Yahoo Tuning Groups Ultimate Backup tuning The Prime Series as Generator (Part 2): Mediants

The Prime Series as Generator (Part 2)
Mediants of The Prime Series Ratios

Here I will reveal something that I noticed recently in my
exploration of the Prime Series Ratios (ratios having both a prime
numerator and denominator).

First allow me to explain the scale upon which the information to
follow is built from. In my work with the prime series ratios, I
found that as the column of prime denominators is moved upward, there
is an interval at the beginning of each transposed series, which
begins to slowly converge toward an interval approaching a 2/1. As we
know, these ratios would never quite close at an exact
mathematical "2/1", but can achieve what would be audibly the same.

The magic prime series ratio which came within 1.427 cents of 2/1,
was 1213/607 @ 1198.573 cents. This ratio was obtained from the 87th
transposition of the prime denominators. Now, there were other higher
ones that better converged toward the 2/1, but to include those
ratios would have made an even larger, and perhaps unwieldy scale for
the study of the Prime Series Ratios, so to keep things a little more
manageable I let 1,213 Prime be my "cut-off" point. I ran the
transposition of the prime denominators to 108 places, seeking the
best convergence, and this one provided the most logical solution.

This gives me a large non-octave scale with 8,479 pitches, with its
first interval above 1/1 being 1153/1151 @ 3.006 cents and its last
being the 1213/607 @ 1198.573 cents. When we look across this vast
landscape, we see many familiar faces, or perhaps, kinds of
audible "clones" of intervals we know from comparing other tuning
systems.

A really compelling and elegant property of the prime series ratios,
(and the purpose of this article) that I discovered while exploring
this scale has to do with applying the Classic Mediant function to
ratios which neighbor directly above and below a "target" lower
number ratio. This operation reveals an interesting and mysterious
world of the geometrical relationships of high primes to lower
numbered ones. In this case though, the results reveal a "mediant of
simplicity", rather than a "mediant of complexity". That this
function seems to work in a reverse order, when applied to complex
prime ratios, becomes a valuable tool for analysis of the behavior of
the prime series. This is something I discovered when I first applied
the mediant function to the harmonic series, with and without phi
weighting.

(i + m)
-------
(j + n)

Let us consider 8 ratios from the "8,479 Tone, Prime Series Ratios,
Non-Octave Scale", which neighbor around what would be the
interval "4/3" @ 498.0449991 cents. Please note that 4/3 is not a
member of this scale.

977/733 497.454
1009/757 497.473
1049/787 497.495
1097/823 497.519

(Our target interval of 4/3 @ 498.0449991 cents, would fall here
between these ratios).

1151/863 498.546
1103/827 498.568
1063/797 498.588
1031/773 498.605

Now if we take in pairs, the ratios lying immediately above and below
the target value; moving outward and stepwise, and perform the
Classic Mediant operation on each pair, we see something extremely
intriguing:

1097/823 497.519
1151/863 498.546
The mediant between the above ratios: 4/3 @ 498.0449991 cents

1049/787 497.495
1103/827 498.568
The mediant between the above ratios: 4/3 @ 498.0449991 cents

1009/757 497.473
1063/797 498.588
The mediant between the above ratios: 4/3 @ 498.0449991 cents

977/733 497.454
1031/773 498.605
The mediant between the above ratios: 4/3 @ 498.0449991 cents

Similarly, if we fix one ratio and find the mediant between it and
the sequence found opposite the target value we get:

1097/823 497.519
1103/827 498.568
The mediant between the above ratios: 4/3 @ 498.0449991 cents

1097/823 497.519
1063/797 498.588
The mediant between the above ratios: 4/3 @ 498.0449991 cents

1097/823 497.519
1031/773 498.605
The mediant between the above ratios: 4/3 @ 498.0449991 cents

There are myriad of such connections that can be seen at this atomic
level. And that there is some seemingly predictable behavior of these
prime ratios being revealed by this process, is extremely
fascinating.

One could go on to show that in many other areas of the scale, the
above process will hold true. For example:

If we take the Prime Series Ratios lying 10 places either side of a
6/5, we get:

181/503
151/419
We find the mediant to be: 6/5 315.641287

If we take the Prime Series Ratios lying 10 places above and below
5/4:
983/997
787/797
We find the mediant to be: 5/4 386.3137139

Now, as one would imagine, there is a point at which this breaks
down. As you keep moving away from the target interval in this
symmetrical manner, you'll reach a point at which it will no longer
return the target value. This is to be expected.

As a matter of fact, if we look at the mediants between the Prime
Ratios, which border the areas of many primary lower numbered
ratios,we find a kind of rippling effect as we move away from center.

I wanted to see if I reduced the number of Prime Ratios in the area
of the lower number primary ratios, if the mediants would extend
further down the pairwise series. What I found was the opposite; with
less Prime Ratios, there was an increase in the fragmentation. My
intuition is that increasing the number would likely extend it,
though I haven't gotten to this point yet.

What perhaps does this show us, what may we infer from it, and how
may we make music with it? A few ideas:

1. One can see that there is some orderly and predictable behavior
of the Prime Series Ratios, which is revealed by the mediant
function; showing a direct relationship to lower number primes.

2. Ratios which produce the simple mediants may have unforeseen
value as scale generators, because of their unique relationship to
lower primary ratios.

