source file: mills2.txt Date: Mon, 14 Jul 1997 23:22:32 +0200 Subject: RE: Basic dimensionality, and related issues From: "Paul H. Erlich" 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> > > >--------------466ADBE552939739789B3CF7 >Content-Type: text/plain; charset-ascii >Content-Transfer-Encoding: 7bit > >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 >Content-Type: text/html; charset-ascii >Content-Transfer-Encoding: 7bit > > >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 >************************* Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 14 Jul 1997 23:51 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA21053; Mon, 14 Jul 1997 23:52:08 +0200 Date: Mon, 14 Jul 1997 23:52:08 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA20965 Received: (qmail 26174 invoked from network); 14 Jul 1997 21:05:42 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 14 Jul 1997 21:05:42 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu