Yahoo Tuning Groups Ultimate Backup tuning Phi Tonality

I haven't read those specific papers before, but I have looked over Sethares' dissonance curves, which indicate the points of maximum harmonic entropy at the peaks. In fact, I've known about Sethares' curves for many many years...they are what got me truly excited about and interested in microtonal in the first place. ;-)
--------------------------------------
-from the article "Pepper's Phi-related function could also be applied to the problem of finding the region of maximum complexity between two simpler ratios, providing a shortcut to the longer process of successive approximation by iterating mediants. In what follows, we show how this function, here termed the "Noble Mediant", can be used to locate regions of maximum complexity. "

------------------------------------------
From the paper, it sounds like you are going for maximum dissonance by finding consecutive areas/plateaus (of highest entropy) between the valleys/areas-of-lowest-entropy in Sethares' dissonance curves.
And, oddly enough, it appears some of the results you get match JI.

But, one thing is for sure...if you are using the formula
Noble_Mediant(i/j, m/n)=
(i + Phi * m) / (j + Phi * n)

To generate your scale...you are going to get something
dramatically different than what I get from my
formula for my tuning
IE
result = (PHI^Y) / (2^X)
where result > 0 and result < PHI^2 and y is
between 0 and 15.
In fact, my version does not even do anything
relative to fractions that represent the peaks
and plateaus of the "harmonic entropy" curve.
-------------------------------------------
This is one thing I'm finding very confusing about
your other PHI tunings...many of them are based on
formulas that have more to do with deriving from
things like whole numbered fractions multiplied by
PHI in some form...than taking a simple incrementing variable
times PHI itself.

So, on the surface, it seems my tuning (at least mathematically)
has more to do with PHI while yours has to do more with
"using PHI do find areas of harmonic entropy relative to something
else".
---------------------------
This difference in use of PHI as a generator potentially means
there are several PHI tunings that look and
sound nothing like each other.

If these papers (all the ones I've read on PHI tunings)
are right...there's is no such thing as a standard
PHI tuning. So again, whether each one sounds better
or worse is likely left up to the listener.
----------------

So, again, to really get a feel for your scale I need an
example. I think we can both agree it's not that I
don't understand a "PHI standard of tuning" (which there
apparently really is none) but, rather, that we are taking
on two completely different methods altogether.

Knowing how your scales work mathematically
vs. Sethares' theories doesn't help, beyond saying that
what you are doing is heavily NOT diatonic (going in
many of the opposite areas of the curve that diatonic
scales do AND just happens to intersect in a few places
with JI).

-Michael

--- On Mon, 2/23/09, Cameron Bobro <misterbobro@...> wrote:

From: Cameron Bobro <misterbobro@...>
Subject: [tuning] Re: Phi Tonality
To: tuning@yahoogroups.com
Date: Monday, February 23, 2009, 6:45 PM

--- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...>

wrote:

> ---Just a little side note- because my use of phi is a bit

different

> ---(I'm just making golden cuts on the frequency level, and linking/

> ---interlocking everything with JI)

> Ah, I kind of figured. So you are using PHI to approximate JI.

No, I'm using phi as a kind of mediant. There are places where it

coincides with JI.

> Actually, the way I've been getting my PHI tuning is simply

> A) To use PHI as a generator for mean-tone-like generation (up to

>tone #15 or so, divided by 2^n). I am pretty sure this has been

done before.

> B) To find a selection from the above tones within the range 0 and

>PHI (I came up with 9 notes the sounded both decent together and

>easily distinguishable)

> C) Duplicating the scale to be from o to PHI^2 (which gives 18

>notes) in order to allow more flexible variation in intervals and

>lessen beating.

> D) To create a scale the minimizes beating as a subset of the notes

found in B, using solely my ear as guide (I came up with 10 notes per

PHI^2 "double octave"...which is the equivalent of about 8 notes per

"normal" 2/1 octave).

Have you read the papers at Anaphoria? And this one:

http://users. bigpond.net. au/d.keenan/ music/noblemedia nt.txt

> ************ ********* ********* ********* ********* ********* ***

> ---Thanks for pointing this out, wonder if anyone else using Phi

will

> ---bother read it? :-)

> Indeed I did. But my larger question is, of course, is how

does it all sound (both does it sound good and does it sound

different, as opposed to sounding "a lot like JI" or "a lot like

existing meantones")?

> I, for one, am more concerning about compositional flexibility

regarding PHI scales than how mathematically ingenious some other

methods to create them may seem upon reading.

> ************ ********* ********* ********* ********* ********* *******

> I can (and have) easily back(ed) my PHI scale with a sound

examples on

> http://www.geocitie s.com/djtrancend ance/

> showing a run of my entire scale with sustain (I am on about

>"version 3" of my scale).

> ------------ -------

> It would be quite interesting to compare my sound example with

>practical sound examples of these other PHI tunings. I'm pretty

>sure my ears will tell me which methods have an obvious advantage on

>those grounds.

Yes, your ears will tell *you*. What my ears tell me might be

something different.

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> I haven't read those specific papers before, but I have looked >over Sethares' dissonance curves, which indicate the points of >maximum harmonic entropy at the peaks. In fact, I've known about >Sethares' curves for many many years...they are what got me truly >excited about and interested in microtonal in the first place. ;-)
> --------------------------------------
> -from the article "Pepper's Phi-related function could also be >applied to the problem of finding the region of maximum complexity >between two simpler ratios, providing a shortcut to the longer >process of successive approximation by iterating mediants. In what >follows, we show how this function, here termed the "Noble Mediant", >can be used to locate regions of maximum complexity. "
>
> ------------------------------------------
> From the paper, it sounds like you are going for maximum >dissonance by finding consecutive areas/plateaus (of highest >entropy) between the valleys/areas-of-lowest-entropy in >Sethares' >dissonance curves.

Hm, I don't know Sethare's dissonance curves beyond what's on his website (his rhythm book is in the library here but I don't know if they've yet ordered "Tuning.." per my request).

But anyway I'm concerned with "dissonance" in terms of the harmonic series. I do get some coincidences with "harmonic entropy" as known on these lists. However, a "universal" dissonance curve can't exist, because spectra vary. Obviously there is going to be less clash with a particular partial if that partial simply isn't there, and a lot more dissonance if there are relatively high-amplitude partials clashing in some particularly sensitive range.

If harmonic entropy charts were published adressing the spectra of some classic "general purpose" waveforms, say saw, triangle, and 1/2 and 1/3 duty cycle pulses, it would be more attractive to more people I suspect. This would be more fun for the guys anyway, I'd think, because for one thing you'd have to figure out a way of dropping specific primes from the reckoning.

> And, oddly enough, it appears some of the results you get match >JI.

Since my whole approach to tuning is based on the ear and singing, there's probably no way I'm ever really going to know if what I'm hearing is, for example, a 17/13 or the golden section of the frequency between 5/4 and 4/3, or a point of "maximum HE", as they're all within a couple of cents. It does matter as far as the color of the tuning as a whole, though, but I just apply Occam's razor: if I monkey all morning with my tunable oscillator and singing, and get a bunch of stuff within a couple of cents of 16/13, 17/13, and 18/13, it's pretty obvious what's going on and it's sensible to build a tuning from there. But if 16/13 and 13/8 sound wrong with an alleged 17/13, but 833 cents sounds great, then I go from there.

>
> But, one thing is for sure...if you are using the formula
> Noble_Mediant(i/j, m/n)=
> (i + Phi * m) / (j + Phi * n)
>
> To generate your scale...you are going to get something
> dramatically different than what I get from my
> formula for my tuning
> IE
> result = (PHI^Y) / (2^X)
> where result > 0 and result < PHI^2 and y is
> between 0 and 15.
> In fact, my version does not even do anything
> relative to fractions that represent the peaks
> and plateaus of the "harmonic entropy" curve.
> -------------------------------------------

Well, the first interval that pops up after phi in your phi-generator/octave-period tuning is in a region of "maximum harmonic entropy"
so who knows? And regardless of HE says, I think 833 cents is definitely in a region of maximum "otherness".

> This is one thing I'm finding very confusing about
> your other PHI tunings...many of them are based on
> formulas that have more to do with deriving from
> things like whole numbered fractions multiplied by
> PHI in some form...than taking a simple incrementing variable
> times PHI itself.
>
> So, on the surface, it seems my tuning (at least mathematically)
> has more to do with PHI while yours has to do more with
> "using PHI do find areas of harmonic entropy relative to something
> else".
> ---------------------------
> This difference in use of PHI as a generator potentially means
> there are several PHI tunings that look and
> sound nothing like each other.
>
> If these papers (all the ones I've read on PHI tunings)
> are right...there's is no such thing as a standard
> PHI tuning. So again, whether each one sounds better
> or worse is likely left up to the listener.
> ----------------
>
> So, again, to really get a feel for your scale I need an
> example. I think we can both agree it's not that I
> don't understand a "PHI standard of tuning" (which there
> apparently really is none) but, rather, that we are taking
> on two completely different methods altogether.
>
>
> Knowing how your scales work mathematically
> vs. Sethares' theories doesn't help, beyond saying that
> what you are doing is heavily NOT diatonic (going in
> many of the opposite areas of the curve that diatonic
> scales do AND just happens to intersect in a few places
> with JI).
>
> -Michael

I don't have a scale, continually using different tunings for mood and color. And I'd say that my music sounds diatonic as all hell, if you include the "neutral" intervals as diatonic; only if you went in measured everything would you find what a "xenharmonic" monstrosity it all really is. I'll float a specific appropriate tune your way, which I think illustrates the effeciveness of combining intervals of great "otherness" with JI (13 and 7 in this case), when I finish it, hopefully in the next couple of days.

take care and pass me a slice of your music links, as I'm having a terrible time digging through the current state of the list and your profile seems to be off limits?

-Cameron Bobro

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> ---Hmm, I don't know Sethare's dissonance curves beyond what's on >his
> website (his rhythm ---book is in the library here but I don't know >if
> they've yet ordered "Tuning.." per my request).
> Neither do I...beyond his website and his website-published >papers. Which aren't all inclusive (his consonance formula is >dyadic and does not include things like triads vs. dyads or things >like the odd fact a major and minor triad sound quite different in >tension despite having the same consonances as derived from his >formula).

Well 5/4 and 6/5 invert within a 3/2, so I agree that the general dissonance/lack of dissonance would be about the same for a major or minor triad, and of course I will probably never be able to judge that "impartially" as I've always heard them as balanced as far as consonance (and in 12-tET the minor triad more consonant, or at least less dissonant).

A 7/6 and a 9/7 within a 3/2, that's a trickier thing, I wonder how Sethares would rate that?

> Which is why I consider his formula a source of good ideas...but >by no means complete or the only intelligent way to calculate >consonance (IE many equally good theories may give drastically >different answers).
> ********************************************************
> ---But anyway I'm concerned with "dissonance" in terms of the >harmonic series.
> ---If harmonic entropy charts were published adressing the spectra >of some
> classic "general ---purpose" waveforms, say saw, triangle, and 1/2 >and 1/3
> duty cycle pulses, it would be ---more attractive to more people I >suspect.
> Right, and most of Sethares' example charts appear to >specifically be attacking the spectrum of the harmonic series (in >fact the valleys/bottom-areas on the chart represent the JI diatonic >scale).
> Although he also gives a few examples of how to make a curve for the spectra of instruments such as bells (which generate drastically different curves vs. "normal" instruments).

Well his book goes much further into that, looking forward to reading the whole thing. At any rate, I find his general approach of addressing various spectra to be more sensible than the sine-based HE theory here at the lists. It must be said, though, that the "abstract" proportions are, in my experience, far stronger than they "should" be- for example we can experience that JI "thing" even with pretty dang "inharmonic" spectra, gapped partials, etc.

> ********************************************************
> ----Since my whole approach to tuning is based on the ear and >singing,
> there's probably no ----way I'm ever really going to know if what >I'm
> hearing is, for example, a 17/13 or the golden ----section of the >frequency
> between 5/4 and 4/3, or a point of "maximum HE", as they're all ---->within
> a couple of cents.
> Right and, as I see it, that wouldn't matter in and of itself. >It seems obvious to me that the relation of notes to each other on >the average is a lot more important than which notes sound most >consonant vs. the root.

