The significance of proportionate beating in PHI and other fractal numbers: with sound examples

Carl asked an excellent question: what is the point of proportionate beating (why I am admittedly somewhat obsessed with)?

Proportionate beating is a huge force behind why I have come to believe fractal numbers, and not the harmonic series, are ideal for the rarely explored challenge of 8 to 10 tone-per-2/1-period chords.

Need proof? Try the following sound bytes and tell me what you think...

8-tone per octave scale made with PHI (using my (1/PHI)^x generator) played with a sine wave:
http://www.geocities.com/djtrancendance/PHI/PHIharmonic8.wav.mp3

8-tone per octave harmonic series fragment (8/8,9/8,10/8....) played with a sine wave
http://www.geocities.com/djtrancendance/PHI/seriesharmonic8.wav.mp3

Notice how the harmonic series version has an almost mechanical (though more "periodic") beating even with pure sine waves...while the PHI version's beating seems much more steady (almost like the subtle warm beating in string section or in a synthesized ambient pad sound).

Whether the more predictable periodic beating in the harmonic series or the more "smooth" beating of the PHI series is more desirable to certain people...the point is each method has its virtues and, I'm pretty sure, the harmonic series doesn't have an all-out monopoly on consonance with 8+ tone scales.

So, I hope, more people (and not just myself, Rick, Chris, and Jacques) will dare to ask the question: can other symmetries beside that of the harmonic series be equally desirable?

-Michael

BTW, the scale used for the PHI series is
1
1.055572
1.09017
1.23607
1.3819
1.5165
1.625
1.86158
2
I choose not to use cents to describe the scale as the construction of my scale has nothing to do with the construction of 12TET (where each of the 12 notes, of course, are 100cents 200cents 300cents etc.)...and the last thing I want is people asking endless questions as to how/how-not my scale compares to 12TET.

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> Carl asked an excellent question: what is the point of
> proportionate beating (why I am admittedly somewhat obsessed
> with)?
>
> Proportionate beating is a huge force behind why I have come
> to believe fractal numbers, and not the harmonic series, are
> ideal for the rarely explored challenge of 8 to 10
> tone-per-2/1-period chords.
>
> Need proof? Try the following sound bytes and tell me what
> you think...
>
> 8-tone per octave scale made with PHI (using my
> (1/PHI)^x generator) played with a sine wave:
> [URL trimmed because of spammy domain]
>
> 8-tone per octave harmonic series fragment (8/8,9/8,10/8....)
> played with a sine wave
> [URL trimmed because of spammy domain]
>
> Notice how the harmonic series version has an almost
> mechanical (though more "periodic") beating even with pure
> sine waves...while the PHI version's beating seems much more
> steady

I hear it. As I suggested earlier today, you are saying you
dislike 'periodicity buzz'. You haven't defined "proportionate
beating", but if either example has it, it must be the harmonic
series example. While not beating in the strictest sense, it
is due to patterns of constructive and destructive interference,
all related by ratios of small whole numbers. In the phi example
the beating is very disorganized. If you're really hitting
phi-based intervals, it is, in fact maximally disorganized.

>Whether the more predictable periodic beating in the harmonic
>series or the more "smooth" beating of the PHI series is more
>desirable to certain people... the point is each method has
>its virtues

Sure. Some folks love periodicity buzz. Others don't.

>So, I hope, more people (and not just myself, Rick, Chris, and
>Jacques) will dare to ask the question: can other symmetries
>beside that of the harmonic series be equally desirable?

Actually many people have asked that question. It's been
discussed quite a bit here over the years.

-Carl

>"I hear it. As I suggested earlier today, you are saying you
dislike 'periodicity buzz'."

  Not so much dislike it as I dislike excessive amounts of it (in the same way I don't dislike beating but do dislike excessive amounts of it).
   And, even with an 8-tone per octave fragment of the harmonic series there is too much periodicity buzz for my ears...to the point of making the result feel rather mechanical to my ears in the same way an organ run through a fast high-depth amplitude modulator does. 

>"You haven't defined "proportionate beating", but if either example has it, it must be the harmonic series example. "
   Haha...well in that case it sounds like you're defining proportionate beating for me and then using that definition against my theory.  I don't see that as quite fair, does anyone else here think it is?

   BTW, compare the two sound examples.  What the harmonic series has I would coin "equivalent beating" (another way to explain periodicity buzz) meaning any two consecutive intervals sound like
they beat/fluctuate in amplitude at the exact same rate (kind of like putting a giant amplitude modulator over the entire instrument).