3. Powerful adaptive capabilities may be explored from these
relationships, allowing one to use these neighboring ratios to
achieve better consonance of chords.

4. A dense spectrum of new rational intervallic variations and
interrelationships become available, giving us myriad of new tuning
flavors.

I have placed a document in our files section
labeled "mediants_PSR.pdf", which includes charts I made with MS
Excel, to help visualize the mediants between the first 50 Prime
Series Ratio pairs around 5 different low number primary ratios. You
may access this here:

/tuning/files/Ligon+FFT/

I seek the input and opinions of the many tuning scholars here about
the findings of this study. I would like to openly ask for the
viewpoints of Dave Keenan, Paul Erlich, Peter Mulkers, Margo
Schulter, Kraig Grady and Joe Monzo on this topic, and invite
creative discussion about it all. Please feel free to tell me what if
any value you may see in this. I would also like to ask if I may
be "recreating the wheel" here? Do you know if this has ever been
noticed before, or explored musically by anyone?

Thanks to all,

Jacky Ligon

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗ligonj@northstate.net

🔗pehrson@pubmedia.com

🔗ligonj@northstate.net

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Jacky wrote,
>
> >I would like to openly ask for the
> >viewpoints of Dave Keenan, Paul Erlich
>
> Since you asked, I don't think you're accomplishing or discovering
anything
> of musical relevance with the prime series methodology that
couldn't be
> achieved by more ordinary means.

Paul,

If the results are audibly the same as "x" tuning, please tell me
what the significant difference in the *sound* of music played with
these tunings is, rather than ones generated by "more ordinary
means". Perhaps your reply will reveal your numerological tendencies
as well. Because likely the only reply, is that it is a mathematical
difference - which is what (?) - numbers baby! I can only turn the
mirror toward you now Paul! : )

It's unfortunate to me that you've failed to consider a new
possibility; meant all along for good explorative fun, and not to
introduce new dogma.

Please become aware that this is not the only kind of tuning I'm
actively exploring.

> I think you're just playing with numbers,
> and ascribing musical significance to these very large numbers or
to their
> absolute primality is what I consider numerology.

See above - we're all "playing with numbers", although I would
prefer the label "exploring with numbers". Who in this forum has
produced the "Grail Tuning System"? Let them step forward, and
introduce theirself. Who in this forum is not exploring tuning
systems with mathematics? Show me one. Please kindly reveal to me
what if any difference there is in the sound of the music, and you'll
have me sold - hook, line and sinker. Don't think you can though.

If the cents values I tune my synth to are identical to ones
from "more ordinary means", how could that, in any stretch of the
imagination disqualify this or that tuning system? Just don't get it.

We give great weight here to the audibility of this and that - so,
what *is* the audible difference? I don't hear numbers - even being
into "numerology" like I am. If the cents are the same, I hear the
same thing.

Insert mathematical reply here:

> I don't find anything
> mysterious about your mediants, and "classically" the mediant of two
> fractions a/b and c/d only "works" if b*c - a*d = 1. Maybe you
shouldn't
> have asked me!

As before, I'd hoped to get your input, to see if there was any value
in this, and I appreciate you honesty - honestly.

Thanks,

Jacky Ligon

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Jacky wrote,

>If the results are audibly the same as "x" tuning, please tell me
>what the significant difference in the *sound* of music played with
>these tunings is, rather than ones generated by "more ordinary
>means". Perhaps your reply will reveal your numerological tendencies
>as well. Because likely the only reply, is that it is a mathematical
>difference - which is what (?) - numbers baby! I can only turn the
>mirror toward you now Paul! : )

>It's unfortunate to me that you've failed to consider a new
>possibility; meant all along for good explorative fun, and not to
>introduce new dogma.

>See above - we're all "playing with numbers", although I would
>prefer the label "exploring with numbers". Who in this forum has
>produced the "Grail Tuning System"? Let them step forward, and
>introduce theirself. Who in this forum is not exploring tuning
>systems with mathematics? Show me one. Please kindly reveal to me
>what if any difference there is in the sound of the music, and you'll
>have me sold - hook, line and sinker. Don't think you can though.

>If the cents values I tune my synth to are identical to ones
>from "more ordinary means", how could that, in any stretch of the
>imagination disqualify this or that tuning system? Just don't get it.

>We give great weight here to the audibility of this and that - so,
>what *is* the audible difference? I don't hear numbers - even being
>into "numerology" like I am. If the cents are the same, I hear the
>same thing.

First of all, you never answered my question as to why your list of 12-tone
MOSs had some notable omissions.

Secondly, I must point out that I never said there was anything wrong with
the sound of these tunings. I just meant that they've already been
discovered by Dave Keenan, Margo Schulter, and others, using direct
mathematical inquiry, and without the irrelevant and potentially confusing
large prime ratios.

Thirdly, I note that Dan Stearns replied,

>Though there are of course many ways to go about things, and different
>ways usually do accomplish different things, I think the most
>efficient and internally consistent way to do this sort of a tour
>through the 'safe zone' is to seed the Stern-Brocot Tree a la Erv
>Wilson's scale tree with a given set of adjacent fractions. This way
>the mediants -- fruits or flowers if you will -- will fill out the
>tree by incrementally filling in the spaces between x, x+y, and y to
>as dense or ventilated a degree as you could possibly want. Now this
>is a logarithmic (i.e., fraction of an octave) method and not a RI
>type method. But I think it will accomplish the grand tour in a much
>more organized and organic fashion.

