y..., "McDougall, Darren Scott - MCDDS001" <MCDDS001@S...> wrote:

> Sorry about the delay in responding (Australian public holiday on Friday).
> I hope this example illustrates the relationship between shared harmonics and
> the numerator. If you consider the harmonics of 300 Hz and 400 Hz:
> 300 400
> 600 800
> 900 1200
> 1200 1600
> 1500 2000
> 1800 2400
> 2100 2800
> 2400 3200
> 2700 3600
> 3000 .
> 3300 .
> 3600
> . you will notice that 1200, 2400 and 3600 Hz are common.
> . This is the overtone series for 1200 Hz. The lowest
> common multiple (LCM) of 300 and 400 is also 1200.
> In my opinion, the more shared harmonics two notes have, the more *substance*
> that interval has. And the lower the LCM, the more shared harmonics there will
> be.
>
> If you calculate the LCM of 'X' and all the integers between 'X' and '2X'
> (unison to an octave) then divide all the LCMs by 'X',

Aha -- that is where your asymmetry is coming in -- you're dividing by the lower
number. That might seem to make sense _if_ you're holding the pitch of the lower note
constant (though it doesn't quite as Dave Keenan and I have discussed). But what if you
hold the pitch of the _upper_ note constant, or hold the _average_ pitch constant?

Anyhow, the LCM of a ratio is lowest terms is simple n*d (numerator times denominator),
and this is the "raw" measure I've come to use (read the archives to see why) for the
harmonic entropy calculation. Note that this agrees in rank-order with Tenney's
harmonic distance measure, log(n*d)/log(2).

But it doesn't really matter much -- if you use just the numerator, as you're doing and as
I originally did (everything on the _On Harmonic Entropy_ page used the numerator
"limit"), you get the same shape of harmonic entropy curve, but with an overall slope
favoring larger intervals. The slightly counterintuitive result is that a numerator
measure going in leads to a denominator measure going out. That's an additional reason
one might favor n*d -- it's the only simple function of n and d that, when used to "seed"
the harmonic entropy calculation, comes out agreeing with the dissonance ranking for
simple ratios according to the output of the calculation.

> Over the weekend I read a bit of harmonic entropy stuff that I printed, I think
> I am ready for the optimal N-tone review. E-mail me privately if it is
> lengthy. Thanks Paul.

I'll attempt to review this when I get back into the office, but it's all in the archives (I
think the title of the thread was "relaxed ETs" or something like that -- see if you can find
it and prepare some questions for me).