The whole idea of consonance "vs." dissonance is way, way too black and white for me, which is why I go on about things like "character".

> That's why the whole PHI-tonality seems so appealing to me...in >such a tuning the notes all have the exact same relationship to each >other and thus multiple notes can be "roots" simultaneously...and >thus I suspect the "average feel" of the relationships between the >notes could over-ride the need to be "consonant about a single root".
>

This is what I've called "floating".

******************************************************************************************
> --Obviously there is going to be less clash with a particular partial if
> that partial simply isn't --there, and a lot more dissonance if there are
> relatively high-amplitude partials clashing in --some particularly
> sensitive range.
> Right but, at least to me, the ultimate consonance test for a new scale is to perform the scale with sine waves and then compare that to a major scale in JI-diatonic (which, for example, sounds better than a major scale under 12TET in sine waves as well as full-harmonic-series-endowed instruments)..
> **********************************************************************

..see above...
******************
>
> --Well, the first interval that pops up after phi in your phi->generator/
> octave-period tuning is in ---a region of "maximum harmonic entropy"
> ---so who knows? And regardless of HE says, I think 833 cents is definitely in a region of ---maximum "otherness".
> Ah, the horror of it all... :-D
> When you say otherness you mean otherness relative to what? A >root tone? The valleys/minimum-entropy-points in some Setharesian >dissonance curves?

Both- floating in relation to the root, and in relation to clearly root-related proportions.
>
> My bet is that another ability of the brain uses the symmetry >of PHI in such a way the brain stops trying to relate everything to >a single root tone.
> Virtually everything I've read about harmonic entropy seems to >have the staple suspicion that all scales should be based on this >"how symmetric is the scale about a single root" idea. Not that >it's a bad idea...but it no longer comes across to me as new ground >but rather a way to experiment with how to "improve certain >intervals and chords at the expense of others" by messing with where >the most consonant roots lie in a scale.
> **********************************************************************

As I said, "consonant/dissonant" isn't that important, it's so context-based anyway. And as I've mentioned many times over the last couple of years, I believe you can find points that are so darn "dissonant" as far as interactions in the spectra that they achieve a kind of fuzzy softness. Pink noise, as an illustration, is far from the most harsh sounding thing, it's very pleasant in fact.

*********************
> --take care and pass me a slice of your music links, as I'm having a
> terrible time digging ---through the current state of the list and your
> profile seems to be off limits?
>    Hehe...my profile is off-limits? I never ever set such a thing....
>     Anyhow, I've probably passed this (my PHI-tuning sound example >link) about 15 times...but far too many people seem interested in >arguing how weird my alternative suspicions on what can define >consonance are to highlight the links. :-D
>    But, regardless of that, here is the link:
>       http://www.geocities.com/djtrancendance/PHINOT.html
>
>    And, yes, the entire scale plays when you load that page...and >an relatively short explanation of the scale follows on that page.

Yes, the Yahoo system seems cantankerous and touchy anyway, I've had many "what the hell just happened there?!" experiences. Thanks for the link, I'll check it out! The scale sounds good.

-Cameron Bobro

🔗djtrancendance@...

🔗Cameron Bobro <misterbobro@...>

🔗djtrancendance@...

I'll respond to the rest at length later but, as a quick note, your example at
http://dl.kibla.org/dl.php?filename=LassoSciFi2.wav came across to me as sounding very natural minus the tones that come in at 27 and 36. I missed to discussion (admittedly passing it off as "yet another JI discussion")...but I'd be interested to hear what mathematical method you used to generate the tuning and what the resulting tuning is so I can look for any patterns that hint at any psychoacoustic phenomena that go "beyond JI". ;-)

And, in short, I agree the problem
---because they don't know what it is in terms of numbers and are afraid
their judgement may
--- radically contradict their theoretical positions.
:-)
really has become a bit of a "writer's block" in tuning where people dump new ideas because it would "render invalid" certain ideas they've spent huge amounts of time working on.

In fact my PHI tuning came up as a fluke; it was meant just to serve as a simple example of how symmetrical one of my JI scales was, but I ended up dumping the entire x/16 JI harmonic series scale project because, math aside, the PHI scale sounded much better to me. One month's work down the drain to switch to 2 months research on the new tuning, yes...but was it worth the 'humiliation' and extra 'redo' work??....of course. :-)
Sometimes, I've found, you have to travel to places you don't know how to explain with math yet to find the most promising results...and ONLY afterwards go back and try your best to summarize it in equations. :-)

--- On Mon, 3/9/09, Cameron Bobro <misterbobro@...>
wrote:

From: Cameron Bobro <misterbobro@...>
Subject: [tuning] Re: Phi Tonality
To: tuning@yahoogroups.com
Date: Monday, March 9, 2009, 4:58 AM

--- In tuning@yahoogroups. com, djtrancendance@ ... wrote:

> --Well 5/4 and 6/5 invert within a 3/2, so I agree that the general

> dissonance/lack of --dissonance would be about the same for a major >or

> minor triad

> But, you see, they are mathematically, but not when you listen >to them.

> Try playing each chord...even with pure sine waves (where >overlapping and spacing of overtones is not even a factor)...the >minor is more tense, even though the "mathematical dissonance" is >the same and the chord comparison is, indeed,.just an inversion. >When you think about it, though, the brain's tolerance for closer >notes decreases at lower frequencies (the critical band gets larger >at lower frequencies) ...thus making the need for a 5/4 and THEN 6/5 >order rather than a 6/5 then 5/4 order apparent.

But I AM basing my judgement on listening. To me, the amount of consonance/dissonan ce between major and minor JI thirds, and triads, is about the same (and minor is more consonant in 12-tET). The characters are simply different.

>as the harmonic series >does) may lead you to think they would not >if you don't consider the >"whole number multiple beating" exception.

> ---Thanks for the link, I'll check it out! The scale sounds good.

> Thank you and it's rather ironic vs. what most theory says it >should sound like, isn't it? :-D

Well I wouldn't know about "most music theory", and my own "music theory", which works with the Gesamtklangfarbe of a piece of music, therefore tuning with an ear on the overall spectrum of the entire piece of music... well, Carl Lumma called it "eye of newt". :-)

> I'm not trying to preach the gospel here and say more everything >tuning theory has generated based on emulating intervals from the >the harmonic series is wrong. It's right. But, apparently, it is >far from the only right...

Have you checked out Erv Wilson's many tunings?

> "Older

> theory" based on replicating intervals from the harmonic series >does a great job at solving the case for consonance via one method >the mind understands (constant rate of beating) and ignores a widely >unrecognized second method (where beating of tones occur in whole >number multiples of each other).

> The conclusion I hope people reach is "Hey, while JI is great >some more people should really try exploring how to take advantage >of the whole number beating exception more often and really develop >it".

> For sure, some things much cooler than what I have done can be >found if, say, some JI tuning wizards try their hands at developing >that theory further. No doubt someone could eventually make the >equivalent of a tonality diamond out of PHI-based tunings and >similar advancements. ..

I wouldn't expect too much to happen here on the tuning list, where the "regular temperament paradigm" is the main thing.

One thing you'll notice is a reluctance to comment on mathmatically unknown music. I suspect that noone is going to comment on the "sci-fi" tuning of the Lasso I put up:

http://dl.kibla. org/dl.php? filename= LassoSciFi2. wav

because they don't know what it is in terms of numbers and are afraid their judgement may radically contradict their theoretical positions. :-)

> ************ ********* ********* ********* ********* ********* ********* ********* ********* ****

> ---Well his book goes much further into that, looking forward to >reading the whole thing.

> If you find any exceptions to the general train of thoughts I >found from his web page in the book, please let me know. :-) But, >in general, I get the odd impression the whole idea of "some types >of beats (same-rate or whole number multiple ones) CAN be >consonant" >and that idea of periodicity is not considered in his >theory: I could not find mention of it even once.

Well, we'll see!

> Mostly listening to Sethares' compositions ALA "Ten Strings" and >"Blue Dabo Girl", it's very easy to tell he's staying well "in key" >using his methods and getting lack-of-dissonance results for >his scales similar to those one would get playing those scales with >pure sine waves (IE the overtones overlap very well in general). >But, at the same time, even

> using the scales he makes with raw sine wave (IE using his scale >made the bell with a sine wave), there is a sense of tension a good >deal higher than something like JI. And as to why that tension is >there...seems to go beyond the scope of just about any dyadic theory >far as I know, because it is triads that decide how the periodicity >of the scale (the one thing Sethares seems not to cover) works...

The work with inharmonic spectra is pretty radical and I don't think is meant to be directly compared with harmonic-spectra music. It's a new and different thing, an approach that friends and colleagues of mine as well who are into synthesis and acoustics and physics and all that good stuff find very interesting. One way you can tell it is a new thing is that Charles Lucy called it the "same old", hahaha!

Well anyway the thing to do is keep making music with whatever tuning turns you on, and time will sort out what people find of value.