With which I wholeheartedly agree.

🔗ligonj@northstate.net

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
>
> First of all, you never answered my question as to why your list of
12-tone
> MOSs had some notable omissions.

Likely because I wasn't looking at it from the point of view that you
have, which I would be delighted to learn about. I was mostly working
from a fixed table of thirds that I wished to feature in the scales.

> Secondly, I must point out that I never said there was anything
wrong with
> the sound of these tunings. I just meant that they've already been
> discovered by Dave Keenan, Margo Schulter, and others, using direct
> mathematical inquiry,

I can promise you that I was not referencing any of this. And I only
used Margo's ratios as examples to show harmonic connections. What's
wrong with that? As above, my goal was targeting a table of thirds -
that's all there was to it. That anything lined up with anything else
that someone else already discovered, was pure happenstance, except
for your suggestion to find 1/4 comma meantone - which I did
consciously try to duplicate with a prime generator. I just hadn't
thought of all this. My "inquiry" was simply into the allignment of
thirds - nothing more.

> and without the irrelevant and potentially confusing
> large prime ratios.

I think this would surely qualify as *your* opinion. It is not my
quest to confuse folks, and I'm sure if they find it "irrelevant",
their fingers will dart toward the old scroll button faster than
lightening (perhaps you will explore this possibility). I'm not
asking anyone to participate in anything that will confuse them, and
only hoped to share some interesting things I'd found. Isn't that
what the old tuning list is all about anyway? Or perhaps you could
pass out some assignments to keep things more focused! : )

>
> Thirdly, I note that Dan Stearns replied,
>
> >Though there are of course many ways to go about things, and
different
> >ways usually do accomplish different things, I think the most
> >efficient and internally consistent way to do this sort of a tour
> >through the 'safe zone' is to seed the Stern-Brocot Tree a la Erv
> >Wilson's scale tree with a given set of adjacent fractions. This
way
> >the mediants -- fruits or flowers if you will -- will fill out the
> >tree by incrementally filling in the spaces between x, x+y, and y
to
> >as dense or ventilated a degree as you could possibly want. Now
this
> >is a logarithmic (i.e., fraction of an octave) method and not a RI
> >type method. But I think it will accomplish the grand tour in a
much
> >more organized and organic fashion.
>
> With which I wholeheartedly agree.

Somewhere in all this you have completely failed to notice that on
several occasions I've stated that I also work weighted intervals,
and I openly have no problem with what Dan is saying here. But on the
same note, I also find primes of interest. I know it's not the end
all, so I don't understand the apparent negativity here. Certainly is
not my intention. In the course of many posts, we all do quite a bit
of comparative work, by taking note of how things allign with other
systems. This is a small part of my interest in this.

You have also failed to notice that on many occasions I've said that
I do not attach significant meaning to the audibility of prime limits
and such to, so your inference about "numerology" is quite unfounded
(it's just a series Paul - I know this). But you still are a cool guy
in my eyes all the same!

Lighten up dude!

This is supposed to be fun!

Jacky Ligon

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗ligonj@northstate.net

🔗D.KEENAN@UQ.NET.AU

Dear Jacky,

I'm sorry to be blunt here, but I'm short on time at the moment. Your
result *might* be mathematically interesting, but I can't see any
musical relevance, because, putting it crudely, "primeness is not
audible". At least I know of no evidence to suggest that anyone can,
by listening, distinguish musical intervals corresponding to rational
frequency ratios whose numerator and denominator are prime, from
nearby ones whose numerator and denominator are not prime, or indeed
nearby irrational ones.

I don't get the point at all, of your insistence on using only ratios
of primes (some of which are quite large) in your explorations of MOS
generators. Your explorations of these generators are interesting in
themselves. Thankyou for writing and posting them. But they would be
just as interesting if you had simply used various numbers of cents
for your generators.

On the question of the purely mathematical interest of your recent
result, I would like to know whether the pairs whose "mediant" fails
to give the most common value, actually *have* a mediant.

As Paul Erlich implied, a pair of ratios X and Y only *have* a mediant
(or at least the special properties of the mediant only occur) if
numerator_of_X * denominator_of_Y differs from numerator_of_Y *
denominator_of_X by exactly one (plus or minus).

My understanding is that the classic mediant always gives the simplest
ratio between two other ratios, but only if the above property (called
adjacency) is satisfied. This property must also be satisfied for the
noble-mediant (which gives the most complex ratio instead of the
simplest). I think I failed to mention this adjacency requirement in
Margo's and my paper, but mentioned it in subsequent posts in the same
thread.

So I think you can actually just choose any pair of ratios on either
side of a target (such as 3/2) and if they are both more complex than
the target (complexity = numerator * denominator), and are adjacent
(by the above test) then their mediant will always be the target.
Primality is irrelevant, and equal distance either side of the target
(in some series) is irrelevant. Yes this is a somewhat surprising
property of the mediant, but well known I think.

Thanks for giving me the opportunity to comment.

Regards,
-- Dave Keenan

🔗ligonj@northstate.net

--- In tuning@y..., D.KEENAN@U... wrote:
> Dear Jacky,
>
> "primeness is not
> audible".

