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RE: Basic dimensionality, and related issues

🔗"Paul H. Erlich" <PErlich@...>

7/14/1997 2:22:32 PM
Graham wrote,

>>On dissonance of composites in general: it is much easier to form
>> consonant chords using composite numbers than prime numbers of
>> about the same size. As an octave invariant example, 1:3:5:15
>> includes the interval 15/1. Try finding an equally consonant 4
>> note octave invariant chord including 13/1. If the consonance of
>> an interval reflects its propensity to form consonant chords, then
>> composites are more consonant than primes and, in some
>> circumstances, the metric should handicap high primes accordingly.
>
>How does your metric accomplish this?
>
>The fact that 15/1 comes up in almost fully consonant chords of the 5-limit,
>while 13/1 does not, is already enough to handicap 13. Specifically, the
>chord 1:3:5:15 has a nice compact appearance in the 5-limit lattice, while no
>chord with 13 in it can be found there at all. (BTW, I think 3:5:9:15 is at
>least as consonant as 1:3:5:15 -- the triangular lattice gives them the same
>shape, while the rectangular lattice appears to treat 1:3:5:15 as simpler).
>
>The consonance of a chord should not be judged solely by the most complex
>interval -- all intervals should be looked at, as well as issues of virtual
>pitch and combination tones. If I was suggesting that the most complex
>interval alone was enough, I apologize profoundly and take it back.
>
>I admit that the metric I was proposing is not very useful if carried over
>from intervals to chords. I don't think yours does any better in that
>respect, though. I don't think the lattice concept is appropriate for
>quantifying the consonance of entire chords, though it might be worthwhile to
>make a few attempts. I think the appropriate measures for chord consonance
>are a combination of the existing roughness algorithms (Plomp&Levelt,
>Kameoka&Kuriyagawa, Sethares) with a "tonalness" algorithm. I have a simple
>version of the latter -- a better version awaits my very intermittent work on
>a concept I call "harmonic entropy."
>
>Anyway, good point Graham, but I wonder how you answer it yourself. BTW, I
>think the consonance of an interval has a meaning different from, and without
>perfect correlation with, the propensity of that interval to occur in
>generally consonant chords.
>
> > > By fibonacci, what I'm referring to is J. Yasser's fibonacci-like
>> >sequence of tunings (5 7 12 19 31 50 81...). (By fibonacci-like I mean
>> >that the next number in the sequence is the sum of the previous two.) 12
>
>> Firstly, wasn't it Kornerup who first identified this sequence?
>
>Sure, but in a very different context. Kornerup identified a subset of the
>ETs consistent with meantone notation and composition and did not include 5
>in the sequence. Yasser proposed that an evolution of tonality takes place
>where at each stage, a given member of the sequence is the number of diatonic
>tones, the previous member is the number of chromatically altered tones, and
>thus the total number of pitches is the next member of the sequence. After
>tonality and atonality are exhaused in a given system, the total set of
>pitches is retuned unequally, and music progresses one Secondly, there are
>lots of Fibonacci-like scale sequences.
>Fibonacci+3, for a start. The special thing about this
>sequence is that the number of steps in a tone and semitone, or
>tone and minor third, or minor third and perfect fourth, are
>neighbouring steps in the Fibonacci series. Another meantone
>type ET is 43TET. 12+31C. Hey ...
>
>------------------------------
>
>Topic No. 4
>
>Date: Sat, 12 Jul 1997 12:40:21 -0400
>From: "Andrew L. Kaye"
>To: tuning@eartha.mills.edu
>Subject: Re: Carlsberg Beer
>Message-ID: <33C7B374.B893DD9F@fast.net>
>
>
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>Daniel Wolf writes:
>In my experience, existing psychophysical theories are either weak or
>culturally local.
>(There are some minor innate behavior mechanisms associated with sounds
>- my favorite
>example is the taste of Carlsberg Beer while listening to particular
>sine wave frequencies
>- but the culturally aquired patterns seem to be overriding; see, for
>example,
>Gilbert Rouget _Music and Trance_).
>
>What is der Karlsberg Effekt? Source?
>
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>
>Daniel Wolf writes:
>
In my experience, existing psychophysical theories are either weak
>or culturally local.
>
(There are some minor innate behavior mechanisms associated with
>sounds - my favorite
>
example is the taste of Carlsberg Beer while listening to particular
>sine wave frequencies
>
- but the culturally aquired patterns seem to be overriding; see,
>for example,
>
Gilbert Rouget _Music and Trance_).
>
>