-Cameron Bobro

🔗Herman Miller <hmiller@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@> wrote:
> >
> > ---Hmm, I don't know Sethare's dissonance curves beyond what's on >his
> > website (his rhythm ---book is in the library here but I don't know >if
> > they've yet ordered "Tuning.." per my request).
> > Neither do I...beyond his website and his website-published >papers. Which aren't all inclusive (his consonance formula is >dyadic and does not include things like triads vs. dyads or things >like the odd fact a major and minor triad sound quite different in >tension despite having the same consonances as derived from his >formula).
>
> Well 5/4 and 6/5 invert within a 3/2, so I agree that the general dissonance/lack of dissonance would be about the same for a major or minor triad, and of course I will probably never be able to judge that "impartially" as I've always heard them as balanced as far as consonance (and in 12-tET the minor triad more consonant, or at least less dissonant).
>
> A 7/6 and a 9/7 within a 3/2, that's a trickier thing, I wonder how Sethares would rate that?
>
>
> > Which is why I consider his formula a source of good ideas...but >by no means complete or the only intelligent way to calculate >consonance (IE many equally good theories may give drastically >different answers).
> > ********************************************************
> > ---But anyway I'm concerned with "dissonance" in terms of the >harmonic series.
> > ---If harmonic entropy charts were published adressing the spectra >of some
> > classic "general ---purpose" waveforms, say saw, triangle, and 1/2 >and 1/3
> > duty cycle pulses, it would be ---more attractive to more people I >suspect.
> > Right, and most of Sethares' example charts appear to >specifically be attacking the spectrum of the harmonic series (in >fact the valleys/bottom-areas on the chart represent the JI diatonic >scale).
> > Although he also gives a few examples of how to make a curve for the spectra of instruments such as bells (which generate drastically different curves vs. "normal" instruments).
>
> Well his book goes much further into that, looking forward to reading the whole thing. At any rate, I find his general approach of addressing various spectra to be more sensible than the sine-based HE theory here at the lists. It must be said, though, that the "abstract" proportions are, in my experience, far stronger than they "should" be- for example we can experience that JI "thing" even with pretty dang "inharmonic" spectra, gapped partials, etc.
>
> > ********************************************************
> > ----Since my whole approach to tuning is based on the ear and >singing,
> > there's probably no ----way I'm ever really going to know if what >I'm
> > hearing is, for example, a 17/13 or the golden ----section of the >frequency
> > between 5/4 and 4/3, or a point of "maximum HE", as they're all ---->within
> > a couple of cents.
> > Right and, as I see it, that wouldn't matter in and of itself. >It seems obvious to me that the relation of notes to each other on >the average is a lot more important than which notes sound most >consonant vs. the root.
>
> The whole idea of consonance "vs." dissonance is way, way too black and white for me, which is why I go on about things like "character".
>
> > That's why the whole PHI-tonality seems so appealing to me...in >such a tuning the notes all have the exact same relationship to each >other and thus multiple notes can be "roots" simultaneously...and >thus I suspect the "average feel" of the relationships between the >notes could over-ride the need to be "consonant about a single root".
> >
>
> This is what I've called "floating".
>
> ******************************************************************************************
> > --Obviously there is going to be less clash with a particular partial if
> > that partial simply isn't --there, and a lot more dissonance if there are
> > relatively high-amplitude partials clashing in --some particularly
> > sensitive range.
> > Right but, at least to me, the ultimate consonance test for a new scale is to perform the scale with sine waves and then compare that to a major scale in JI-diatonic (which, for example, sounds better than a major scale under 12TET in sine waves as well as full-harmonic-series-endowed instruments)..
> > **********************************************************************
>
> ..see above...
> ******************
> >
> > --Well, the first interval that pops up after phi in your phi->generator/
> > octave-period tuning is in ---a region of "maximum harmonic entropy"
> > ---so who knows? And regardless of HE says, I think 833 cents is definitely in a region of ---maximum "otherness".
> > Ah, the horror of it all... :-D
> > When you say otherness you mean otherness relative to what? A >root tone? The valleys/minimum-entropy-points in some Setharesian >dissonance curves?
>
> Both- floating in relation to the root, and in relation to clearly root-related proportions.
> >
> > My bet is that another ability of the brain uses the symmetry >of PHI in such a way the brain stops trying to relate everything to >a single root tone.
> > Virtually everything I've read about harmonic entropy seems to >have the staple suspicion that all scales should be based on this >"how symmetric is the scale about a single root" idea. Not that >it's a bad idea...but it no longer comes across to me as new ground >but rather a way to experiment with how to "improve certain >intervals and chords at the expense of others" by messing with where >the most consonant roots lie in a scale.
> > **********************************************************************
>
> As I said, "consonant/dissonant" isn't that important, it's so context-based anyway. And as I've mentioned many times over the last couple of years, I believe you can find points that are so darn "dissonant" as far as interactions in the spectra that they achieve a kind of fuzzy softness. Pink noise, as an illustration, is far from the most harsh sounding thing, it's very pleasant in fact.
>
> *********************
> > --take care and pass me a slice of your music links, as I'm having a
> > terrible time digging ---through the current state of the list and your
> > profile seems to be off limits?
> > Hehe...my profile is off-limits? I never ever set such a thing....
> > Anyhow, I've probably passed this (my PHI-tuning sound example >link) about 15 times...but far too many people seem interested in >arguing how weird my alternative suspicions on what can define >consonance are to highlight the links. :-D
> > But, regardless of that, here is the link:
> > http://www.geocities.com/djtrancendance/PHINOT.html
> >
> > And, yes, the entire scale plays when you load that page...and >an relatively short explanation of the scale follows on that page.
>
> Yes, the Yahoo system seems cantankerous and touchy anyway, I've had many "what the hell just happened there?!" experiences. Thanks for the link, I'll check it out! The scale sounds good.
>
> -Cameron Bobro
>
"A 7/6 and a 9/7 within a 3/2, that's a trickier thing." The problem with this triad, as opposed to 4:5:6, is that it's outer fifth 3/2 is from the 21st harmonic i.e. 7/6 X 9/7 = 63/42 = 3/2, such that 42 and 63 are the 2nd and 3rd harmonic of root 21, respectively. 4:5:6 on the other hand gives the root 4 as the tonic. The question is whether consonance/dissonance, taken out of harmonic context, are really adequate concepts to describe musical harmony. Since there are an infinite number of ratios we could consider, my fear is that without some guide from existing musical practice we could waste entire lifetimes chasing rabbits down the hole and being none the wiser for it.

-Rick

🔗Michael Sheiman <djtrancendance@...>

🔗djtrancendance@...

🔗Graham Breed <gbreed@...>

🔗rick_ballan <rick_ballan@...>

🔗djtrancendance@...

--Hi Michael,

---I remember reading a quote by Stravinsky somewhere that, contrary to
what we might ---expect, infinite possibilities can become so overwhelming
that all creativity eventually ---grinds to a halt (which was why he
adopted serialism).
Right. For example if we try to establish an infinite number of ways to make ratios between sets of frequencies beat at multiples of each other we could have thousands of different results. That's why such experiments would likely need to be reduced to functions, such as that from using PHI as a meantone-type generator, to limit experimentation to a more finite set.

---Consider this: if all periodic waves require
rational numbers then irrational numbers are ---aperiodic. (Recall for eg
Pythagoras' proof that the sq root of 2, or b5th interval, is not a
---rational number). This means that, theoretically speaking at least, no
two cycles will ever ---repeat and an irrational wave will never repeat
itself all the way to infinity.
Right, it may not repeat...but, on the other hand, it may come SO close to repeating, in some places, that the ear can not recognize the difference.

---Now add to this the fact that there are an uncountably infinite number
of these waves ---(aleph 1, see Georg Cantor), that this number is so
large compared with countable infinity ---(aleph 0, the amount of rational
numbers) that if we threw a dart on a number line the odds ---of hitting a
rational is so small as to be practically zero.
But, again, I don't care about hitting rational numbers but rather numbers of frequencies whose beating is proportionate. For example 100hz 122hz and 166hz (where the beating rates are all irrational numbers, but all multiples of each other IE 122-100hz = 22hz and 166-122hz=44hz which = 22hz * 2)!

--The rational numbers
and musical harmony become like a precious and rare gem in the --midst of
a universe of chaos.
I would argue rational numbers become "one way of achieving something the mind can recognize beautiful patterns in". And irrational numbers become another one which seems far more delicate and harder to get right but still possible (in the same way using a derivative in calculus to find a slope is vs. the equation of a slope in algebra).

Far as the belief that "only the rare gem of rational numbers can form great consonance" you think http://www.geocities.com/djtrancendance/PHIxcellence2.mp3 (a scale formed purely from irrational numbers) sounds "unorganized" or "far less beautiful than rational tones"...please let me know.

My point again is NOT that I have found a "perfect scale" but, rather, that irrational numbers can produce easily interpret-able music.

---This is what I had at the back of my mind when
writing the rabbit hole.
Which is...where exactly? Or is that somehow related to the movie series "into the Rabbit hole", about quantum physics, how aging works, and other exotic thoughts?
---But I was wrong ---saying we'd be none the wiser
for it, for exploring things like phi etc...
---I have regained a new
---
appreciation of rational harmony.
Isn't that basically saying it was only worth exploring irrational harmony to explain how many ways there are NOT to get rational harmony thus helping prove a pet-hypothesis that rational harmony is the only answer? If so...so much for open-mindedness... :-(

-Michael

--- On Tue, 3/10/09, rick_ballan <rick_ballan@...> wrote:

From: rick_ballan <rick_ballan@...>
Subject: [tuning] Re: Phi Tonality
To: tuning@yahoogroups.com
Date: Tuesday, March 10, 2009, 7:50 PM

--- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...> wrote:

> Rick,

> --That's why the whole PHI-tonality seems so appealing to me...in such --a tuning the notes all have the exact same relationship to each --other and thus multiple notes can be "roots" simultaneously. ..and --thus I suspect the "average feel" of the relationships between the --notes could over-ride the need to be "consonant about a single --root".

> > This is what I've called "floating". (RICK)

> In that case, my point is that the ability to "float" in a tuning is worth investigation. Furthermore the adaptive JI case of "make everything consonant by matching it to a root" appears to fall far short of covering all the consonant scales that could be made if people took advantage of "floating" when making scales instead of always doing JI-style "matching to the exact root note".

> ************ ********* ********* ********* ********* ********* ********* ********* ********* ********* **

> > When you say otherness you mean otherness relative to what? A

> >root tone? The valleys/minimum- entropy-points in some Setharesian >dissonance curves?

> --Both- floating in relation to the root, and in relation to clearly

> --root-related proportions. (RICK)

> No problem...my best guess is, in the case of my PHI scale, such a "root related proportion" is the same a+b/a = a/b relationship that PHI has.

> ************ ********* ********* ********* ********* ********* *

> ---And as I've mentioned many times over the last couple of years, I

> ---believe you can find points that are so darn "dissonant" as far as

> ---interactions in the spectra that they achieve a kind of fuzzy

> ---softness. (RICK)

> Maybe, but is it really such a tragedy to exploit that? :-D In fact, I've found scale schemes that make instrument overtones in chords nearly overlap but not quite sometimes sound better than having the overtones match exactly.

> Actually Sethares' theory seems to state your "suspicion" as well...that two notes that are too close meld into a relaxing "fuzzy softness" chorus-type effect...probably quite similar to what happens with pink noise.

> -Michael

Hi Michael,

I remember reading a quote by Stravinsky somewhere that, contrary to what we might expect, infinite possibilities can become so overwhelming that all creativity eventually grinds to a halt (which was why he adopted serialism). Consider this: if all periodic waves require rational numbers then irrational numbers are aperiodic. (Recall for eg Pythagoras' proof that the sq root of 2, or b5th interval, is not a rational number). This means that, theoretically speaking at least, no two cycles will ever repeat and an irrational wave will never repeat itself all the way to infinity.

Now add to this the fact that there are an uncountably infinite number of these waves (aleph 1, see Georg Cantor), that this number is so large compared with countable infinity (aleph 0, the amount of rational numbers) that if we threw a dart on a number line the odds of hitting a rational is so small as to be practically zero. The rational numbers and musical harmony become like a precious and rare gem in the midst of a universe of chaos. This is what I had at the back of my mind when writing the rabbit hole. But I was wrong saying we'd be none the wiser for it, for exploring things like phi etc...I have regained a new appreciation of rational harmony.

-Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> --Hi Michael,
>
> ---I remember reading a quote by Stravinsky somewhere that, contrary to
> what we might ---expect, infinite possibilities can become so overwhelming
> that all creativity eventually ---grinds to a halt (which was why he
> adopted serialism).
> Right. For example if we try to establish an infinite number of ways to make ratios between sets of frequencies beat at multiples of each other we could have thousands of different results. That's why such experiments would likely need to be reduced to functions, such as that from using PHI as a meantone-type generator, to limit experimentation to a more finite set.
>
> ---Consider this: if all periodic waves require
> rational numbers then irrational numbers are ---aperiodic. (Recall for eg
> Pythagoras' proof that the sq root of 2, or b5th interval, is not a
> ---rational number). This means that, theoretically speaking at least, no
> two cycles will ever ---repeat and an irrational wave will never repeat
> itself all the way to infinity.
> Right, it may not repeat...but, on the other hand, it may come SO close to repeating, in some places, that the ear can not recognize the difference.
>
>
> ---Now add to this the fact that there are an uncountably infinite number
> of these waves ---(aleph 1, see Georg Cantor), that this number is so
> large compared with countable infinity ---(aleph 0, the amount of rational
> numbers) that if we threw a dart on a number line the odds ---of hitting a
> rational is so small as to be practically zero.
> But, again, I don't care about hitting rational numbers but rather numbers of frequencies whose beating is proportionate. For example 100hz 122hz and 166hz (where the beating rates are all irrational numbers, but all multiples of each other IE 122-100hz = 22hz and 166-122hz=44hz which = 22hz * 2)!
>
> --The rational numbers
> and musical harmony become like a precious and rare gem in the --midst of
> a universe of chaos.
> I would argue rational numbers become "one way of achieving something the mind can recognize beautiful patterns in". And irrational numbers become another one which seems far more delicate and harder to get right but still possible (in the same way using a derivative in calculus to find a slope is vs. the equation of a slope in algebra).
>
> Far as the belief that "only the rare gem of rational numbers can form great consonance" you think http://www.geocities.com/djtrancendance/PHIxcellence2.mp3 (a scale formed purely from irrational numbers) sounds "unorganized" or "far less beautiful than rational tones"...please let me know.
>
> My point again is NOT that I have found a "perfect scale" but, rather, that irrational numbers can produce easily interpret-able music.
>
> ---This is what I had at the back of my mind when
> writing the rabbit hole.
> Which is...where exactly? Or is that somehow related to the movie series "into the Rabbit hole", about quantum physics, how aging works, and other exotic thoughts?
> ---But I was wrong ---saying we'd be none the wiser
> for it, for exploring things like phi etc...
> ---I have regained a new
> ---
> appreciation of rational harmony.
> Isn't that basically saying it was only worth exploring irrational harmony to explain how many ways there are NOT to get rational harmony thus helping prove a pet-hypothesis that rational harmony is the only answer? If so...so much for open-mindedness... :-(
>
> -Michael
>
>
>
>
> --- On Tue, 3/10/09, rick_ballan <rick_ballan@...> wrote:
>
> From: rick_ballan <rick_ballan@...>
> Subject: [tuning] Re: Phi Tonality
> To: tuning@yahoogroups.com
> Date: Tuesday, March 10, 2009, 7:50 PM
>
>
>
>
>
>
>
>
>
>
>
>
> --- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...> wrote:
>
> >
>
> > Rick,
>
> >
>
> > --That's why the whole PHI-tonality seems so appealing to me...in such --a tuning the notes all have the exact same relationship to each --other and thus multiple notes can be "roots" simultaneously. ..and --thus I suspect the "average feel" of the relationships between the --notes could over-ride the need to be "consonant about a single --root".
>
> > > This is what I've called "floating". (RICK)
>
> >
>
> > In that case, my point is that the ability to "float" in a tuning is worth investigation. Furthermore the adaptive JI case of "make everything consonant by matching it to a root" appears to fall far short of covering all the consonant scales that could be made if people took advantage of "floating" when making scales instead of always doing JI-style "matching to the exact root note".
>
> > ************ ********* ********* ********* ********* ********* ********* ********* ********* ********* **
>
> >
>
> > > When you say otherness you mean otherness relative to what? A
>
> > >root tone? The valleys/minimum- entropy-points in some Setharesian >dissonance curves?
>
> >
>
> > --Both- floating in relation to the root, and in relation to clearly
>
> > --root-related proportions. (RICK)
>
> > No problem...my best guess is, in the case of my PHI scale, such a "root related proportion" is the same a+b/a = a/b relationship that PHI has.
>
> > ************ ********* ********* ********* ********* ********* *
>
> >
>
> > ---And as I've mentioned many times over the last couple of years, I
>
> > ---believe you can find points that are so darn "dissonant" as far as
>
> > ---interactions in the spectra that they achieve a kind of fuzzy
>
> > ---softness. (RICK)
>
> > Maybe, but is it really such a tragedy to exploit that? :-D In fact, I've found scale schemes that make instrument overtones in chords nearly overlap but not quite sometimes sound better than having the overtones match exactly.
>
> > Actually Sethares' theory seems to state your "suspicion" as well...that two notes that are too close meld into a relaxing "fuzzy softness" chorus-type effect...probably quite similar to what happens with pink noise.
>
> >
>
> > -Michael
>
> >
>
> Hi Michael,
>
>
>
> I remember reading a quote by Stravinsky somewhere that, contrary to what we might expect, infinite possibilities can become so overwhelming that all creativity eventually grinds to a halt (which was why he adopted serialism). Consider this: if all periodic waves require rational numbers then irrational numbers are aperiodic. (Recall for eg Pythagoras' proof that the sq root of 2, or b5th interval, is not a rational number). This means that, theoretically speaking at least, no two cycles will ever repeat and an irrational wave will never repeat itself all the way to infinity.
>
>
>
> Now add to this the fact that there are an uncountably infinite number of these waves (aleph 1, see Georg Cantor), that this number is so large compared with countable infinity (aleph 0, the amount of rational numbers) that if we threw a dart on a number line the odds of hitting a rational is so small as to be practically zero. The rational numbers and musical harmony become like a precious and rare gem in the midst of a universe of chaos. This is what I had at the back of my mind when writing the rabbit hole. But I was wrong saying we'd be none the wiser for it, for exploring things like phi etc...I have regained a new appreciation of rational harmony.
>
>
>
> -Rick
>
Michael,

Firstly, the 12 tone system is already irrational. The difference is that it evolved using the rational numbers as a guide, trying to distribute them evenly over each key. True, they might be almost periodic, or they might be harmonics in the higher octaves e.g. 645/512 = 1.2597... is a very good approximation to the tempered maj 3, 2 power of 1/3 = 1.25992.... So I am not against irrationals per se, which would be absurd, but am merely bringing certain 'awkward' points into the open.

Secondly, the example you gave of beating are not irrational numbers but whole numbers i.e. 122-100Hz = 22Hz etc so I haven't got a clue what you mean here. But as I have also pointed out many times in this group, the very concept of frequency itself relies on the GCD between rational numbers, which is also a model of musical harmony, and beat 'frequencies' are a secondary concept. To use your example, 122/100 = 61/50, therefore GCD = 2, 100 and 122 are the 50th and 61st harmonics of tonic 2, and the beat freq (btw, half the difference) is 1/2(122-100) = 11 and therefore is the 11th harmonic.

-Rick

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Michael Sheiman <djtrancendance@...>

--the very concept of frequency itself relies on the GCD between rational
numbers,
Huh? Sounds more like the concept of the harmonic series to me...

---and beat
'frequencies' are a secondary concept.
Beating is IF you are modeling you scales after the harmonic series.

*****************************************************************************************
--Firstly, the 12 tone system is already irrational. The difference is
that it evolved using the --rational numbers as a guide, trying to
distribute them evenly over each key.
Trust me, even distribution doesn't solve anything alone. Try playing the 10TET tuning. It sounds terribly dissonant, yet is very evenly spaced. You can partly help against that problem by shifting the timbre/instrument-overtones using irrational intervals to better match the tuning (ALA Sethares' tuning-to-timbre method)...but then we go back to using irrational intervals. :-D

---!!!!!!True, they --might be almost
periodic!!!!!!!!!!
    In fact, in the case of 12TET, so close they are almost indistinguishable from each rational intervals to the human ear. Which is why you seem to still be basically arguing again "irrational intervals don't work".

BTW,
---http://www.geocities.com/djtrancendance/PHIxcellence2.mp3
    I've given you this link to you several times...a link to an example of an irrational scale (that gets nowhere near rational intervals). You seem to go on and on about why irrational intervals can't work, yet you can't seem to site, say, anything in my example that sounds dissonant. Is it too much to ask you to listen to the file and give it a chance (or, on the other hand, perhaps let me know what about it doesn't work coming from your ears and not your pre-determined theoretical biases)?

--- On Thu, 3/12/09, rick_ballan <rick_ballan@...m.au> wrote:
From: rick_ballan <rick_ballan@...>
Subject: [tuning] Re: Phi Tonality
To: tuning@yahoogroups.com
Date: Thursday, March 12, 2009, 5:00 PM

--- In tuning@yahoogroups. com, djtrancendance@ ... wrote:

> --Hi Michael,

> ---I remember reading a quote by Stravinsky somewhere that, contrary to

> what we might ---expect, infinite possibilities can become so overwhelming

> that all creativity eventually ---grinds to a halt (which was why he

> adopted serialism).

> Right. For example if we try to establish an infinite number of ways to make ratios between sets of frequencies beat at multiples of each other we could have thousands of different results. That's why such experiments would likely need to be reduced to functions, such as that from using PHI as a meantone-type generator, to limit experimentation to a more finite set.

> ---Consider this: if all periodic waves require

> rational numbers then irrational numbers are ---aperiodic. (Recall for eg

> Pythagoras' proof that the sq root of 2, or b5th interval, is not a

> ---rational number). This means that, theoretically speaking at least, no

> two cycles will ever ---repeat and an irrational wave will never repeat

> itself all the way to infinity.

> Right, it may not repeat...but, on the other hand, it may come SO close to repeating, in some places, that the ear can not recognize the difference.

> ---Now add to this the fact that there are an uncountably infinite number

> of these waves ---(aleph 1, see Georg Cantor), that this number is so

> large compared with countable infinity ---(aleph 0, the amount of rational

> numbers) that if we threw a dart on a number line the odds ---of hitting a

> rational is so small as to be practically zero.

> But, again, I don't care about hitting rational numbers but rather numbers of frequencies whose beating is proportionate. For example 100hz 122hz and 166hz (where the beating rates are all irrational numbers, but all multiples of each other IE 122-100hz = 22hz and 166-122hz=44hz which = 22hz * 2)!

> --The rational numbers

> and musical harmony become like a precious and rare gem in the --midst of

> a universe of chaos.

> I would argue rational numbers become "one way of achieving something the mind can recognize beautiful patterns in". And irrational numbers become another one which seems far more delicate and harder to get right but still possible (in the same way using a derivative in calculus to find a slope is vs. the equation of a slope in algebra).

> Far as the belief that "only the rare gem of rational numbers can form great consonance" you think http://www.geocitie s.com/djtrancend ance/PHIxcellenc e2.mp3 (a scale formed purely from irrational numbers) sounds "unorganized" or "far less beautiful than rational tones"...please let me know.

> My point again is NOT that I have found a "perfect scale" but, rather, that irrational numbers can produce easily interpret-able music.

> ---This is what I had at the back of my mind when

> writing the rabbit hole.

> Which is...where exactly? Or is that somehow related to the movie series "into the Rabbit hole", about quantum physics, how aging works, and other exotic thoughts?

> ---But I was wrong ---saying we'd be none the wiser

> for it, for exploring things like phi etc...

> ---I have regained a new

> ---

> appreciation of rational harmony.

> Isn't that basically saying it was only worth exploring irrational harmony to explain how many ways there are NOT to get rational harmony thus helping prove a pet-hypothesis that rational harmony is the only answer? If so...so much for open-mindedness. .. :-(

> -Michael

> --- On Tue, 3/10/09, rick_ballan <rick_ballan@ ...> wrote:

> From: rick_ballan <rick_ballan@ ...>

> Subject: [tuning] Re: Phi Tonality

> To: tuning@yahoogroups. com

> Date: Tuesday, March 10, 2009, 7:50 PM

> --- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...> wrote:

> >

> > Rick,

> >

> > --That's why the whole PHI-tonality seems so appealing to me...in such --a tuning the notes all have the exact same relationship to each --other and thus multiple notes can be "roots" simultaneously. ..and --thus I suspect the "average feel" of the relationships between the --notes could over-ride the need to be "consonant about a single --root".

> > > This is what I've called "floating". (RICK)

> >

> > In that case, my point is that the ability to "float" in a tuning is worth investigation. Furthermore the adaptive JI case of "make everything consonant by matching it to a root" appears to fall far short of covering all the consonant scales that could be made if people took advantage of "floating" when making scales instead of always doing JI-style "matching to the exact root note".

> > ************ ********* ********* ********* ********* ********* ********* ********* ********* ********* **

> >

> > > When you say otherness you mean otherness relative to what? A

> > >root tone? The valleys/minimum- entropy-points in some Setharesian >dissonance curves?

> >

> > --Both- floating in relation to the root, and in relation to clearly

> > --root-related proportions. (RICK)

> > No problem...my best guess is, in the case of my PHI scale, such a "root related proportion" is the same a+b/a = a/b relationship that PHI has.

> > ************ ********* ********* ********* ********* ********* *

> >

> > ---And as I've mentioned many times over the last couple of years, I

> > ---believe you can find points that are so darn "dissonant" as far as

> > ---interactions in the spectra that they achieve a kind of fuzzy

> > ---softness. (RICK)

> > Maybe, but is it really such a tragedy to exploit that? :-D In fact, I've found scale schemes that make instrument overtones in chords nearly overlap but not quite sometimes sound better than having the overtones match exactly.

> > Actually Sethares' theory seems to state your "suspicion" as well...that two notes that are too close meld into a relaxing "fuzzy softness" chorus-type effect...probably quite similar to what happens with pink noise.

> >

> > -Michael

> >

> Hi Michael,

> I remember reading a quote by Stravinsky somewhere that, contrary to what we might expect, infinite possibilities can become so overwhelming that all creativity eventually grinds to a halt (which was why he adopted serialism). Consider this: if all periodic waves require rational numbers then irrational numbers are aperiodic. (Recall for eg Pythagoras' proof that the sq root of 2, or b5th interval, is not a rational number). This means that, theoretically speaking at least, no two cycles will ever repeat and an irrational wave will never repeat itself all the way to infinity.