Dave,

Thanks for your input on this.

I would like to clarify once again (perhaps you've missed this while
away), that I'm not making any claims, either explicit of inferred
about the audibility of primes. This is a dead argument to me and one
which I'm in agreement with you and many others.

> At least I know of no evidence to suggest that anyone can,
> by listening, distinguish musical intervals corresponding to
rational
> frequency ratios whose numerator and denominator are prime, from
> nearby ones whose numerator and denominator are not prime, or
indeed
> nearby irrational ones.

See above. This is *not* what I was attempting to do at all.

> I don't get the point at all, of your insistence on using only
ratios
> of primes (some of which are quite large) in your explorations of
MOS
> generators. Your explorations of these generators are interesting
in
> themselves. Thank you for writing and posting them. But they would
be
> just as interesting if you had simply used various numbers of cents
> for your generators.

It so simple Dave; I just wanted to see what could be revealed by
looking into and playing scales from the prime series; that's all.
Not much different in my mind from many other explorative things of
this nature. I glad that you don't seem to have a problem with folks
sharing ideas outside the mainstream.

Maybe my curiosity does take me into areas that are
considered "irrelevant", by some. Perhaps we should appoint a list
moderator to filter out explorative and irrelevant tuning work like
this, and perhaps boot violators who inject trivial ideas into our
dead serious game here, just to avoid the potential "confusion" of
others. If it is the consensus that I should keep things to myself,
then I will happily do it (as I have been already), and spare folks
the waste of their time. What may be fun exploration for me, may be
agonizing for others, which I will respectfully accept.

> On the question of the purely mathematical interest of your recent
> result, I would like to know whether the pairs whose "mediant"
fails
> to give the most common value, actually *have* a mediant.

Yes.

> Primality is irrelevant, and equal distance either side of the
target
> (in some series) is irrelevant. Yes this is a somewhat surprising
> property of the mediant, but well known I think.
>
> Thanks for giving me the opportunity to comment.
>

I thought it was interesting too, and I appreciate you taking time to
look at it.
And thank you for your usually courteous and insightful reply.

Jacky Ligon

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗ligonj@northstate.net

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
>
> Wasn't 5:4 in your list of thirds? If not, why?

It was, but scales you enjoy with it likely weren't. Had I thought in
*your* manner, and perhaps if I *was you*, I'd have produced all that
you desire, but by force of nature this isn't so.

Please tell me what the omissions were, so that I may learn further.
The omission of them was not intentional, as said before, because I
wasn't thinking in the terms you speak of. My goal was not to
reproduce other scales, but to see what lies in the primes. I'm sure
I have no clue what you are talking about, except for that it might
be other meantone scales.

> Jacky, that's precisely my point! If it's not audible, then it's
numerology!
> The only numbers I care about are the ones that explain what you
_hear_!

This is hilarious! I have absolutely *no* difficulty hearing the
thirds that were the target of the MOS scales. Don't care much what
primes they were - I could hear them!

Do you not *hear* 3/2 BTW? Doesn't 3/2 have a distinctly recognizable
sound to you?

You have still failed to answer my question about what is the audible
difference between thirds achieved in the manner, than ones produced
by more conventional means. As we know there is none. Only in a few
cases were there was around a 3 cents deviation from target, could
you possibly hear the difference, and even then you would have to be
able to a&b between them (not the intended musical idea I had in
mind).

> OK, I'm smiling -- I find vigorous debate entertaining, actually.

I really don't see the point, because I'm not trying to debate
anything here. Just share tunings and make music - a humble goal.

Jacky Ligon

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Jacky wrote,

>It was, but scales you enjoy with it likely weren't. Had I thought in
>*your* manner, and perhaps if I *was you*, I'd have produced all that
>you desire, but by force of nature this isn't so.

>Please tell me what the omissions were, so that I may learn further.
>The omission of them was not intentional, as said before, because I
>wasn't thinking in the terms you speak of. My goal was not to
>reproduce other scales, but to see what lies in the primes. I'm sure
>I have no clue what you are talking about, except for that it might
>be other meantone scales.

As you know, I was specifically thinking of 1/4-comma meantone. You included
1/3-comma meantone (which gets you 9 6:5s), but not 1/4-comma meantone which
gets you 8 5:4s -- actually, would 2/7-comma meantone get you _both_ 8 5:4s
_and_ 9 6:5s by your reckoning?

>> Jacky, that's precisely my point! If it's not audible, then it's
numerology!
>> The only numbers I care about are the ones that explain what you
_hear_!

>This is hilarious! I have absolutely *no* difficulty hearing the
>thirds that were the target of the MOS scales. Don't care much what
>primes they were - I could hear them!

Of course, I can hear most of these thirds too -- I wasn't talking about the
numbers you used in the thirds, which were small -- I was talking about the
numbers you used in the fifths, which were large!

>Do you not *hear* 3/2 BTW? Doesn't 3/2 have a distinctly recognizable
>sound to you?

Yes, that's the only fifth from your list that I can "hear" as such.

>You have still failed to answer my question about what is the audible
>difference between thirds achieved in the manner, than ones produced
>by more conventional means. As we know there is none. Only in a few
>cases were there was around a 3 cents deviation from target, could
>you possibly hear the difference, and even then you would have to be
>able to a&b between them (not the intended musical idea I had in
>mind).

