Yahoo Tuning Groups Ultimate Backup tuning perceived consonance, and Kolinski's stretch tuning

KOLINSKI'S NON-'8ve' EQUAL-TEMPERED 'STRETCH TUNING'
----------------------------------------------------

Mieczyslaw Kolinski, in 'A New Equidistant 12-Tone Temperament'
(Journal of the American Musicological Society, v 12, #2-3,
summer-fall 1959, p 210-214), proposes a non-8ve equal-temperament
calculated with a basic step-size of (3/2)^(1/7), which gives
'perceived 8ves' of ~1203.35142587 cents.

Kolinski gives the decimal part of the '8ve' cents-value
as an rational approximation (i.e., fraction), presumably
for ease in mental calculation (not many general readers
had calculators or computers in 1959).

His figure of 1203&3/7 cents (~22% error) arises from his
rounding of the Pythagorean comma to 24 cents. A far better
rational approximation of the cents value is 1203&13/37 cents
(0.02% error), and a good one with low integer-limit and low
prime-limit in the terms of the fraction (and thus a bit easier
to calculate mentally) is 1203&7/20 cents (0.4% error). Finally,
1203&1/3 cents (5.1% error) is both much more accurate and also
much easier to calculate than Kolinski's figure.

Kolinski justifies this tuning as follows:

>
> ... a basic aural phenomenon challenges this axiomatic concept
> of the integrity of the ratio 1:2: acoustical tests on the
> evaluation of the octave [*2] have disclosed that the vibration
> ratio 1:2 is perceived as a slightly flat octave and that a
> widening of the interval corresponding to about two vibrations
> per second in the middle register is necessary in order to
> arrive at the perception of a perfect octave. Further tests
> undertaken by Otto Abraham and Erich von Hornbostel [*3] have
> shown that the correction required for the double octave is
> larger than that for the single one. This is not surprising,
> for if the adjustment needed for one octave equals _x_, one
> might assume that the correction needed for the double octave
> has the value _2x_, and that this value increases with the
> increasing number of octave enlargements.
>
>
> [*2] Carl Stumpf and Max Meyer, "Massbestimmungen u"ber die
> Reinheit konsonanter Intervalle", _Beitra"ge zur Akustik und
> Musikwissenschaft_ II (Leipzig, 1898), p. 84.
>
> *[3] Von Hornbostel, personal communication to the author.
>

Is anyone familiar with these references? Or any more modern
experiments in the same vein? I'd like to have more exact
data on the perception of consonance; Kolinski here is rather
vague... 'a widening of the interval corresponding to about
two vibrations per second in the middle register' doesn't
quite give me enough to go on.

MY CALCULATION OF THE STRETCHED 'PERCEIVED 8ve'
-----------------------------------------------

I've calculated some figures on the assumption that by 'middle
register' Kolinski means the center of the audible range of
hearing. Of course, these frequencies vary for each individual
and during a lifetime. I've pretty much just taken very rough
approximate frequency-limits 'from the air'.

If I assume 20000 Hz as the top limit _t_ of hearing, and
10 '8ves' (measured as 2/1 ratios) below it, 19&17/32 Hz, as
the bottom limit _b_ (these numbers should be adjusted according
to the more accurate data I'm seeking!), the exact midpoint _m_
is 625 Hz:

m = (SQRT(t/b))*b

Now if I take that 625 Hz _m_ as the midpoint (i.e., 'tritone')
of the central '8ve', I can calculate the top and bottom limits
of that central '8ve' (measuring the '8ve' here first as a 2/1)
as the SQRT(2) above and below _m_ respectively. So I the top
note of the central '8ve' is 625*(SQRT(2)) = 883&53/60 Hz, and
the bottom note is 625/(SQRT(2)) = 441&113/120 Hz.

According to Kolinski:

> ratio 1:2 is perceived as a slightly flat octave and that a
> widening of the interval corresponding to about two vibrations
> per second in the middle register is necessary in order to
> arrive at the perception of a perfect octave.

So I add 1 Hz to the top frequency, and subtract 1 Hz from the
bottom frequency, to keep the midpoint of the central *perceived*
'8ve' at 625 Hz. This gives ~884&53/60 Hz for the top note and
~440&113/120 Hz for the bottom note of the perceived 'perfect'
central '8ve'. This interval is ~1205.879335 cents (about
1205&7/8 cents), which is a little wider than Kolinski's '8ve',
but somewhat close to it.

MY CALCULATION OF STRETCHED 'PERCEIVED BASIC JI RATIOS'
-------------------------------------------------------

*If* one assumes that the stretching function for this 'audible
consonance' is constant (but there's no reason to assume that!
... let's see that accurate data!!), this result yields a
constant stretch factor or ratio of ~1.0034014 = ~295/294.