> Now add to this the fact that there are an uncountably infinite number of these waves (aleph 1, see Georg Cantor), that this number is so large compared with countable infinity (aleph 0, the amount of rational numbers) that if we threw a dart on a number line the odds of hitting a rational is so small as to be practically zero. The rational numbers and musical harmony become like a precious and rare gem in the midst of a universe of chaos. This is what I had at the back of my mind when writing the rabbit hole. But I was wrong saying we'd be none the wiser for it, for exploring things like phi etc...I have regained a new appreciation of rational harmony.

> -Rick

Michael,

-Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
> --the very concept of frequency itself relies on the GCD between rational
> numbers,
> Huh? Sounds more like the concept of the harmonic series to me...
>
> ---and beat
> 'frequencies' are a secondary concept.
> Beating is IF you are modeling you scales after the harmonic series.
>
> *****************************************************************************************
> --Firstly, the 12 tone system is already irrational. The difference is
> that it evolved using the --rational numbers as a guide, trying to
> distribute them evenly over each key.
> Trust me, even distribution doesn't solve anything alone. Try playing the 10TET tuning. It sounds terribly dissonant, yet is very evenly spaced. You can partly help against that problem by shifting the timbre/instrument-overtones using irrational intervals to better match the tuning (ALA Sethares' tuning-to-timbre method)...but then we go back to using irrational intervals. :-D
>
> ---!!!!!!True, they --might be almost
> periodic!!!!!!!!!!
> In fact, in the case of 12TET, so close they are almost indistinguishable from each rational intervals to the human ear. Which is why you seem to still be basically arguing again "irrational intervals don't work".
>
> BTW,
> ---http://www.geocities.com/djtrancendance/PHIxcellence2.mp3
> I've given you this link to you several times...a link to an example of an irrational scale (that gets nowhere near rational intervals). You seem to go on and on about why irrational intervals can't work, yet you can't seem to site, say, anything in my example that sounds dissonant. Is it too much to ask you to listen to the file and give it a chance (or, on the other hand, perhaps let me know what about it doesn't work coming from your ears and not your pre-determined theoretical biases)?
>
>
>
>
>
> --- On Thu, 3/12/09, rick_ballan <rick_ballan@...> wrote:
> From: rick_ballan <rick_ballan@...>
> Subject: [tuning] Re: Phi Tonality
> To: tuning@yahoogroups.com
> Date: Thursday, March 12, 2009, 5:00 PM
>
>
>
>
>
>
>
>
>
>
>
>
> --- In tuning@yahoogroups. com, djtrancendance@ ... wrote:
>
> >
>
> > --Hi Michael,
>
> >
>
> > ---I remember reading a quote by Stravinsky somewhere that, contrary to
>
> > what we might ---expect, infinite possibilities can become so overwhelming
>
> > that all creativity eventually ---grinds to a halt (which was why he
>
> > adopted serialism).
>
> > Right. For example if we try to establish an infinite number of ways to make ratios between sets of frequencies beat at multiples of each other we could have thousands of different results. That's why such experiments would likely need to be reduced to functions, such as that from using PHI as a meantone-type generator, to limit experimentation to a more finite set.
>
> >
>
> > ---Consider this: if all periodic waves require
>
> > rational numbers then irrational numbers are ---aperiodic. (Recall for eg
>
> > Pythagoras' proof that the sq root of 2, or b5th interval, is not a
>
> > ---rational number). This means that, theoretically speaking at least, no
>
> > two cycles will ever ---repeat and an irrational wave will never repeat
>
> > itself all the way to infinity.
>
> > Right, it may not repeat...but, on the other hand, it may come SO close to repeating, in some places, that the ear can not recognize the difference.
>
> >
>
> >
>
> > ---Now add to this the fact that there are an uncountably infinite number
>
> > of these waves ---(aleph 1, see Georg Cantor), that this number is so
>
> > large compared with countable infinity ---(aleph 0, the amount of rational
>
> > numbers) that if we threw a dart on a number line the odds ---of hitting a
>
> > rational is so small as to be practically zero.
>
> > But, again, I don't care about hitting rational numbers but rather numbers of frequencies whose beating is proportionate. For example 100hz 122hz and 166hz (where the beating rates are all irrational numbers, but all multiples of each other IE 122-100hz = 22hz and 166-122hz=44hz which = 22hz * 2)!
>
> >
>
> > --The rational numbers
>
> > and musical harmony become like a precious and rare gem in the --midst of
>
> > a universe of chaos.
>
> > I would argue rational numbers become "one way of achieving something the mind can recognize beautiful patterns in". And irrational numbers become another one which seems far more delicate and harder to get right but still possible (in the same way using a derivative in calculus to find a slope is vs. the equation of a slope in algebra).
>
> >
>
> > Far as the belief that "only the rare gem of rational numbers can form great consonance" you think http://www.geocitie s.com/djtrancend ance/PHIxcellenc e2.mp3 (a scale formed purely from irrational numbers) sounds "unorganized" or "far less beautiful than rational tones"...please let me know.
>
> >
>
> > My point again is NOT that I have found a "perfect scale" but, rather, that irrational numbers can produce easily interpret-able music.
>
> >
>
> > ---This is what I had at the back of my mind when
>
> > writing the rabbit hole.
>
> > Which is...where exactly? Or is that somehow related to the movie series "into the Rabbit hole", about quantum physics, how aging works, and other exotic thoughts?
>
> > ---But I was wrong ---saying we'd be none the wiser
>
> > for it, for exploring things like phi etc...
>
> > ---I have regained a new
>
> > ---
>
> > appreciation of rational harmony.
>
> > Isn't that basically saying it was only worth exploring irrational harmony to explain how many ways there are NOT to get rational harmony thus helping prove a pet-hypothesis that rational harmony is the only answer? If so...so much for open-mindedness. .. :-(
>
> >
>
> > -Michael
>
> >
>
> >
>
> >
>
> >
>
> > --- On Tue, 3/10/09, rick_ballan <rick_ballan@ ...> wrote:
>
> >
>
> > From: rick_ballan <rick_ballan@ ...>
>
> > Subject: [tuning] Re: Phi Tonality
>
> > To: tuning@yahoogroups. com
>
> > Date: Tuesday, March 10, 2009, 7:50 PM
>
> >
>
> >
>
> >
>
> >
>
> >
>
> >
>
> >
>
> >
>
> >
>
> >
>
> >
>
> >
>
> > --- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...> wrote:
>
> >
>
> > >
>
> >
>
> > > Rick,
>
> >
>
> > >
>
> >
>
> > > --That's why the whole PHI-tonality seems so appealing to me...in such --a tuning the notes all have the exact same relationship to each --other and thus multiple notes can be "roots" simultaneously. ..and --thus I suspect the "average feel" of the relationships between the --notes could over-ride the need to be "consonant about a single --root".
>
> >
>
> > > > This is what I've called "floating". (RICK)
>
> >
>
> > >
>
> >
>
> > > In that case, my point is that the ability to "float" in a tuning is worth investigation. Furthermore the adaptive JI case of "make everything consonant by matching it to a root" appears to fall far short of covering all the consonant scales that could be made if people took advantage of "floating" when making scales instead of always doing JI-style "matching to the exact root note".
>
> >
>
> > > ************ ********* ********* ********* ********* ********* ********* ********* ********* ********* **
>
> >
>
> > >
>
> >
>
> > > > When you say otherness you mean otherness relative to what? A
>
> >
>
> > > >root tone? The valleys/minimum- entropy-points in some Setharesian >dissonance curves?
>
> >
>
> > >
>
> >
>
> > > --Both- floating in relation to the root, and in relation to clearly
>
> >
>
> > > --root-related proportions. (RICK)
>
> >
>
> > > No problem...my best guess is, in the case of my PHI scale, such a "root related proportion" is the same a+b/a = a/b relationship that PHI has.
>
> >
>
> > > ************ ********* ********* ********* ********* ********* *
>
> >
>
> > >
>
> >
>
> > > ---And as I've mentioned many times over the last couple of years, I
>
> >
>
> > > ---believe you can find points that are so darn "dissonant" as far as
>
> >
>
> > > ---interactions in the spectra that they achieve a kind of fuzzy
>
> >
>
> > > ---softness. (RICK)
>
> >
>
> > > Maybe, but is it really such a tragedy to exploit that? :-D In fact, I've found scale schemes that make instrument overtones in chords nearly overlap but not quite sometimes sound better than having the overtones match exactly.
>
> >
>
> > > Actually Sethares' theory seems to state your "suspicion" as well...that two notes that are too close meld into a relaxing "fuzzy softness" chorus-type effect...probably quite similar to what happens with pink noise.
>
> >
>
> > >
>
> >
>
> > > -Michael
>
> >
>
> > >
>
> >
>
> > Hi Michael,
>
> >
>
> >
>
> >
>
> > I remember reading a quote by Stravinsky somewhere that, contrary to what we might expect, infinite possibilities can become so overwhelming that all creativity eventually grinds to a halt (which was why he adopted serialism). Consider this: if all periodic waves require rational numbers then irrational numbers are aperiodic. (Recall for eg Pythagoras' proof that the sq root of 2, or b5th interval, is not a rational number). This means that, theoretically speaking at least, no two cycles will ever repeat and an irrational wave will never repeat itself all the way to infinity.
>
> >
>
> >
>
> >
>
> > Now add to this the fact that there are an uncountably infinite number of these waves (aleph 1, see Georg Cantor), that this number is so large compared with countable infinity (aleph 0, the amount of rational numbers) that if we threw a dart on a number line the odds of hitting a rational is so small as to be practically zero. The rational numbers and musical harmony become like a precious and rare gem in the midst of a universe of chaos. This is what I had at the back of my mind when writing the rabbit hole. But I was wrong saying we'd be none the wiser for it, for exploring things like phi etc...I have regained a new appreciation of rational harmony.
>
> >
>
> >
>
> >
>
> > -Rick
>
> >
>
> Michael,
>
>
>
> Firstly, the 12 tone system is already irrational. The difference is that it evolved using the rational numbers as a guide, trying to distribute them evenly over each key. True, they might be almost periodic, or they might be harmonics in the higher octaves e.g. 645/512 = 1.2597... is a very good approximation to the tempered maj 3, 2 power of 1/3 = 1.25992.... So I am not against irrationals per se, which would be absurd, but am merely bringing certain 'awkward' points into the open.
>
>
>
> Secondly, the example you gave of beating are not irrational numbers but whole numbers i.e. 122-100Hz = 22Hz etc so I haven't got a clue what you mean here. But as I have also pointed out many times in this group, the very concept of frequency itself relies on the GCD between rational numbers, which is also a model of musical harmony, and beat 'frequencies' are a secondary concept. To use your example, 122/100 = 61/50, therefore GCD = 2, 100 and 122 are the 50th and 61st harmonics of tonic 2, and the beat freq (btw, half the difference) is 1/2(122-100) = 11 and therefore is the 11th harmonic.
>
>
>
> -Rick

Michael,

I keep on going on about it to you because you keep on missing the point. And I have listened to the site and already said that it sounds out of tune. What does the term 'alternate' mean in this 'alternate tuning' site, alternative to what? It is alternatives to the harmonic series, 12 edo, meantone, in short all of those systems which experience has taught us to trust. And how can we find valid options if we do not understand the benefits and downfalls of those systems and take them as our starting point?

"--the very concept of frequency itself relies on the GCD between rational numbers, Huh? Sounds more like the concept of the harmonic series to me...---and beat 'frequencies' are a secondary concept.
Beating is IF you are modeling you scales after the harmonic series."

Well it is called the "harmonic" series for a reason. It is a natural phenomenon which occurs whenever we pluck a stretched string or blow air through a pipe. Its discovery dates back to the very beginnings of science with Pythagoras (570 B.C.). Without this spiritual aspect of "harmonia", which meant universal order, mathematics probably would have remained a form of accounting.

The difference b/w the harmonic series and the GCD is that we can play upper harmonics on two different instruments and create a new fundamental frequency which is not played on the original two. This phenomenon realises itself as the key and is called tonality.
And no, beating is independent of the harmonic series. It is half the difference b/w ANY two frequencies. It dissappears when two strings (say) are in tune because the beat freq becomes an harmonic.