  "Proportionate beating", therefore, I'm attempting to coin as a term to represent beating at different rates...but rates the are related to each other (for example, in the same way as PHI related two split part of a line to each other where large / small part = large + small part / large part.  The synthesizer terminolgy for this would be putting several different types of amplitude modulators/LFO's moving at different but related rates on different frequency ranges of a stagnant sound (like an organ) in order to make it sound more like a soft string/pad sound.

>"Sure. Some folks love periodicity buzz. Others don't."
   Which is why I am trying to pose the idea of a fractal-number based scale as an alternative to the harmonic series rather than an inferior or superior to it.
   I've shown the PHI scale to several (around 12) people in comparison to the harmonic series and consistently over 40% of them actually preferred the PHI scale to the harmonic series.  You could simply say it creates "a different kind of buzz" than periodicity buzz...and that a good deal of people could prefer either type of buzz in the same way some people prefer the sound of an organ under a single amplitude modulator to a string ensemble section playing the same note and vice-versa.

   But, hopefully, it at least gives enough proof to make people ask the following question on a regular basis, are there comparably good sounding paths (on the average) to the harmonic
series?

   And if I don't have the skills to forge the path myself...the least I can do is perfect it enough so someone more expert than myself will take over and carry the torch for me, turning it into something the average musician can use.

Mike/Me>"So, I hope, more people (and not just myself, Rick, Chris, and Jacques) will dare to ask the question: can other symmetries

beside that of the harmonic series be equally desirable?"

Carl>"Actually many people have asked that question. It's been

discussed quite a bit here over the years."

   I believe it, though I haven't caught sight of much of it and would appreciate some links/references. Again, 99% of what I see here is either about JI, mean-tone tunings, MOS scales, TET tunings, or mentioning of odd non-whole-number-ratio-based scales but only for the sake of exploring dissonance (rather than consonance).  And there's nothing wrong with having a lot of expert info on such matters, of course, but it does bother me that often such topics blurt out any attempts/progress on alternatives to harmonic series symmetry. 

    In fact only recently have I had a good 4-5 people on here start to really consistantly probe the "what if?" side of if alternative symmetries can work and how (rather than the "what the he-double-hockey-sticks" response).
   I have also recently seen many people on hear create many different methods to do so (with
several of them sounding better than historic PHI scales I've tried, for example).  So far what I am learning is just how many un-turned stones there are in this process since so many of us (and not just myself) can make good original scales using it.  My point: we could make a lot more progress if many more of us, including stubborn experts like yourself, actually gave it an honest shot rather than dissing it left and right and often for reasons that seem to boil down to "it's just wrong and weird...period" without actually trying to work with the scales mentioned (be they mine, or Rick's, or Chris's, or anyone else's).

   That being said...if you could show me links to some non-whole number ratio based scales based on a goal of consonance (and/or proportionate beating, if you understand my term/definition above), I would greatly appreciate it.

-Michael

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
>> I hear it. As I suggested earlier today, you are saying you
>> dislike 'periodicity buzz'.
>
>   Not so much dislike it as I dislike excessive amounts of
> it (in the same way I don't dislike beating but do dislike
> excessive amounts of it).

In case it matters, you won't get nearly as much with acoustic
instruments playing the harmonic series as you do with digital
synthesis. You can also reduce it by microtempering -- temperaments
that have errors of around 2 cents, such as ennealimal.

>> You haven't defined "proportionate beating", but if either
>> example has it, it must be the harmonic series example.
>
>    Haha...well in that case it sounds like you're defining
> proportionate beating for me

I'm trying to help you make sense of what you have said.

> and then using that definition against my theory.

Not at all.

> I don't see that as quite fair, does anyone else here think it is?

All you have to do is define "proportionate beating" as you've
been asked to do on numerous occasions, and we can talk about it.
Since you're having so much trouble, I thought I'd try to help.
One thing I won't do is let you blast this list with pseudoscience
day in and day out without calling you on it.

>    BTW, compare the two sound examples.  What the harmonic
> series has I would coin "equivalent beating" (another way to
> explain periodicity buzz)

Explain? Coining a term doesn't explain anything. You can
call it equivalent beating or periodicity buzz -- what's the
difference?

> meaning any two consecutive intervals sound like they
> beat/fluctuate in amplitude at the exact same rate (kind of like
> putting a giant amplitude modulator over the entire instrument).

But that's not what's happening. They 'beat' at different rates,
but the zero-crossings line up. That's all. They do this because
the 'beat rates' lie in small whole number relationships
(proportions).

>   "Proportionate beating", therefore, I'm attempting to coin as
> a term to represent beating at different rates...but rates the
> are related to each other (for example, in the same way as PHI
> related two split part of a line to each other where large / small
> part = large + small part / large part.