This coefficient multiplied by the standard basic low-integer
JI ratios gives perceived perfect consonances as follows:
(== means 'equivalent [not equal] to)

'8ve' == 2/1 == ~1206 cents
'perfect 5th' == 3/2 == ~ 708 cents
'major 3rd' == 5/4 == ~ 392 cents
'harmonic 7th' == 7/4 == ~ 975 cents

All figures are subject to further adjustment when better
experimental data becomes available.

Also, what about the timbres of the sounds used in these
experiments? Wouldn't they have a dramatic effect on the
size of intervals perceived as consonant?

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Joseph Pehrson <josephpehrson@compuserve.com>

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

🔗Herman Miller <hmiller@IO.COM>

On Sat, 23 Sep 2000 09:25:18 -0000, " Monz" <MONZ@JUNO.COM> wrote:

>KOLINSKI'S NON-'8ve' EQUAL-TEMPERED 'STRETCH TUNING'
>----------------------------------------------------
>
>Mieczyslaw Kolinski, in 'A New Equidistant 12-Tone Temperament'
>(Journal of the American Musicological Society, v 12, #2-3,
>summer-fall 1959, p 210-214), proposes a non-8ve equal-temperament
>calculated with a basic step-size of (3/2)^(1/7), which gives
>'perceived 8ves' of ~1203.35142587 cents.

What a coincidence! I was playing around with this exact scale just the
other day. I actually generated the scale the hard way, by taking a series
of perfect fifths and reducing the scale by a 1/7-Pythagorean comma
sharpened octave. (I've been experimenting with various combinations of
tempered octaves and fifths.) Later I realized that it was actually an
equal scale (and wondered why I didn't figure that out in the first place).

With the octaves sharpened by 1/7-Pythagorean comma, you can flatten half
the fifths by some amount and sharpen the rest of the fifths by the same
amount for a stretched scale that has properties of a well-tempered scale.
The advantage of stretching the octaves is that you don't need to flatten
the fifths as much in order to improve the thirds.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/music.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Joseph Pehrson" wrote:
> http://www.egroups.com/message/tuning/13324
>
> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
> >
> > http://www.egroups.com/message/tuning/13319
>
> Perhaps, then, music theory will have to be "rewritten" since with
> "stretched octaves" the harmonics of two pitches do not exactly
> coincide... so, therefore, "parallel octaves" shouldn't be quite so
> onerous (??)

Joe, it's a very complicated situation. Most experiments on
psychoacoustics have been done with pure sine tines, which only
a very few composers like to use for actual music (list-member
David Beardsley is one of those few).

For most actual compositons, the timbres are far more complex,
and the way people react to intervals played in those timbres
hasn't been studied anywhere near as much. Sethares probably
has done the most work in this area.

But a more important point to make is that mathematics and science
may never be able to uncover *everything* about musical perception.
Perception of music is entirely bound up with the fact the we
humans are extremely complicated living creatures, with ear/brain
systems that respond to stimuli not only in specifically
quantifiable ways that have a physiological basis (Helmholtz was
the one who really got the ball rolling on that), but also
in ways that are based on past learning and experience, on
cultural and societal norms, on individual personal expectations
and inherent natural abilities, etc. etc.

I think in a lot of ways that we here on this list are making
some of the most valuable contributions to 'rewriting' music-
theory. Intonation is an aspect of music-making that was largely
ignored over the past 100 years of theorizing, with the glaring
exception of Partch's theoretical work (and others of course,
but Partch stands out like a BIG 'sore thumb' - and I think
he liked it that way!).

A lot of theorizing that still goes on in the 'standard' music
journals and halls of academia, and has been going on for
millenia, is based on assumptions about hearing that are plainly
and simply *not true*. The entire edifice of 12-tET serial
theory is based on the assumption of '8ve'-equivalence, so if
it turns out that this is not an accurate reflection of what
we hear, then how much is that theory really explaining?

I'm not saying that there is no value in that theory at all;
I'm an avid reader of those journals, and the theories do provide
mathematical constructs that can have interesting applications
in composition. But these theories are *not* often accompanied
by that kind of a disclaimer, and instead are purported to
represent the 'truth' about musical perception.

We've talked a bit here in the past about the discrepancy between
proscriptive and descriptive music-theory. Much (probably
the vast majority, actually) of past theory was proscriptive,
and a lot of current academic theory is too. But a lot of
what microtonalists are doing is descriptive, based on analysis
of actual performances of actual pieces - at least, this is
how I try to ground my work. See my analysis of a Robert Johnson
blues vocal for an example:
http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm

So yes, I think music-theory will have to be rewritten, and it
is already in the process of happening. In general, what I expect
we'll find is that there were important societal/cultural reasons
why composers and listeners felt it imperative to follow certain
'rules' in their music, and that these rules are forever in a
process of evolution, because so is human society.