You also said that you tried 10 tet and it sounded out of tune. But nobody has ever claimed (especially not Bill) that all equal distributions will be the same or that disproving one will automatically discount all the others. Again, the 12 tet and meantone systems evolved out of attempting to distribute the harmonic series over all keys. We find similar types of tunings in other cultures such as the 7 (??) tet of Turkish music.

Finally, I'll ask you for the last time how can you explain PHI musically? Given (a + b)/a = a/b, which gives a quadratic equation the solution of which is PHI, then what is the possible meaning of the addition of a + b? It is twice the average freq, which means that it is an octave above. So we have :the octave of the average freq divided by the larger freq equals the ratio b/w the two freq's. What are we achieving here?

-Rick

🔗djtrancendance@...

--I keep on going on about it to you because you keep on missing the
point. And I have --listened to the site and already said that it sounds
out of tune.
   Funny...I've never heard you say it. Out of tune to what, exactly?
   Anyway, my point is not to sound like 12TET or harmonic-series-like intervals, but to sound consonant/relaxed. I never challenged you to prove my scale is/isn't out of tune...but, instead, to prove it is/isn't consonant. I asked you if you believed something didn't work, then which notes clash?
   But, apparently, you found it more convenient to simply change the topic to the idea of "keys", which I never mentioned...

---What does the term 'alternate' mean in this 'alternate
tuning' site, alternative to what?
To me, it simply means anything which can generate intelligent sounding easy to listen to musical expression. Which may or may not follow intervals from the harmonic series...

--It is alternatives to the harmonic
series,
Saying all of micro-tonal and music in general is ONLY based on the idea of mimicking the harmonic series seems blatantly ignorant to me. Sure, most micro-tonal is "variations on the theme of the harmonic series", but people like Sethares and Schubert have messed at least somewhat successfully with the idea of scale systems and compositional styles that sound beautiful yet royally break the harmonic series.

---12 edo, meantone, in short all of those systems
-- which
experience has taught us to trust.
Nothing wrong with those, but nothing really new about them either: like you said they all pretty much boil down to the same "monopolistic" thing. A very easy counter example: Sethares' use of 10TET in his song "Ten Strings" where he re-aligns the timbre of the instruments to something that looks completely unlike the harmonic series.

---And how can we find valid options if
we do not understand the benefits and downfalls of ---those systems and
take them as our starting point?
Simple...we can realize that there is more than one pathway to consonance. Sethares found one in making "out of key" scales not based on the harmonic series and matching them with timbres that create a fair degree of consonance.

--Well it is called the "harmonic" series for a reason. It is a natural
phenomenon which ---occurs whenever we pluck a stretched string or blow
air through a pipe. Its discovery --dates back to the very beginnings of
science with Pythagoras (570 B.C.). Without this --spiritual aspect of
"harmonia", which meant universal order, mathematics probably would
--have remained a form of accounting.
Right...and, note, Sethares use of timbre becomes IMPOSSIBLE with instruments in nature because those in nature produce overtones aligned with the harmonic series...yet his "unnatural/bent harmonic series" can also produce consonance to a fair degree. Sure, Pythagoras found something very natural and useful...but that doesn't mean it's the ONLY way the mind can easily organize sound artistically so as to smoothly communicate emotion.

---The difference b/w the harmonic series and the GCD is that we can play
upper harmonics --on two different instruments and create a new
fundamental frequency which is not played --on the original two. This
phenomenon realises itself as the key and is called tonality.
Meaning...two different tones point to a tone below those two with is equal to the GCD of those tones. You act like it's something I'm just learning now...but I've known that concept ever since I started studying micro-tonal music.
And, yes, It's a good and working theory but, again, I don't think it's the only way the mind can easily organize tones. To me that's like saying imaginary numbers are useless because the square root of -1 is impossible in nature.

When people sound-engineer/program PADS with their synthesizers...they often use frequency modulations on overtone that deviate from the harmonic that actually increase the pleasantness of the sound...now compare that to the essentially perfect harmonic series formed by a pluck of a guitar string: the PADS, in many cases, actually sound more relaxed.

--And no, beating is independent of the harmonic series. It is half the
difference b/w ANY --two frequencies. It dissappears when two strings
(say) are in tune because the beat freq --becomes an harmonic.
If you are saying "the harmonic series resolves beating"...try playing harmonics 25,26, and 27 of any note together. Note they beat so much...that even though they point to the same harmonic they still sound un-relaxed. Even the harmonic series has its faults so far as consonance. My point is, again, there are two ways to generate "beating consonance" relatively well...one is to make the beating the same between notes IE 200hz,300hz,400hz (IE basically, the harmonic series; the beating in this example is of course 100hz) and the other is to make all beating between tones multiples of each other IE 200hz,400hz,600hz (multiples = 1, 2, 3), but that can also be 100hz 162hz 262hz (multiples = 1, 1.618, 1.618^2). When you think about it, the multiples method uses a generator in a >>>mean-tone<<<-like way...but it certainly doesn't have to be
stuck with mean-tone like or near-rational. intervals

--You also said that you tried 10 tet and it sounded out of tune. But
nobody has ever --claimed (especially not Bill) that all equal
distributions will be the same or that disproving --one will
automatically discount all the others.
Agreed, nobody has...before you said "this is because the distribution among notes is equal"...and now you seem to be admitting "not always"...which was my point in the first place.

--Again, the 12 tet and meantone
systems evolved out of attempting to distribute the --harmonic series
over all keys. We find similar types of tunings in other cultures such
as --the 7 (??) tet of Turkish music.

So what? Yes, they work, but it surely is not the only way toward easily listen-able music, just the most obvious (both in nature and in the ease of calculating how to make real-world/non-electronic instruments that match it).
Again, Sethares' timbre-matched use of 10TET already seems to hint at that: even if my attempts to twist the use of consonance sounds bad in your mind, there have still been a few people who have successfully pushed for alternatives to the "equal-speed beating" ideal of the harmonic series.

The difference between the old and new...at least to me...is that now we have the advantage of technology to test things that used to be very hard to calculate quickly and thus build more scales by experimentation and more abstract patterns than 1,2,3 (1,2,3 (as multiples)...is exactly what the harmonic series is).

🔗Michael Sheiman <djtrancendance@...>

Rick,

--Finally, I'll ask you for the last time how can you explain PHI
musically? Given (a + b)/a = --a/b, which gives a quadratic equation the
solution of which is PHI, then what is the possible --meaning of the
addition of a + b? It is twice the average freq, which means that it is
an --octave above.

--So we have :the octave of the average freq divided by
the larger freq equals the ratio b/w the ---two freq's. What are we
achieving here?
Good question...the beating between any two frequencies is proportionate to the beating of any other two frequencies...they are NOT beating at the same rates but, rather, rates that follow a distinct pattern. The PHI^x/2(octave)^y mean-tone-generation-like formula for making a tuning gives numbers that generate proportionate beating rates.
*****************************************************
This is in contrast to the harmonic series which, IMVHO, works because the beating between all frequencies in the series are proportionate (and, in the most obvious way: the beating rate between are two frequencies in the harmonic series is constant IE 100,200,300,400hz all have a beating rate relative to each other that a whole number multiple/division of 100hz).

-Michael

--- On Sat, 3/14/09, rick_ballan <rick_ballan@...> wrote:

From: rick_ballan <rick_ballan@yahoo.com.au>
Subject: [tuning] Re: Phi Tonality
To: tuning@yahoogroups.com
Date: Saturday, March 14, 2009, 7:37 PM

--- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...> wrote:

> --the very concept of frequency itself relies on the GCD between rational

> numbers,

> Huh? Sounds more like the concept of the harmonic series to me...

> ---and beat

> 'frequencies' are a secondary concept.

> Beating is IF you are modeling you scales after the harmonic series.

> ************ ********* ********* ********* ********* ********* ********* ********* ********* *****

> --Firstly, the 12 tone system is already irrational. The difference is

> that it evolved using the --rational numbers as a guide, trying to

> distribute them evenly over each key.

> Trust me, even distribution doesn't solve anything alone. Try playing the 10TET tuning. It sounds terribly dissonant, yet is very evenly spaced. You can partly help against that problem by shifting the timbre/instrument- overtones using irrational intervals to better match the tuning (ALA Sethares' tuning-to-timbre method)...but then we go back to using irrational intervals. :-D

> ---!!!!!!True, they --might be almost

> periodic!!!! !!!!!!

> In fact, in the case of 12TET, so close they are almost indistinguishable from each rational intervals to the human ear. Which is why you seem to still be basically arguing again "irrational intervals don't work".

> BTW,

> ---http://www.geocitie s.com/djtrancend ance/PHIxcellenc e2.mp3

> I've given you this link to you several times...a link to an example of an irrational scale (that gets nowhere near rational intervals). You seem to go on and on about why irrational intervals can't work, yet you can't seem to site, say, anything in my example that sounds dissonant. Is it too much to ask you to listen to the file and give it a chance (or, on the other hand, perhaps let me know what about it doesn't work coming from your ears and not your pre-determined theoretical biases)?

> --- On Thu, 3/12/09, rick_ballan <rick_ballan@ ...> wrote:

> From: rick_ballan <rick_ballan@ ...>

> Subject: [tuning] Re: Phi Tonality

> To: tuning@yahoogroups. com

> Date: Thursday, March 12, 2009, 5:00 PM

> --- In tuning@yahoogroups. com, djtrancendance@ ... wrote:

> >

> > --Hi Michael,

> >

> > ---I remember reading a quote by Stravinsky somewhere that, contrary to

> > what we might ---expect, infinite possibilities can become so overwhelming

> > that all creativity eventually ---grinds to a halt (which was why he

> > adopted serialism).

> > Right. For example if we try to establish an infinite number of ways to make ratios between sets of frequencies beat at multiples of each other we could have thousands of different results. That's why such experiments would likely need to be reduced to functions, such as that from using PHI as a meantone-type generator, to limit experimentation to a more finite set.

> >

> > ---Consider this: if all periodic waves require

> > rational numbers then irrational numbers are ---aperiodic. (Recall for eg

> > Pythagoras' proof that the sq root of 2, or b5th interval, is not a

> > ---rational number). This means that, theoretically speaking at least, no

> > two cycles will ever ---repeat and an irrational wave will never repeat

> > itself all the way to infinity.

> > Right, it may not repeat...but, on the other hand, it may come SO close to repeating, in some places, that the ear can not recognize the difference.

> >

> > ---Now add to this the fact that there are an uncountably infinite number

> > of these waves ---(aleph 1, see Georg Cantor), that this number is so

> > large compared with countable infinity ---(aleph 0, the amount of rational

> > numbers) that if we threw a dart on a number line the odds ---of hitting a

> > rational is so small as to be practically zero.

> > But, again, I don't care about hitting rational numbers but rather numbers of frequencies whose beating is proportionate. For example 100hz 122hz and 166hz (where the beating rates are all irrational numbers, but all multiples of each other IE 122-100hz = 22hz and 166-122hz=44hz which = 22hz * 2)!

> >

> > --The rational numbers

> > and musical harmony become like a precious and rare gem in the --midst of

> > a universe of chaos.

> > I would argue rational numbers become "one way of achieving something the mind can recognize beautiful patterns in". And irrational numbers become another one which seems far more delicate and harder to get right but still possible (in the same way using a derivative in calculus to find a slope is vs. the equation of a slope in algebra).

> >

> > Far as the belief that "only the rare gem of rational numbers can form great consonance" you think http://www.geocitie s.com/djtrancend ance/PHIxcellenc e2.mp3 (a scale formed purely from irrational numbers) sounds "unorganized" or "far less beautiful than rational tones"...please let me know.

> >

> > My point again is NOT that I have found a "perfect scale" but, rather, that irrational numbers can produce easily interpret-able music.

> >

> > ---This is what I had at the back of my mind when

> > writing the rabbit hole.

> > Which is...where exactly? Or is that somehow related to the movie series "into the Rabbit hole", about quantum physics, how aging works, and other exotic thoughts?

> > ---But I was wrong ---saying we'd be none the wiser

> > for it, for exploring things like phi etc...

> > ---I have regained a new

> > ---

> > appreciation of rational harmony.