Howabout noble beating? Or irrational beating? Or metastable
beating?

> The synthesizer terminolgy for this would be putting several
> different types of amplitude modulators/LFO's moving at different
> but related rates on different frequency ranges of a stagnant
> sound (like an organ) in order to make it sound more like a soft
> string/pad sound.

You won't get the same effect doing that.

>    Which is why I am trying to pose the idea of a fractal-number
> based scale as an alternative to the harmonic series rather than
> an inferior or superior to it.

Accepted. I never disagreed with this. Except the term fractal
number, which doesn't make much sense and doesn't appear to be
an accepted term.

>    And if I don't have the skills to forge the path myself...the
> least I can do is perfect it enough so someone more expert than
> myself will take over and carry the torch for me, turning it into
> something the average musician can use.

The average musician can use any one of the 5,000 scales in the
archive. Why should they use yours? I'll answer that for you.
Because it's an interesting scale and you think it sounds cool.
That's all you've given us so far.

>That being said...if you could show me links to some non-whole
>number ratio based scales based on a goal of consonance (and/or
>proportionate beating, if you understand my term/definition above),
>I would greatly appreciate it.

I have already. Let's start with you reading Erlich (pick any
of his three main papers, already cited numerous times for you)
and post your comments back here. After that, I'd be happy to
show you some scales again. It helps to understand them to use
them musically, which I'm also hoping you'll do.

-Carl

Carl>"The average musician can use any one of the 5,000 scales in the
archive. Why should they use yours? I'll answer that for you.
Because it's an interesting scale and you think it sounds cool.
That's all you've given us so far."

No, because >they< think it sounds cool is the main reason (and should be).
For example, already a decent handful of people here have told me my scales sound cool...not just 'interesting'.
Heck, I've posted compositions with my scales that have considerably better ratings than my 12TET ones...and I don't even mention they are micro-tonal. I certainly am not the only one who likes the sound of my scales for their own sake (regardless of the math, or even if they recognize it's micro-tonal or not).

And the main thing, perhaps, that my scales offer is the approach to making the entire scale a chord, with many different tonal colors of expression available rather than focusing on one single tone/tonality (something Mike B picked up on). This should be hugely tempting to musician because the learning curve is so low vs. other scales in the SCALA list...the problem of finding where the sweet chords are no longer exists when virtually all combinations of tones form fairly sweet chords. Get it?

Also, I am quite convince musicians aren't going to choose a scale because it fits a certain well-established mathematic theory....unless the gain in expression is higher than the hit they take for having to re-learn music theory to accommodate for the new scale.
Out of the 5,000+ scales/tunings in the SCALA archive a huge majority are going to follow or be intended to be "geared" into some sort of diatonic theory. Which, IMVHO, is why so many musicians don't care, they figure "why bother to learn something that sounds 80-90% like 12TET". The other problem is a whole lot of the scales, such as Pelog scales, are very very hard to use in chord-based music thus highly compromising degree of expression by the musician.
The ones that do work, IMVHO, include things like deca-tonic scales, 7-limit JI scales, and MOS scales (namely those up to 6-notes per octave like the one Marcus Satellite
uses). And, still, those go back to the same general feel as diatonic harmony and trade some extra degree of expression for the hassle of learning new chord theory to use them. Not exactly so easy to get "average joe musician" to use, in my opinion...although I believe there lies some good potential in 6-tone MOS for bright-sounding simple catchy pop music (it is very easy to find sweet intervals and chords in 6-note MOS, I've found, but not 7+) and deca-tonic for jazz musicians who are willing to take the time to learn it for that extra degree of chordal expression.

-Michael

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

> Out of the 5,000+ scales/tunings in the SCALA archive a huge
> majority are going to follow or be intended to be "geared"
> into some sort of diatonic theory.

I haven't done a survey of them, have you? I've sampled around
here and there it it seems like the majority of scales in there
are _not_ constructed according to what I would call a
diatonic theory.

-Carl

Can't resist to make few comments here to do a devil's advocate:

On 8 May 2009, at 8:50 AM, djtrancendance@... wrote:

> No, because >they< think it sounds cool is the main reason (and > should be).
> For example, already a decent handful of people here have told me > my scales sound cool...not just 'interesting'.
> Heck, I've posted compositions with my scales that have > considerably better ratings than my 12TET ones...and I don't even > mention they are micro-tonal. I certainly am not the only one who > likes the sound of my scales for their own sake (regardless of the > math, or even if they recognize it's micro-tonal or not).
>
Scale is just scale, and music composition is something different. You mix two different things together.