Jacky, I thought I (along with Dan and Dave K.) has made it clear by now
that there is no audible difference, and one is of course free to target
thirds with whatever cents deviation you want, but your method gives a
rather arbitrary and incomplete view of the possibilities, which a
systematic survey of fifth sizes (in cents), as Margo and Dave have done,
covers the cases you brought up as part of a _continuum_.

🔗ligonj@northstate.net

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
>
> As you know, I was specifically thinking of 1/4-comma meantone. You
included
> 1/3-comma meantone (which gets you 9 6:5s), but not 1/4-comma
meantone which
> gets you 8 5:4s -- actually, would 2/7-comma meantone get you
_both_ 8 5:4s
> _and_ 9 6:5s by your reckoning?

Thanks for this Paul!

See, I just was not thinking in these terms, but I am interested in
learning all I can about meantone tunings, and will endeavor to grasp
them better, as well as the actual way they are constructed. Perhaps
I just have to come around to all this in some back door way. I will
study these possibilities more closely - rest assured. It was
fascinating and revealing to me when you brought the 1/4 CMT to my
attention, and I went through the exercise of creating it with the
PS. I realize my approach is a little unorthodox, but heck that's me -
flaws and all, what more can I say. I am the perpetual tuning and
music student, and the day I start declaring that I know it all, I
hope they've got the padded room ready, with all the thorazine I may
need.

> Of course, I can hear most of these thirds too -- I wasn't talking
about the
> numbers you used in the thirds, which were small -- I was talking
about the
> numbers you used in the fifths, which were large!

Again, I make no special or outrageous claims about the audibility of
primes - especially these high ones, but please try to grasp my angle
too; I was only presenting what was nakedly there, without any of the
implied ideas about primeness. Perhaps I am in error here as well,
but that's what I though tuning science, and science in general was
supposed to be about. I just tried to humbly put forth what I found,
without attempting to put any numerological spin on any portion of
it. Where did I make this error? Please let me know where you detect
it, so that I may avoid the mistake again.

> Jacky, I thought I (along with Dan and Dave K.) has made it clear
by now
> that there is no audible difference, and one is of course free to
target
> thirds with whatever cents deviation you want, but your method
gives a
> rather arbitrary and incomplete view of the possibilities, which a
> systematic survey of fifth sizes (in cents), as Margo and Dave have
done,
> covers the cases you brought up as part of a _continuum_.

I agree with you here, and really have not been in disagreement with
you ever about the many ways to do things. I understand these other
methods, but what good what it do any of us, to learn about those
darn prime numbers, if I was trying to exactly duplicate someone
else's achievements?

What you call "arbitrary and incomplete", is what I have been
metaphorically referring to as the inherent "noise"
and "imperfection" in the prime series. You, Dan and Dave are right
about it not smoothly dividing up the MOS space; and it may be highly
ironic to you that this "imperfection", is exactly what attracts me
to this series. I love "imperfect" systems like this, which create
their own logic (or illogic), and one can easily see that this is
also present in myriad of natural objects too. That it is a
disobedient system, is what tweaks my interest. Why must all things
be orderly? Actually they rarely are; chaos is lurking just a step
away - maybe even closer, whether we perceive it or not.

Forgive me if I am an inadequate debater. But just let me get you on
that chess board, and you'll recite the Uncle Mantra in my honor!

Jacky Ligon

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗ligonj@northstate.net

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Jacky Ligon wrote,
>
> <<it may be highly ironic to you that this "imperfection", is
exactly
> what attracts me to this series. I love "imperfect" systems like
this,
> which create their own logic (or illogic),>>
>
> I'm sure you could fill up the space with prime generators if you
had
> the patience, skill and burning desire to do so <grinning thing>!,
but
> I think the sort of thing that Paul is trying to say here is 'what's
> the point if you can do it this way with an inherent and elegant why
> it works built right into it'. But that's just my take and I really
> don't want to be putting words into other peoples mouths and
whatnot.

Dan,

To even try the prime ratios as MOS generators, was something that
only occurred to me months after my initial study of this began. It
was not my goal, just an afterthought actually, but the idea
presented itself during the course of several parallel
investigations. I can't underline this point sufficiently enough -
this is not what I set out to do, and only saw the possibility
revealed months after I began looking into this. And please
understand that I've not even for an instant tried to refute, or be
in opposition to the Fibo method of finding MOS generators or any
other way for that matter. To infer this just shows that we have
ignored what I have written, in an attempt to place me in some kind
of frame which I do not fit. I have not sought to diminish any method
with my own work, but only to present what I found. Opposition to
this is extremely curious to me.

> So to get back to mouthing some of my own mind, I think it would be
> very interesting to search out something special about co-prime
ratios
> or the related like... something they have to offer that is somehow
> symbiotically or organically particular or tied to their own
> construction.

That's what I was pointing to with the mediants. Whether there is
anything of value there, remains to be seen. I will not claim that
there is at this point either. It has just been an attempt to
stimulate creative and constructive discussion, which I hope will
come forward as we move ahead.

>
> As it seems like things might be getting dangerously close to the
> boiling point here, I'd just like to say that I wouldn't take Paul's
> responses here to grimly (i.e., "personally") if I were you, after
all
> you did specifically ask for it!