What is der Karlsberg Effekt? Source?
>
>--------------466ADBE552939739789B3CF7--
>
>------------------------------
>
>Topic No. 5
>
>Date: Sat, 12 Jul 1997 09:48:24 -0700 (PDT)
>From: John Chalmers
>To: Alternative Tuning List
>Subject: ET, Limits
>Message-ID:
>
>Gary: I think the context makes it clear that I was talking about equal
>temperaments and contrasting closed cyclic systems to open, infinite ones.
>
>Partch defined limit implicitly and used it in the titles of two chapters
>(Chapter Seven, Analysis of the 5 Limit, and Chapter Six, Application
>of the 11 limit). However, I can find no place where he explicitly
>defines it, though it is clear from context (to me, at least) that the
>Prime Limit of a tuning is determined by the highest prime number used to
>define ratios in that tuning. Partch does use 9 and mentions 15 as odd
>numbers
>which define ratios, but does not use them to define Limits. He appears
>to consider ratios such as 9/8, 81/64, 27/16 and their inversions to be
>at the 3 limit.
>
>One of the problems with HP's exposition is his technical vocabulary.
>"Identity" as in the definitions of ratios, e.g., "Ratios of 9: those
>ratios with identities no larger than 9, in which 9 is present: 9/8,
>16/9, 9/5, 10/9, 9/7, 14/9" is not very meaningful as few people would
>consider 9 as identical to 1 or any other number (except under the modulo
>N operation). What he means is "odd multiple" and he carefully
>distinguishes his concept of identity from that of partial or _"ingredient
>of Harmonic Content"_.
>
>Calling the identities "correlatives" does not make the concept
>appreciably clearer, and he defines the correlatives as the
>set 1, 3, 5, 7, 9, 11 ....
>
> He further defines Odentity and Udentity according to whether the
>odd number appears in the numerator (over numbers) or denominator.
>
>At least in Genesis of a Music, HP seems to have restricted the term Limit
>to prime numbers, though odd numbers serve as "Identities" in
>defining ratios.
>
>The definition of a Ratio of N is analogous to that of 9 given above.
>Replace 9 by N in the first portion and list all ratios in the tuning
>containing N, but no larger prime, as a factor. It is not necessary to
>assume octave equivalence, though Partch does as he uses ratios both as
>a labels for tones of his scale and for the relation to a 1/1. Two is thus
>not considered a ratio defining number in his theory.
>
>I can only say that with exposure, his technical vocabulary becomes
>clearer.
>
>Since the LCM is a function of all prime factors, including 2, and
>their powers, the concept of LCM and Prime Limit are very different.
>All inversions of both the major and minor triads are at the 5 Limit as
>5 is the largest Prime Number that appears in any of them (1:3:5, 4:5:6,
>5:6:8, 6:8:10, 10:12:15, 12:15:20, 15:20:24, /1:/3:/5 etc.).
>
>--John
>
>------------------------------
>
>Topic No. 6
>
>Date: Sat, 12 Jul 1997 15:25:07 -0400
>From: Daniel Wolf
>To: "INTERNET:tuning@eartha.mills.edu"
>Subject: Re: Carlsberg Beer
>Message-ID: <199707121525_MC2-1AB9-D028@compuserve.com>
>
>According to Craig Adcock, in _James Turrell: The Art of Light and Space_>,
>an experiment carried out by Kristian Holt-Hansen in 1968 involving ''Tas>te
>and Pitch'' tested how the tastes of Carlsberg Lager and Carlsberg Elepha>nt
>beer were affect by sound: at 650 Hz. for the Elephant and 10-15 Hz. for
>the Lager, the beer tasted distinctly better. I believe that L. Wechsler
>gives a different version of the results in his monograph on Robert Irwin>. >
>
>I have no idea if or how the taste of Carlsberg Beer has changed over the>
>years (and personally, I stick to stouts and Roggenbiers), but I anxiousl>y
>await verification or revision of the results present above. A serious ta>sk
>for tuning scholars, at last!
>
>------------------------------
>
>Topic No. 7
>
>Date: Sat, 12 Jul 1997 23:26:36 -0400 (EDT)
>From: Joseph Downing
>To: tuning@eartha.mills.edu
>Subject: Re: Perishable tunings and other permutations?
>Message-ID:
>
>On Sat, 12 Jul 1997, Charles Lucy wrote:
>> All this chat about meantone tuning is very interesting, yet it
>> would seem very difficult to affirm the details of which tunings
>> were used on keyboard instruments, as the tuning is perishable.
>> Does anyone have access to physical fretted instruments from the
>> periods in question, so that fret, nut and bridge distances may
>> be measured?
>
>Of course, the other instrument that if fairly stable in tuning is the
>organ. If pipes are 'dead-cut' to 'correct'length,(which was not uncommon
>in earlier times, and not uncommon now amongst better builders) then it is
>possilbe to get a fair idea of the builders ideas about tuning. In fact,
>a lot of the info about tuning does come from organs, especially organs in
>poor areas, where the resources were not available to 'up-date' the organ
>to the latest fancy.
>
>Joe Downing,
>in Syracuse
>
>------------------------------
>
>End of TUNING Digest 1132
>*************************