How "scale" sounds depends a lot on the way how it's used compositionally.

To play some games with scale tones can't be called a composition.

>
> And the main thing, perhaps, that my scales offer is the approach > to making the entire scale a chord, with many different tonal > colors of expression available rather than focusing on one single > tone/tonality (something Mike B picked up on). This should be > hugely tempting to musician because the learning curve is so low > vs. other scales in the SCALA list...the problem of finding where > the sweet chords are no longer exists when virtually all > combinations of tones form fairly sweet chords. Get it?
>

If it's so easy to use, then it's very limiting, and all works made from it will sound very similar.

If all tones combinations sound sweet and nice, where are your "different tonal colors"? Where's a potential to input some contrast in music if only consonance is there?
>
> Also, I am quite convince musicians aren't going to choose a scale > because it fits a certain well-established mathematic > theory....unless the gain in expression is higher than the hit they > take for having to re-learn music theory to accommodate for the new > scale.
>
If there's some math in the way how scale (tuning) was created, then there's a possibility to derive from it also the other musical parameters necessary for composition. And always it's possible to create new theories. Not necessarily it must have anything to do with traditional theory.

Or it can be used freely, without any theory if somebody prefers.

> Out of the 5,000+ scales/tunings in the SCALA archive a huge > majority are going to follow or be intended to be "geared" into > some sort of diatonic theory. Which, IMVHO, is why so many > musicians don't care, they figure "why bother to learn something > that sounds 80-90% like 12TET". The other problem is a whole lot of > the scales, such as Pelog scales, are very very hard to use in > chord-based music thus highly compromising degree of expression by > the musician.

"Chord based music" is only one of many possibilities how music can look.

And why not to use pelog in chords? Yes, they will be out of tune, that's all. This is nothing wrong on it if composer wants it.
> The ones that do work, IMVHO, include things like deca-tonic > scales, 7-limit JI scales, and MOS scales (namely those up to 6-> notes per octave like the one Marcus Satellite
> uses). And, still, those go back to the same general feel as > diatonic harmony and trade some extra degree of expression for the > hassle of learning new chord theory to use them. Not exactly so > easy to get "average joe musician" to use, in my opinion...although > I believe there lies some good potential in 6-tone MOS for bright-> sounding simple catchy pop music (it is very easy to find sweet > intervals and chords in 6-note MOS, I've found, but not 7+) and > deca-tonic for jazz musicians who are willing to take the time to > learn it for that extra degree of chordal expression.
>
> -Michael
>
Do you really think an average musician will and should use microtones? Why? He has no reason. Nothing to say about pop or jazz. They don't need it, it's not part of those styles and in my opinion never will be. Also instruments which they use can't make microtones perfectly.

Besides my opinion is that microtones (except all historical or ethnical scales) are logical future development of contemporary music, and should be used only by experienced musicians who know quite well all the past and recent music theory and want to continue in new ways and refine further their music. There's no reason why some unskilled person should polute the world with his/her poor compositional attempts with microtones, when he/she is not able to work in traditional systems.

Daniel Forro

>"Scale is just scale, and music composition is something different.
You mix two different things together."

True enough, but making a scale succeed in a polyphonic composition requires at least a fairly good degree of ability for consonance built into the scale, right?
The songs weren't posted on some funky avant-garde site, but a reputable music site with a majority of the music on there being non-experimental. And it still managed 4.5+ of 5 rating.
And true, composition matters, but it's very rare someone can "compose their way out" into making an only decent scale sound beautiful...and I certainly am no Debussy so far as compositional skill.

>"If it's so easy to use, then it's very limiting, and all works made
from it will sound very similar."
>"If all tones combinations sound sweet and nice, where are your
"different tonal colors"?

I'll have to test against that assumption but, especially following from Mike B's statement that my PHI scales seem to have a huge degree of tonal color, I highly doubt that will be the case. I know easy to use often means less tonal color...such as it does in 5TET where it's virtually impossible to make very sour chords...but that's a very narrow 5-tone scale and this is a fully fledged 8-tone scale (an extra tone over diatonic).

>" Where's a potential to input some contrast in music if only consonance is there?"
Number of notes used, amplitude of certain notes....these can both be used to control the degree of tension. Not to mention rhythm and staccato, as always, can be used to build tension.
>"If there's some math in the way how scale (tuning) was created, then
there's a possibility to derive from it also the other musical
parameters necessary for composition."

You still have parameters, such as those mentioned above. And you still have the large palette of tonal colors. Only the whole "math of finding non-sour chords" problem is virtually eliminated so you can concentrate on other, often more exciting musical parameters instead.