Oh, I don't. I realize that Paul loves to debate about tunings, but
it just happens to not be my forte. Actually in some perverse way, I
also find it stimulating (not particularly "entertaining" to me
though). But the difficult thing, is I felt I had taken every
precaution to repeatedly point out that I do not necessarily attach
any import to primes over anything else. To infer this was a
misconstruence of what I wrote. It was just an effort to take a look
at the prime series. I still don't really grasp why there should be
any revulsion for tuning investigations of any variety.

<please insert a liberal dose of
> tension relieving smiling things at this point...> By all means,
keep
> posting exactly as you see fit... I think everyone learns from the
> discourse; well so long as no one explodes and things get gruesome
> anyway!

Thanks Dan! I didn't really see things approaching the boiling point.
Not to worry, it's all in good fun! And I'm sure that Paul's kind
computer side manner will inspire generations of microtonalists to
come forward to speak about their creative work on the Tuning List.

(profusion of smiley things inserted)

:) : ) : ) : )

Thanks,

Jacky Ligon

🔗jpehrson@rcn.com

🔗ligonj@northstate.net

--- In tuning@y..., jpehrson@r... wrote:

> Hi Jacky....
>
> Just because you were told that your theory had no musical and
> acoustical validity and was not of any interest mathematically
either
> is no reason to get upset!

Joseph,

Hello!

I think "no musical and acoustical validity", would not do justice to
the fact that I produced intervals which are audibly identical
to "acoustic" thirds. No matter what the manner of generation, they
are still *acoustical intervals*, and therefore of *musical value*.
This I know from playing them myself. Please note that Paul could
find no difference with the acoustic thirds I produced, and only
found difference with my method of scale generation.

Please consider if I had not mentioned the generative method I used
and only revealed cents values, there would be no confusion as to
whether there was any acoustical relevance to the goal of the tunings
I used (as stated by Dave Keenan). It is the opposition to the way I
generated these scales which has been pushed to the fore; then over a
cliff. : )

Oh, don't misperceive that I was upset - just trying to clear up the
pointless inferences about things I never said or implied. Imagine if
you had taken every precaution to avoid confusion, and then the very
things you felt you had made clear on a number of things, suddenly
became "issues". Perhaps anyone would see the need to even more
forcefully underline the facts about their intentions.

> Congrats to you for your great courage to post something like this!
> Perhaps the problem was asking our knowledgeables for er...
> "criticism."

Every day is a learning experience, and thank you for not having
issues with posting things which are a little "off-beat" (you ain't
seen nothing yet baby!). And does being a "knowledgeable" mean that
they think they know it all? I automatically would be highly suspect
of any individuals who made such a declaration. I submit to you that
the real "formula of relevancy" has to do with how many compositions
one has made with microtonal scales. All the debating, theorizing and
yakking about scales does not "significant works" of music make. Like
Partch said: "It's just papers blowing in the wind". He knew the
truth, just like Darreg and many others did (and do).

> For my own part, if I posted something I would hope that nobody
would
> notice it... so congrats on your work and courage...

Again, thank you Joseph! I would devour anything you post with great
interest, since I don't think I've ever seen a scale you've
constructed.

I would like to ask you if you can speak about your JI research that
I've gotten wind of. Especially if it's not too top secret; please
share the tunings with us. I love JI/RI creative scales! Are the
ones you've spoke of, ones that you have designed yourself?

You may notice that scale design is a hugely important activity for
me. I have been designing my own scales for a very long time - and JI
is the area I know best.

Please, if you see fit, reveal your tunings and speak about the
musical attributes. I would be delighted to learn about them! If you
worry that they would be ridiculed here, we can chat privately if you
like.

Thanks,

Jacky Ligon

🔗jpehrson@rcn.com

--- In tuning@y..., ligonj@n... wrote:

/tuning/topicId_18060.html#18163

> Again, thank you Joseph! I would devour anything you post with
great interest, since I don't think I've ever seen a scale you've
> constructed.
>

That would seem quite reasonable, since I haven't done any...

> I would like to ask you if you can speak about your JI research
that I've gotten wind of. Especially if it's not too top secret;
please share the tunings with us. I love JI/RI creative scales! Are
the ones you've spoke of, ones that you have designed yourself?
>

Thanks so much, Jacky, for your invitation for me to invade more
"post-space..."

One of the experiments I did, did not involved ORIGINAL scales, but
was a comparison of various forms of 19-tone scales. This
investigation was part of the process of my trying to find out which
scale would work best for my piece VERKLARTE NEUNZEHN.

Basically, this piece used motivic material that, at least to me,
seemed a little reminiscent of the Second Viennese School. (c.f. Joe
Monzo, our resident "expert" on this period...)

However, the tuning was in various forms of 19-tones per octave.
Actually Manuel op de Coul helped me out a bit with this, since it
turned out that one of the scales that I was looking at in Scala was
actually INCORRECT. He fixed it, and it was his corrected scale I
ultimately used... an extraction of 19 tones from a 31 tone Just
Intonation scale.

This was all stuff that we discussed, I believe, prior to your
arrival on this list...

I also compared this scale with various other forms of 19 before
making a choice as to which one to use...

The other options, as you will see on the following webpage, included
a 19-tone scale extracted from 31-tET, a 19-tone scale derived from
quarter comma meantone, a 7-limit 19-tone scale and, finally, 19-tET.