A book that I can't recommend enough, which will open your eyes
to a lot of what I'm saying here, is Richard Norton, _Tonality
in Western Culture_, ISBN# 0-271-00359-6. It attempts to present
a Marxist view of the evolution of tonal perception from the
ancient Greeks thru Schoenberg and his followers, with the
emphasis on the *subjective*.

Another good one is Eugene Narmour's _Beyond Schenkerism_,
ISBN# 0-226-56848-2. This book presents Narmour's 'implication-
realization' model of musical perception, which I personally find
to be *far* more relevant to the way I actually perceive musical
compositions than anything in the millions of words Schenker and
his followers ever wrote.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗M. Edward Borasky <znmeb@teleport.com>

> -----Original Message-----
> From: Monz [mailto:MONZ@JUNO.COM]
> Sent: Sunday, September 24, 2000 11:12 AM
> To: tuning@egroups.com
> Subject: [tuning] Re: perceived consonance, and Kolinski's stretch
> tuning

> Joe, it's a very complicated situation. Most experiments on
> psychoacoustics have been done with pure sine tines, which only
> a very few composers like to use for actual music (list-member
> David Beardsley is one of those few).

I like "pure" square waves (odd harmonics only, very clarinet-like) myself.
I also like rectangular waves, which contain all harmonics. These happen to
be easy to generate on a computer, have a "fast" transform (the Fast Walsh
Transform, if you care :-) and theoretically can produce any sound. I had
quite a few digital instruments based on these principles "on the drawing
boards" in (and before) the days when microprocessors were slow, had
eight-bit registers and 64 Kbyte address spaces and were programmable only
with hex keypads. "Conventional" additive synthesis is now feasible at
reasonable cost, so these little toys of mine will remain on the drawing
board, I suppose :-).

> For most actual compositons, the timbres are far more complex,
> and the way people react to intervals played in those timbres
> hasn't been studied anywhere near as much. Sethares probably
> has done the most work in this area.

The basis of Sethares' work is the Plomp - Levelt studies of *non-musical*
listeners using pure sine waves. Sethares builds his theories on this
foundation.

> But a more important point to make is that mathematics and science
> may never be able to uncover *everything* about musical perception.
> Perception of music is entirely bound up with the fact the we
> humans are extremely complicated living creatures, with ear/brain
> systems that respond to stimuli not only in specifically
> quantifiable ways that have a physiological basis (Helmholtz was
> the one who really got the ball rolling on that), but also
> in ways that are based on past learning and experience, on
> cultural and societal norms, on individual personal expectations
> and inherent natural abilities, etc. etc.

A great deal of work has been done in this area. I can recommend Lerdahl and
Jackendoff's "Generative Theory of Tonal Music"

http://www1.fatbrain.com/asp/bookinfo/bookinfo.asp?theisbn=026262107X

and Schwanauer / Levitt's "Machine Models of Music"

http://www1.fatbrain.com/asp/bookinfo/bookinfo.asp?theisbn=0262193191

Music is a popular area of investigation in cognitive psychology precisely
because of the fact that "we humans are extremely complicated living
creatures"; by limiting one's study to music, one, in the words of Dijkstra,
"limits the amount of detailed reasoning required to a doable amount".
--
M. Edward Borasky
mailto:znmeb@teleport.com
http://www.borasky-research.com

🔗M. Edward Borasky <znmeb@teleport.com>

🔗Alison Monteith <alison.monteith3@which.net>

Monz wrote:

> I think in a lot of ways that we here on this list are making
> some of the most valuable contributions to 'rewriting' music-
> theory.

You most certainly are and I hope that the work will go from strength to strength in the face
of resistance, indifference and possibly fear from some conventional quarters of the musical
establishment in my neck of the woods. When I talk of my interest in tuning matters to
established professionals in Edinburgh they often look at me as if I had two heads. In these
situations I thank God for the integrity of the research and dedication of the researchers in
our field.

> We've talked a bit here in the past about the discrepancy between
> proscriptive and descriptive music-theory. Much (probably
> the vast majority, actually) of past theory was proscriptive,
> and a lot of current academic theory is too. But a lot of
> what microtonalists are doing is descriptive, based on analysis
> of actual performances of actual pieces - at least, this is
> how I try to ground my work.