> > Isn't that basically saying it was only worth exploring irrational harmony to explain how many ways there are NOT to get rational harmony thus helping prove a pet-hypothesis that rational harmony is the only answer? If so...so much for open-mindedness. .. :-(

> >

> > -Michael

> >

> > --- On Tue, 3/10/09, rick_ballan <rick_ballan@ ...> wrote:

> >

> > From: rick_ballan <rick_ballan@ ...>

> > Subject: [tuning] Re: Phi Tonality

> > To: tuning@yahoogroups. com

> > Date: Tuesday, March 10, 2009, 7:50 PM

> >

> > --- In tuning@yahoogroups. com, Michael Sheiman <djtrancendance@ ...> wrote:

> >

> > >

> >

> > > Rick,

> >

> > >

> >

> > > --That's why the whole PHI-tonality seems so appealing to me...in such --a tuning the notes all have the exact same relationship to each --other and thus multiple notes can be "roots" simultaneously. ..and --thus I suspect the "average feel" of the relationships between the --notes could over-ride the need to be "consonant about a single --root".

> >

> > > > This is what I've called "floating". (RICK)

> >

> > >

> >

> > > In that case, my point is that the ability to "float" in a tuning is worth investigation. Furthermore the adaptive JI case of "make everything consonant by matching it to a root" appears to fall far short of covering all the consonant scales that could be made if people took advantage of "floating" when making scales instead of always doing JI-style "matching to the exact root note".

> >

> > > ************ ********* ********* ********* ********* ********* ********* ********* ********* ********* **

> >

> > >

> >

> > > > When you say otherness you mean otherness relative to what? A

> >

> > > >root tone? The valleys/minimum- entropy-points in some Setharesian >dissonance curves?

> >

> > >

> >

> > > --Both- floating in relation to the root, and in relation to clearly

> >

> > > --root-related proportions. (RICK)

> >

> > > No problem...my best guess is, in the case of my PHI scale, such a "root related proportion" is the same a+b/a = a/b relationship that PHI has.

> >

> > > ************ ********* ********* ********* ********* ********* *

> >

> > >

> >

> > > ---And as I've mentioned many times over the last couple of years, I

> >

> > > ---believe you can find points that are so darn "dissonant" as far as

> >

> > > ---interactions in the spectra that they achieve a kind of fuzzy

> >

> > > ---softness. (RICK)

> >

> > > Maybe, but is it really such a tragedy to exploit that? :-D In fact, I've found scale schemes that make instrument overtones in chords nearly overlap but not quite sometimes sound better than having the overtones match exactly.

> >

> > > Actually Sethares' theory seems to state your "suspicion" as well...that two notes that are too close meld into a relaxing "fuzzy softness" chorus-type effect...probably quite similar to what happens with pink noise.

> >

> > >

> >

> > > -Michael

> >

> > >

> >

> > Hi Michael,

> >

> > I remember reading a quote by Stravinsky somewhere that, contrary to what we might expect, infinite possibilities can become so overwhelming that all creativity eventually grinds to a halt (which was why he adopted serialism). Consider this: if all periodic waves require rational numbers then irrational numbers are aperiodic. (Recall for eg Pythagoras' proof that the sq root of 2, or b5th interval, is not a rational number). This means that, theoretically speaking at least, no two cycles will ever repeat and an irrational wave will never repeat itself all the way to infinity.

> >

> > Now add to this the fact that there are an uncountably infinite number of these waves (aleph 1, see Georg Cantor), that this number is so large compared with countable infinity (aleph 0, the amount of rational numbers) that if we threw a dart on a number line the odds of hitting a rational is so small as to be practically zero. The rational numbers and musical harmony become like a precious and rare gem in the midst of a universe of chaos. This is what I had at the back of my mind when writing the rabbit hole. But I was wrong saying we'd be none the wiser for it, for exploring things like phi etc...I have regained a new appreciation of rational harmony.

> >

> > -Rick

> >

> Michael,

> Firstly, the 12 tone system is already irrational. The difference is that it evolved using the rational numbers as a guide, trying to distribute them evenly over each key. True, they might be almost periodic, or they might be harmonics in the higher octaves e.g. 645/512 = 1.2597... is a very good approximation to the tempered maj 3, 2 power of 1/3 = 1.25992.... So I am not against irrationals per se, which would be absurd, but am merely bringing certain 'awkward' points into the open.

> Secondly, the example you gave of beating are not irrational numbers but whole numbers i.e. 122-100Hz = 22Hz etc so I haven't got a clue what you mean here. But as I have also pointed out many times in this group, the very concept of frequency itself relies on the GCD between rational numbers, which is also a model of musical harmony, and beat 'frequencies' are a secondary concept. To use your example, 122/100 = 61/50, therefore GCD = 2, 100 and 122 are the 50th and 61st harmonics of tonic 2, and the beat freq (btw, half the difference) is 1/2(122-100) = 11 and therefore is the 11th harmonic.

> -Rick

Michael,

"--the very concept of frequency itself relies on the GCD between rational numbers, Huh? Sounds more like the concept of the harmonic series to me...---and beat 'frequencies' are a secondary concept.

Beating is IF you are modeling you scales after the harmonic series."

And no, beating is independent of the harmonic series. It is half the difference b/w ANY two frequencies. It dissappears when two strings (say) are in tune because the beat freq becomes an harmonic.

-Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> --I keep on going on about it to you because you keep on missing the
> point. And I have --listened to the site and already said that it sounds
> out of tune.
> Funny...I've never heard you say it. Out of tune to what, exactly?
> Anyway, my point is not to sound like 12TET or harmonic-series-like intervals, but to sound consonant/relaxed. I never challenged you to prove my scale is/isn't out of tune...but, instead, to prove it is/isn't consonant. I asked you if you believed something didn't work, then which notes clash?
> But, apparently, you found it more convenient to simply change the topic to the idea of "keys", which I never mentioned...
>
> ---What does the term 'alternate' mean in this 'alternate
> tuning' site, alternative to what?
> To me, it simply means anything which can generate intelligent sounding easy to listen to musical expression. Which may or may not follow intervals from the harmonic series...
>
> --It is alternatives to the harmonic
> series,
> Saying all of micro-tonal and music in general is ONLY based on the idea of mimicking the harmonic series seems blatantly ignorant to me. Sure, most micro-tonal is "variations on the theme of the harmonic series", but people like Sethares and Schubert have messed at least somewhat successfully with the idea of scale systems and compositional styles that sound beautiful yet royally break the harmonic series.
>
> ---12 edo, meantone, in short all of those systems
> -- which
> experience has taught us to trust.
> Nothing wrong with those, but nothing really new about them either: like you said they all pretty much boil down to the same "monopolistic" thing. A very easy counter example: Sethares' use of 10TET in his song "Ten Strings" where he re-aligns the timbre of the instruments to something that looks completely unlike the harmonic series.
>
>
> ---And how can we find valid options if
> we do not understand the benefits and downfalls of ---those systems and
> take them as our starting point?
> Simple...we can realize that there is more than one pathway to consonance. Sethares found one in making "out of key" scales not based on the harmonic series and matching them with timbres that create a fair degree of consonance.
>
>
>
> --Well it is called the "harmonic" series for a reason. It is a natural
> phenomenon which ---occurs whenever we pluck a stretched string or blow
> air through a pipe. Its discovery --dates back to the very beginnings of
> science with Pythagoras (570 B.C.). Without this --spiritual aspect of
> "harmonia", which meant universal order, mathematics probably would
> --have remained a form of accounting.
> Right...and, note, Sethares use of timbre becomes IMPOSSIBLE with instruments in nature because those in nature produce overtones aligned with the harmonic series...yet his "unnatural/bent harmonic series" can also produce consonance to a fair degree. Sure, Pythagoras found something very natural and useful...but that doesn't mean it's the ONLY way the mind can easily organize sound artistically so as to smoothly communicate emotion.
>
>
> ---The difference b/w the harmonic series and the GCD is that we can play
> upper harmonics --on two different instruments and create a new
> fundamental frequency which is not played --on the original two. This
> phenomenon realises itself as the key and is called tonality.
> Meaning...two different tones point to a tone below those two with is equal to the GCD of those tones. You act like it's something I'm just learning now...but I've known that concept ever since I started studying micro-tonal music.
> And, yes, It's a good and working theory but, again, I don't think it's the only way the mind can easily organize tones. To me that's like saying imaginary numbers are useless because the square root of -1 is impossible in nature.
>
> When people sound-engineer/program PADS with their synthesizers...they often use frequency modulations on overtone that deviate from the harmonic that actually increase the pleasantness of the sound...now compare that to the essentially perfect harmonic series formed by a pluck of a guitar string: the PADS, in many cases, actually sound more relaxed.
>
>
> --And no, beating is independent of the harmonic series. It is half the
> difference b/w ANY --two frequencies. It dissappears when two strings
> (say) are in tune because the beat freq --becomes an harmonic.
> If you are saying "the harmonic series resolves beating"...try playing harmonics 25,26, and 27 of any note together. Note they beat so much...that even though they point to the same harmonic they still sound un-relaxed. Even the harmonic series has its faults so far as consonance. My point is, again, there are two ways to generate "beating consonance" relatively well...one is to make the beating the same between notes IE 200hz,300hz,400hz (IE basically, the harmonic series; the beating in this example is of course 100hz) and the other is to make all beating between tones multiples of each other IE 200hz,400hz,600hz (multiples = 1, 2, 3), but that can also be 100hz 162hz 262hz (multiples = 1, 1.618, 1.618^2). When you think about it, the multiples method uses a generator in a >>>mean-tone<<<-like way...but it certainly doesn't have to be
> stuck with mean-tone like or near-rational. intervals
>
>
>
> --You also said that you tried 10 tet and it sounded out of tune. But
> nobody has ever --claimed (especially not Bill) that all equal
> distributions will be the same or that disproving --one will
> automatically discount all the others.
> Agreed, nobody has...before you said "this is because the distribution among notes is equal"...and now you seem to be admitting "not always"...which was my point in the first place.
>
> --Again, the 12 tet and meantone
> systems evolved out of attempting to distribute the --harmonic series
> over all keys. We find similar types of tunings in other cultures such
> as --the 7 (??) tet of Turkish music.
>
> So what? Yes, they work, but it surely is not the only way toward easily listen-able music, just the most obvious (both in nature and in the ease of calculating how to make real-world/non-electronic instruments that match it).
> Again, Sethares' timbre-matched use of 10TET already seems to hint at that: even if my attempts to twist the use of consonance sounds bad in your mind, there have still been a few people who have successfully pushed for alternatives to the "equal-speed beating" ideal of the harmonic series.
>
> The difference between the old and new...at least to me...is that now we have the advantage of technology to test things that used to be very hard to calculate quickly and thus build more scales by experimentation and more abstract patterns than 1,2,3 (1,2,3 (as multiples)...is exactly what the harmonic series is).

>
Michael,

The GCD frequency isn't restricted to acoustic instruments. It is a fact of wave theory and Fourier analysis. It will apply whenever you generate sounds via any method whatsoever. In fact in the early 90's I worked closely with a scientist named Tony Childs who worked for the company which invented many of the programs you take for granted. My point has been that the traditional beating/consonance type approach which made popular by for eg Helmholtz deals with + and - actually misses the point of harmony, which is ratio based (division, multiplication). As a jazz musician, I use 'dissonances' all the time and yet it is still in tune. Similarly, I find Bill's music quite beautiful and have told him so many times. And in fact Bill's idea that certain timbres (which is still based on harmonics) fit with certain tuning systems fits well with the fact that a single note already brings with it it own tuning system (for notes are also chords, the idea of a single note besides a single sine wave being an illusion).