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🔗Carter Scholz <csz@...>

7/14/1997 3:04:33 PM
I've been following the thread on harmonic spaces and metrics with
interest, and although it's been a while since I've done any work in
this area I'd like to comment briefly.

I've found it somewhat fruitful to think of any given harmonic metric --
like any given tuning -- as one way of hearing among many ways.

In fact I'm skeptical that there can be any such thing as a single
metric (in the strict mathematical sense) for "consonance" or
"concordance" or "harmonicity", because these concepts are so highly
contextual. I don't believe that ANY metric can state categorically
that a 15/1 is more or less "consonant" than a 5/3, or a 15/8 than a
16/15, or a root-position major triad than a second-inversion triad, or
any such comparison, without carefully circumscribing its assertion by a
precise definition of what "consonance" means in its context. (Tenney's
_History of 'Consonance' and 'Dissonance'_ is a valuable look at how
notions of "consonance" have changed over the centuries in European art
music.) In designing a metric, there is an understandable tendency to
shuttle between number and intuition, to say "my ears tell me..." about
this quality, while also striving for quantitative objectivity.

Which is NOT to say that the quality can't be measured, or that the
design of metrics is therefore futile. But it does seem inevitable that
any given metric will necessarily contain highly subjective decisions --
e.g. how to weight the various dimensions, whether or not it's octave
invariant, whether the limit-system is based on primes or odd numbers or
odd composites, et cetera -- decisions that carry with them theoretical
assumptions about the nature of harmony and tonality, tending to
radically simplify and somewhat falsify the processes by which we hear
and interpret.

I believe that a somewhat fuller and more useful description of
harmonicity might be reached by using a GROUP of metrics, each
descriptive of some particular aspect of pitch and harmonic space,
instead of seeking a single "best" metric for a quality as multivalent
and contextual as "consonance".

Various metrics based on small-number-ratio theories of consonance have
been discussed extensively on the list. I would like to see more
discussion of what I consider "problem areas", the first two of them
psychoacoustic, the third a more general agenda:

1) Critical bandwidth. Though there has been much discussion of octave
invariance, there has been little of critical bandwidth, which requires
that we bring absolute frequency, not just interval ratios, into any
metric that claims psychoacoustic validity. (Sethares's extension of
Plomp-Levelt does this, and takes steady-state timbre into account as
well.)

2) Non-integers/approximations. Small-number-ratio metrics fail with
irrationals. E.g. 300001/200000 is much more "dissonant" than 3/2;
indeed, the closer the approximation, the more distant it's ranked by
any small-number metric. The ear has a much more adaptable threshold.
Some metric that describes near-misses could be quite useful. (Again,
by incorporating critical bandwidth, Sethares's metric avoids this
particular pitfall. Tenney alludes to, but does not incorporate, a
"tolerance range" in his harmonic distance metric.)

3) Acculturation. A catch-all by which I mean any sort of bias or
assumption or preference that may be hiding behind a veneer of
objectivity. Octave-invariance has been discussed. Some discussion of
where & how qualities become quantities might be helpful, e.g. the
importance of 5-limit intervals in tertial harmony. Not that bias or
preference could or should be banished, but it ought to be identified.
Thus, a metric designed to describe the closeness of a pitch set to a
harmonic series might reasonably rank a 7/1 closer than a 3/2. Et cetera.

The goal might be to arrive at some set of simple metrics that do not
confuse a multi-dimensional subjective quality like "consonance" with a
single quantitative measurement. One could then choose the qualities
that one deems important and use those metrics appropriate to the
analytical or compositional task at hand.