My feeling is that many theoreticians of the past began by describing current practice and
rather quickly turned to telling us how it should be. This is a prominent feature of Medieval
theoristetical practice.Much later the unfortunately named Mr. Fux began by describing the
contrapuntal technique of Palestrina and others, though at the same time he urged his readers
to follow the 'rules' of counterpoint that he had extracted. And many others since, Piston
comes to mind, have made fine careers out of telling us what Bach did (and what he did next)
and exhorting us to follow his example as far as possible. This is very educational. However,
the problem we face is that the wonderful JI/microtonal repertoire is small and often
inaccessible (I write as a European) . We need more to describe. Then one day our theorists
might prescribe and the wheel will turn once more, schools will spring up and young people
will learn the wonders of JI/microtonal music.

> So yes, I think music-theory will have to be rewritten, and it
> is already in the process of happening. In general, what I expect
> we'll find is that there were important societal/cultural reasons
> why composers and listeners felt it imperative to follow certain
> 'rules' in their music, and that these rules are forever in a
> process of evolution, because so is human society.

Change is the only unchanging principle.

🔗phv40@hotmail.com

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
>
> > Is anyone familiar with these references? Or any more modern
> > experiments in the same vein?
>
> Yes, I've seen experiments published that seem to
> suggest that for sine waves, 1209 cents is perceived as
> an in-tune octave; but for timbres with harmonic
> partials, the partials become more important. That's
> why you don't see stretch tuning proposed for horns,
> reeds, or bowed strings but you do see it for pianos
> (and Chapman Sticks).

Wow, I was about to ask this: Why do I hear of stretch tuning for
pianos and attempts to do the same on Stick and other fretted
instruments (e.g. The Novak fretboard for fretted 12t-ET guitars and
bass guitars) but not for other pitched instruments such as winds and
bowed strings?

Thanks for answering my question before I even asked it! :)

Monz, I also want to thank you for your responses which I have
forwarded to my friend. He lives in the San Diego area too, so who
knows, he might actually try to contact you. He's been asking
everyone he knows about stretch tuning. :)

Paolo

🔗David Beardsley <xouoxno@virtulink.com>

phv40@hotmail.com wrote:

> --- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> >
> > > Is anyone familiar with these references? Or any more modern
> > > experiments in the same vein?
> >
> > Yes, I've seen experiments published that seem to
> > suggest that for sine waves, 1209 cents is perceived as
> > an in-tune octave; but for timbres with harmonic
> > partials, the partials become more important. That's
> > why you don't see stretch tuning proposed for horns,
> > reeds, or bowed strings but you do see it for pianos
> > (and Chapman Sticks).
>
> Wow, I was about to ask this: Why do I hear of stretch tuning for
> pianos and attempts to do the same on Stick and other fretted
> instruments (e.g. The Novak fretboard for fretted 12t-ET guitars and
> bass guitars) but not for other pitched instruments such as winds and
> bowed strings?

I think Stretch tunings work on pianos (strings like
metal bars because of tension) but not an instrument
like the Stick, Bass or Guitar.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

🔗Can Akkoc <akkoc@asms.net>

🔗phv40@hotmail.com

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
> phv40@h... wrote:
> > Wow, I was about to ask this: Why do I hear of stretch tuning for
> > pianos and attempts to do the same on Stick and other fretted
> > instruments (e.g. The Novak fretboard for fretted 12t-ET guitars
and
> > bass guitars) but not for other pitched instruments such as winds
and
> > bowed strings?
>
> I think Stretch tunings work on pianos (strings like
> metal bars because of tension) but not an instrument
> like the Stick, Bass or Guitar.

Yet, for some reason, my Stick playing friend _and_ his guitarist
bandmate were both compelled by their perception of intonation on the
highest and lowest strings of their respective instruments to attempt
just that. :)

So far my friend seems to be happier with the pseudo stretch-tuning
of his Stick. I reminded him that a literal stretch tuning would
move the frets in such a way that the melody side of his Stick would
look like a Ralph Novak guitar fretboard (the infamous "fanned-fret"
look).

What this has got me wondering now is what if my buddy's Stick was
31t-ET instead of 12t-ET? Would he still be compelled to attempt
stretch-tuning on it?

Paolo

perceived consonance, and Kolinski's stretch tuning

🔗Monz <MONZ@JUNO.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

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

🔗Herman Miller <hmiller@IO.COM>

🔗Monz <MONZ@JUNO.COM>

🔗M. Edward Borasky <znmeb@teleport.com>

🔗M. Edward Borasky <znmeb@teleport.com>

🔗Alison Monteith <alison.monteith3@which.net>

🔗phv40@hotmail.com

🔗David Beardsley <xouoxno@virtulink.com>

🔗Can Akkoc <akkoc@asms.net>

🔗phv40@hotmail.com

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