In this case I didn't really have the desire to create my OWN scale,
since there was so much to explore right here!!!

I made some commentary on these scales on the following webpage, and
several listers also listened to them. Some people found little
difference between the 19-tone scales, BUT others found HUGH
differences....notably Carl Lumma.

Manuel op de Coul also suggested that some of the discrepencies could
be attributed to the lack of definition of the TX81Z... only 2.5
cents, so perhaps that contributed to an unintended "difference of
effects..."

Here is the webpage:

http://users.rcn.com/jpehrson/tuning.html

Do you hear any differences?? I bet you like the ones extracted from
JI best (??) How did I know.....! :)

_________ _______ ____ _
Joseph Pehrson

🔗ligonj@northstate.net

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., ligonj@n... wrote:
>
> /tuning/topicId_18060.html#18163
>
> > Again, thank you Joseph! I would devour anything you post with
> great interest, since I don't think I've ever seen a scale you've
> > constructed.
>
> That would seem quite reasonable, since I haven't done any...
>
> > I would like to ask you if you can speak about your JI research
> that I've gotten wind of. Especially if it's not too top secret;
> please share the tunings with us. I love JI/RI creative scales!
Are
> the ones you've spoke of, ones that you have designed yourself?
>
> Thanks so much, Jacky, for your invitation for me to invade more
> "post-space..."
>
> One of the experiments I did, did not involved ORIGINAL scales, but
> was a comparison of various forms of 19-tone scales. This
> investigation was part of the process of my trying to find out
which
> scale would work best for my piece VERKLARTE NEUNZEHN.

Joseph,

OK, I've listened to all of your examples and I think my ears prefer:

1. Verklarte Neunzehn example 1: 19-31ji-coul.mp3

2. Verklarte Neunzehn example 3: meanquar_19.mp3

3. Verklarte Neunzehn example 4: ji_19a.mp3

And very interesting to me, was that 19 tET sounded the most rough -
wouldn't have guessed that! Maybe it's the 19tET fifths as compared
to the JI fifths that reinforced this perception.

> Basically, this piece used motivic material that, at least to me,
> seemed a little reminiscent of the Second Viennese School. (c.f.
Joe
> Monzo, our resident "expert" on this period...)

I've always like this piece! Got my attention when I first came here.
Which tuning does the one on the Tuning Punks page use?

> I also compared this scale with various other forms of 19 before
> making a choice as to which one to use...
>
> The other options, as you will see on the following webpage,
included
> a 19-tone scale extracted from 31-tET, a 19-tone scale derived from
> quarter comma meantone, a 7-limit 19-tone scale and, finally, 19-
tET.

So the 19 from 1/4 CMT, was taken from a 1/4 CMT chain of 31 fifths @
around 697 cents?

> In this case I didn't really have the desire to create my OWN
scale,
> since there was so much to explore right here!!!

But you know what (?) this is very creative too (!), and your
exploration here is very interesting, as well as being tangible (it
jumped off the page into a musical reality!). I like the idea of this
kind of comparative study.

> Manuel op de Coul also suggested that some of the discrepancies
could
> be attributed to the lack of definition of the TX81Z... only 2.5
> cents, so perhaps that contributed to an unintended "difference of
> effects..."

I think you mean here 1.56.

> http://users.rcn.com/jpehrson/tuning.html
>
> Do you hear any differences?? I bet you like the ones extracted
from
> JI best (??) How did I know.....! :)

You're right! But I did agree that the 1/4 CMT was also very smooth
sounding too. Very interesting scale!

Please explain a little about the 31 tone 1/4 CMT scale or direct me
to web pages where I may learn more about this.

This is great Joseph!

Thanks for posting this,

Jacky Ligon

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗D.KEENAN@UQ.NET.AU

Dear Jacky,

We "knowledgeables" are only too aware of the vast yawning chasms of
our ignorance on matters psychoacoustic.

So it is perhaps understandable that we should want to clearly
separate what little has been gleaned over the centuries by the
scientific method, from things we see as "mere numerology". That is
the ascribing of special musical or psychoacoustic (i.e. perceptible,
even if only subliminal) properties on the sole basis that the numbers
have special mathematical properties, or because they have special
properties in other application areas (such as the visual arts).

Please forgive us if we are sometimes a little overzealous in this
regard.

You have now made it clear that you are not claiming this for ratios
of large primes as scale generators. But I think it is undersrtandable
that we should think you were, despite earlier advice to the contrary,
given that we're still not sure why you were expressing your
generators as large prime ratios at all.

I guess you were wanting to see if there was any simple
mathematical relationship between the numbers in the generator (when
expressed as a ratio of large primes) and the corresponding generated
Just thirds (or at least thirds expressed as ratios of small whole
numbers).

I guess that we failed to understand that that was what you were doing
because it was intuitively obvious to us (through mathematical
experience) that there would be no such mathematical relationship. We
are always happy to have our intuitions demolished by the facts (this
is one of the most exciting things a scientist or mathematician can
experience), but I hope it is clear that this hasn't happened in this
case.

For example the sizes of generator (as a frequency ratio) at which the
Just thirds appear (and the transition points between better
approximations of one or another Just third) are all irrational. By
choosing large enough primes one can always find a ratio of primes
that is imperceptibly close to a given irrational. So there is no
significance to the prime ratios unless they are always much smaller
than one would expect for randomly chosen irrationals using the same
tolerance. They are not.