"Yes, they (12 tet/meantone) work, but it surely is not the only way toward easily listen-able music, just the most obvious (both in nature and in the ease of calculating how to make real-world/non-electronic instruments that match it)." But as Sherlock Holmes said, "Watson, everything is obvious upon explanation". And in fact there is nothing obvious or easy about them, as there is nothing obvious or easy about Bill's work (which you don't understand either). eg the term "12 tet system" itself can be a bit reductionist. It includes 6, 4, 3 and 2 tets as subsets, therefore giving the only equal intervals, triads or 4 note chords. It can be seen as equal 4ths over 5 8ve's, equal 5th's over 7 8ve's, and closer to the actual practice of 12 tet music, successive minor and major thirds and vice-versa. But to describe the model exactly, we would have to describe the study of jazz/classical musical harmony which is a lifetime pursuit. In other words, it is much, much more difficult than simply downloading scalar and reading a few articles on micro-tuning and then imagining that you are at the cutting-edge of the "neuveax open-minded".

-Rick

🔗djtrancendance@...

--The GCD frequency isn't restricted to acoustic instruments. It is a fact of wave theory and --Fourier analysis.
I have worked with Fourier analysis and know how it divides a sound into several measures of amplitudes of sine wave (or the reverse in an inverse Fourier transform).
But what does that have to do with the GCD? For example if you take a 1048576 bin sized FFT all the frequencies you get will not be reduce-able to simple rational fractions, guaranteed.

--And in fact Bill's idea that certain timbres (which is still based on
harmonics) fit with ---certain tuning systems fits well
   Hmmm...he messed with the re-aligning the overtones of timbre, not the harmonics (unless your are using these terms interchange-ably in which case it makes sense).   But his theories are based on beating and roughness...in fact his scales/timbre ties are located from troughs of the curve from and beating/roughness based consonance formula.
******************************************************************
---with the fact that
a single note already brings with it it own tuning system
Bill's theories certainly don't seem to have the same tuning system because they start on the same note...otherwise how did he come up with using the 10TET and other things the royally don't obey the harmonic series? So a scale starts with C5 and ends with C6...but he can bend that into using 9,10,11...notes per octave and not the 12 that approximate the harmonic series well.
******************************************************************
--It includes 6, 4, 3 and 2 tets as subsets, therefore giving the only equal intervals, triads or --4 note chords.
     Right, but that would be basis for chord theory...and I never claimed to be an expert at that. My brother makes a living as a jazz musician and I would never say composition or "splitting the whole into parts" is easy.
   But the general ideas behind this are not complicated: the
areas where overtones of all notes in the chord intersect or near-intersect both higher notes in the scale AND other overtones from the chord well (either that or are far enough from each other not to cause much beating) form chords.
Memorizing and finding out where ALL those possible combinations are can take a lifetime of practice, agreed...but the mathematical concept isn't that hard...in the same way knowing how to take a derivative in calculus isn't that hard, but knowing all the applications of derivatives (including making your own derivative equations to explain mechanics) is incredibly hard.
But I'm not trying to say I know everything about every way to slice-and-dice diatonic systems into chord theories...I'm talking about a completely different system (one where virtually all combinations of notes form chords and the scale is itself a chord, so such chord-matching theories are not needed).

---there is nothing
obvious or easy about Bill's work (which you don't understand either).
You haven't given any practical examples of why Bill's work is so different from how I explain it. Numerous times on his website he mentions dissonance curves and beating.
First he says on http://eceserv0.ece.wisc.edu/~sethares/consemi.html
"To explain perceptions of musical
intervals, Plomp and Levelt note that most traditional musical tones
have a spectrum consisting of a root or fundamental frequency, and a
series of sine wave partials that occur at integer multiples of the
fundamental. Figure 2 depicts one such timbre. If this timbre is
sounded at various intervals, the dissonance of the intervals can be
calculated by adding up all of the dissonances between all pairs of
partials."

So his theory seems to be based in the idea of eliminating beating and DOES (I agree) have a lower-numbered-ratio basis FOR STANDARD ACOUSTIC timbres. But still he sites Plomp and Levelt and a roughness curve (based on beating) as the main generator for his theory. And what about other timbres?

That's my point: for other timbres his research seems to point at beating as the basis moreover harmonic matching to low-integer ratios...even if many of the notes he arrives at are low-integer ratios. Here's an example of an exception:
In "Tuning for FM Timbres" he comes up with 2.11 as being a maximum point of consonance for an FM timbre instrument, which translates to about (but not exactly) 19/9 (which = 2.11111). You can argue how much it relates to JI or the harmonic
series...but 19/9 is apparently more than 7-limit JI and certainly the beyond 5-limit JI result he got for his "acoustic timbre" experiment. And the final point is...he FM timbre, although much weirder than 5-limit JI, is also quite consonant.

I'm interested to continue this discussion but, this time around, since I'm using concrete examples, I would appreciate if you do so as well?

-Michael

--- On Sun, 3/15/09, rick_ballan <rick_ballan@...> wrote:

From: rick_ballan <rick_ballan@...>
Subject: [tuning] Re: Phi Tonality
To: tuning@yahoogroups.com
Date: Sunday, March 15, 2009, 8:47 PM

--- In tuning@yahoogroups. com, djtrancendance@ ... wrote:

> --I keep on going on about it to you because you keep on missing the

> point. And I have --listened to the site and already said that it sounds

> out of tune.

> Funny...I've never heard you say it. Out of tune to what, exactly?

> Anyway, my point is not to sound like 12TET or harmonic-series- like intervals, but to sound consonant/relaxed. I never challenged you to prove my scale is/isn't out of tune...but, instead, to prove it is/isn't consonant. I asked you if you believed something didn't work, then which notes clash?

> But, apparently, you found it more convenient to simply change the topic to the idea of "keys", which I never mentioned...

> ---What does the term 'alternate' mean in this 'alternate

> tuning' site, alternative to what?

> To me, it simply means anything which can generate intelligent sounding easy to listen to musical expression. Which may or may not follow intervals from the harmonic series...

> --It is alternatives to the harmonic

> series,

> Saying all of micro-tonal and music in general is ONLY based on the idea of mimicking the harmonic series seems blatantly ignorant to me. Sure, most micro-tonal is "variations on the theme of the harmonic series", but people like Sethares and Schubert have messed at least somewhat successfully with the idea of scale systems and compositional styles that sound beautiful yet royally break the harmonic series.

> ---12 edo, meantone, in short all of those systems

> -- which

> experience has taught us to trust.

> Nothing wrong with those, but nothing really new about them either: like you said they all pretty much boil down to the same "monopolistic" thing. A very easy counter example: Sethares' use of 10TET in his song "Ten Strings" where he re-aligns the timbre of the instruments to something that looks completely unlike the harmonic series.

> ---And how can we find valid options if

> we do not understand the benefits and downfalls of ---those systems and

> take them as our starting point?

> Simple...we can realize that there is more than one pathway to consonance. Sethares found one in making "out of key" scales not based on the harmonic series and matching them with timbres that create a fair degree of consonance.

> --Well it is called the "harmonic" series for a reason. It is a natural

> phenomenon which ---occurs whenever we pluck a stretched string or blow

> air through a pipe. Its discovery --dates back to the very beginnings of

> science with Pythagoras (570 B.C.). Without this --spiritual aspect of

> "harmonia", which meant universal order, mathematics probably would

> --have remained a form of accounting.

> Right...and, note, Sethares use of timbre becomes IMPOSSIBLE with instruments in nature because those in nature produce overtones aligned with the harmonic series...yet his "unnatural/bent harmonic series" can also produce consonance to a fair degree. Sure, Pythagoras found something very natural and useful...but that doesn't mean it's the ONLY way the mind can easily organize sound artistically so as to smoothly communicate emotion.

> ---The difference b/w the harmonic series and the GCD is that we can play

> upper harmonics --on two different instruments and create a new

> fundamental frequency which is not played --on the original two. This

> phenomenon realises itself as the key and is called tonality.

> Meaning...two different tones point to a tone below those two with is equal to the GCD of those tones. You act like it's something I'm just learning now...but I've known that concept ever since I started studying micro-tonal music.

> And, yes, It's a good and working theory but, again, I don't think it's the only way the mind can easily organize tones. To me that's like saying imaginary numbers are useless because the square root of -1 is impossible in nature.

> When people sound-engineer/ program PADS with their synthesizers. ..they often use frequency modulations on overtone that deviate from the harmonic that actually increase the pleasantness of the sound...now compare that to the essentially perfect harmonic series formed by a pluck of a guitar string: the PADS, in many cases, actually sound more relaxed.

> --And no, beating is independent of the harmonic series. It is half the

> difference b/w ANY --two frequencies. It dissappears when two strings

> (say) are in tune because the beat freq --becomes an harmonic.

> If you are saying "the harmonic series resolves beating"...try playing harmonics 25,26, and 27 of any note together. Note they beat so much...that even though they point to the same harmonic they still sound un-relaxed. Even the harmonic series has its faults so far as consonance. My point is, again, there are two ways to generate "beating consonance" relatively well...one is to make the beating the same between notes IE 200hz,300hz, 400hz (IE basically, the harmonic series; the beating in this example is of course 100hz) and the other is to make all beating between tones multiples of each other IE 200hz,400hz, 600hz (multiples = 1, 2, 3), but that can also be 100hz 162hz 262hz (multiples = 1, 1.618, 1.618^2). When you think about it, the multiples method uses a generator in a >>>mean-tone< <<-like way...but it certainly doesn't have to be

> stuck with mean-tone like or near-rational. intervals

> --You also said that you tried 10 tet and it sounded out of tune. But

> nobody has ever --claimed (especially not Bill) that all equal

> distributions will be the same or that disproving --one will

> automatically discount all the others.

> Agreed, nobody has...before you said "this is because the distribution among notes is equal"...and now you seem to be admitting "not always"...which was my point in the first place.

> --Again, the 12 tet and meantone

> systems evolved out of attempting to distribute the --harmonic series

> over all keys. We find similar types of tunings in other cultures such

> as --the 7 (??) tet of Turkish music.

> So what? Yes, they work, but it surely is not the only way toward easily listen-able music, just the most obvious (both in nature and in the ease of calculating how to make real-world/non- electronic instruments that match it).

> Again, Sethares' timbre-matched use of 10TET already seems to hint at that: even if my attempts to twist the use of consonance sounds bad in your mind, there have still been a few people who have successfully pushed for alternatives to the "equal-speed beating" ideal of the harmonic series.

> The difference between the old and new...at least to me...is that now we have the advantage of technology to test things that used to be very hard to calculate quickly and thus build more scales by experimentation and more abstract patterns than 1,2,3 (1,2,3 (as multiples).. .is exactly what the harmonic series is).

Michael,

The GCD frequency isn't restricted to acoustic instruments. It is a fact of wave theory and Fourier analysis. It will apply whenever you generate sounds via any method whatsoever. In fact in the early 90's I worked closely with a scientist named Tony Childs who worked for the company which invented many of the programs you take for granted. My point has been that the traditional beating/consonance type approach which made popular by for eg Helmholtz deals with + and - actually misses the point of harmony, which is ratio based (division, multiplication) . As a jazz musician, I use 'dissonances' all the time and yet it is still in tune. Similarly, I find Bill's music quite beautiful and have told him so many times. And in fact Bill's idea that certain timbres (which is still based on harmonics) fit with certain tuning systems fits well with the fact that a single note already brings with it it own tuning system (for notes are also chords, the idea of a
single note besides a single sine wave being an illusion).

"Yes, they (12 tet/meantone) work, but it surely is not the only way toward easily listen-able music, just the most obvious (both in nature and in the ease of calculating how to make real-world/non- electronic instruments that match it)." But as Sherlock Holmes said, "Watson, everything is obvious upon explanation" . And in fact there is nothing obvious or easy about them, as there is nothing obvious or easy about Bill's work (which you don't understand either). eg the term "12 tet system" itself can be a bit reductionist. It includes 6, 4, 3 and 2 tets as subsets, therefore giving the only equal intervals, triads or 4 note chords. It can be seen as equal 4ths over 5 8ve's, equal 5th's over 7 8ve's, and closer to the actual practice of 12 tet music, successive minor and major thirds and vice-versa. But to describe the model exactly, we would have to describe the study of jazz/classical musical harmony which is a lifetime pursuit. In other words, it is
much, much more difficult than simply downloading scalar and reading a few articles on micro-tuning and then imagining that you are at the cutting-edge of the "neuveax open-minded" .

-Rick

Phi Tonality

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