>"And why not to use pelog in chords? Yes, they will be out of tune, that's all. This is nothing wrong on it if composer wants it."
There is nothing wrong with it. But, the flip side is, don't expect many major artists to be able to get away with it. Even quirky virtuoso ones like Leo Kottke. I try to keep one ear into personal expression and the other into making something the average musician will want to make their music with as well.

>"and should be used only by experienced musicians who know
quite well all the past and recent music theory..there's no reason why
some unskilled person should polute the world with his/her poor
compositional attempts with microtones, when he/she is not able to
work in traditional systems."

Ugh...now that is pure elitism. You know (sadly), most 12TET musicians will actually say the same thing about micro-tonal, usually phrased as something like "why are you polluting our beautiful tonal music with your un-emotional a-tonal micro-tonal crap and then showing us mathematical equations about why we should have to like it?" And both sides can do a very intelligent job of proving the other side is wrong and both end up looking like fools.
My point: the whole situation of traditional musicians dissing micro-tonal ones as not advanced enough and vice-versa is like the eye for an eye that makes the whole world blind...

-Michael

Look Mike, I'm just gonna put it out there... Why are you so intent on
arguing with people? Your scale sounds great. It can be used to make
music that sounds good. Why don't you just write some music and leave
it at that?

I have my own ideas why your scales sound good. And I don't know if I
agree with your hypotheses as to why your scale works, to be honest.
Nonetheless, is the important thing here not to be making music? As
soon as one person says some sort of humorous comment at your expense
you write these huge replies... Who cares?

-Mike

On Thu, May 7, 2009 at 11:06 PM, <djtrancendance@...> wrote:
>
>
>
>>"Scale is just scale, and music composition is something different.
> You mix two different things together."
>
> True enough, but making a scale succeed in a polyphonic composition requires
> at least a fairly good degree of ability for consonance built into the
> scale, right?
> The songs weren't posted on some funky avant-garde site, but a reputable
> music site with a majority of the music on there being non-experimental. And
> it still managed 4.5+ of 5 rating.
> And true, composition matters, but it's very rare someone can "compose their
> way out" into making an only decent scale sound beautiful...and I certainly
> am no Debussy so far as compositional skill.
>
>>"If it's so easy to use, then it's very limiting, and all works made
> from it will sound very similar."
>>"If all tones combinations sound sweet and nice, where are your
> "different tonal colors"?
>
> I'll have to test against that assumption but, especially following from
> Mike B's statement that my PHI scales seem to have a huge degree of tonal
> color, I highly doubt that will be the case. I know easy to use often means
> less tonal color...such as it does in 5TET where it's virtually impossible
> to make very sour chords...but that's a very narrow 5-tone scale and this is
> a fully fledged 8-tone scale (an extra tone over diatonic).
>
>>" Where's a potential to input some contrast in music if only consonance is
>> there?"
> Number of notes used, amplitude of certain notes....these can both be used
> to control the degree of tension. Not to mention rhythm and staccato, as
> always, can be used to build tension.
>>"If there's some math in the way how scale (tuning) was created, then
> there's a possibility to derive from it also the other musical
> parameters necessary for composition."
>
> You still have parameters, such as those mentioned above. And you still have
> the large palette of tonal colors. Only the whole "math of finding non-sour
> chords" problem is virtually eliminated so you can concentrate on other,
> often more exciting musical parameters instead.
>
>>"And why not to use pelog in chords? Yes, they will be out of tune, that's
>> all. This is nothing wrong on it if composer wants it."
> There is nothing wrong with it. But, the flip side is, don't expect many
> major artists to be able to get away with it. Even quirky virtuoso ones like
> Leo Kottke. I try to keep one ear into personal expression and the other
> into making something the average musician will want to make their music
> with as well.
>
>>"and should be used only by experienced musicians who know
> quite well all the past and recent music theory..there's no reason why
> some unskilled person should polute the world with his/her poor
> compositional attempts with microtones, when he/she is not able to
> work in traditional systems."
>
> Ugh...now that is pure elitism. You know (sadly), most 12TET musicians will
> actually say the same thing about micro-tonal, usually phrased as something
> like "why are you polluting our beautiful tonal music with your un-emotional
> a-tonal micro-tonal crap and then showing us mathematical equations about
> why we should have to like it?" And both sides can do a very intelligent job
> of proving the other side is wrong and both end up looking like fools.
> My point: the whole situation of traditional musicians dissing micro-tonal
> ones as not advanced enough and vice-versa is like the eye for an eye that
> makes the whole world blind...
>
> -Michael
>
>