Regards,
-- Dave Keenan

🔗ligonj@northstate.net

🔗jpehrson@rcn.com

--- In tuning@y..., ligonj@n... wrote:

/tuning/topicId_18060.html#18170

>
> Joseph,
>
> OK, I've listened to all of your examples and I think my ears
prefer:
>
> 1. Verklarte Neunzehn example 1: 19-31ji-coul.mp3
>
> 2. Verklarte Neunzehn example 3: meanquar_19.mp3
>
> 3. Verklarte Neunzehn example 4: ji_19a.mp3

Hmmm... curious, you've chosen mostly the ones with ratios rather
than cents values...

Just joking. Frankly, these are exactly the ones I preferred as well.
The two "final contenders" for the tuning of this piece were the op
de Coul 19 out of 31 JI scale and the 7-limit 19 note scale.

Ultimately, I chose the 7-limit 19 note scale for the piece, but it
was pretty much a "toss up." The other "cents value" scales seemed
to have more distracting beating in the sonorities... Was this
basically psychological, a result of seeing decimal numbers rather
than ratios?.. Dunno. Possibly, but I don't think so. It seemed I
was actually HEARING it.

A caveat is in order for all of this. When this study first surfaced
on the tuning list, somebody remarked... maybe Paul, but possibly
somebody else... that there was so much going on in this piece that
it was really difficult to come to ANY objective conclusion about
what was going on with the tuning.

Additionally, and this was the point that Manuel op de Coul brought
up, the limits of resolution of the TX81Z, 1.56 cents (yes, of
course, I don't know why I was thinking 2.5... it's not THAT bad!)
would mean that the "rounding factor" could create discrepancies in
the tunings depending on the particular frequencies that were
required. So, maybe, ideally, they wouldn't be quite as different as
they sounded in this exercise.

Well, regardless, the point is that a process like this would not
really be a scientific experiment, for better or worse, but it still
would obviously be possible to listen to the piece in the different
tunings and make a preference! And, ultimately, that's the important
part anyway if the point of doing it is finding an optimal tuning for
a piece...

>
> And very interesting to me, was that 19 tET sounded the most rough
-
> wouldn't have guessed that! Maybe it's the 19tET fifths as compared
> to the JI fifths that reinforced this perception.

Without a doubt, I had absolutely the same reaction to it...

>
> > Basically, this piece used motivic material that, at least to me,
> > seemed a little reminiscent of the Second Viennese School. (c.f.
> Joe Monzo, our resident "expert" on this period...)
>
> I've always like this piece! Got my attention when I first came
here.

Thank you so very much Jacky. Actually, I am happy to report that it
is one of the electronic works selected for performance in the "audio
room" at the Microfest at Claremont, CA April 6-8!

I was also asked to attend this conference, and it will be a great
joy and honor to be there...

> Which tuning does the one on the Tuning Punks page use?
>

I finally settled on the 7-limit 19-tone scale...

>
> So the 19 from 1/4 CMT, was taken from a 1/4 CMT chain of 31 fifths
@ around 697 cents?

I believe so... I see a 696.578...

>
> Please explain a little about the 31 tone 1/4 CMT scale or direct
me to web pages where I may learn more about this.
>

I notice that Paul Erlich, in his usual prompt fashion, has already
provided a link for this study!

Thanks again, Jacky, for your interest!

_______ _____ ______ ____
Joseph Pehrson

🔗D.KEENAN@UQ.NET.AU

🔗MONZ@JUNO.COM

🔗graham@microtonal.co.uk

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Hi Dave (Keenan)

> So I think you can actually just choose any pair of ratios on either
> side of a target (such as 3/2) and if they are both more complex than
> the target (complexity = numerator * denominator), and are adjacent
> (by the above test) then their mediant will always be the target.
> Primality is irrelevant, and equal distance either side of the target
> (in some series) is irrelevant. Yes this is a somewhat surprising
> property of the mediant, but well known I think.

I'm not sure that Jacky's results are explained yet, and I find them rather
intriguing.

Ex.
1151*823-1097*863=562, so 1097/823 and 1151/863 aren't adjacent.
1097+1151 = 2248
823+863 = 1686

and 2248/1686 = 4/3 exactly.

Similarly, looking at 1097/823 and 1103/827
1103*823-1097*827=550
1103+1097 =2200
823+827 = 1650

and 2200/1650 = 4/3 exactly again.

Rather surprising!

If I understand the article correctly,

977/733, 1009/757, 1049/787, and 1097/823
are simply the four closest large prime ratios below 4/3 up to prime limit 1213

and 1151/863, 1103/827, 1063/797, and 1031/773
are the four closest large prime ratios above 4/3 up to the same prime limit.

I wonder what explanation there can be for this?

Also, I think it suggests that using large prime ratios may well be a helpful way
to explore temperaments, because of the possibility of working backwards
and finding small ratios in this way.

Choose a particular prime limit, and the small ratio 4/3 will correspond to
a particular zone of the large scale constructed in this way - the zone
of prime ratios to either side of it that yield 4/3 when the classic
mediant operation is applied (ignoring any questions of adjacency).

This 4/3 zone may have some significance in scale construction. It is at
least a possibility that is worth investigating!

Anyone got any ideas about why Jacky's construction works?

: - )

Robert