source file: mills2.txt Date: Thu, 27 Feb 1997 11:04:21 -0800 Subject: Re: Meantone Explanation From: kollos@cavehill.dnet.co.uk (Jonathan Walker) Gary Morrison wrote (2/2/97): > Or relative to the tonic anyway; 15:8 is of course is mediant > of the dominant, so perhaps one could argue it that way. Perhaps > one could argue that fifth comma meantone was devised to make the > third of V-chord just in the same sense that third-comma meantone > makes the IV-chord just. That combined with the fact that the > total error between the third of I (the mediant) and the third of > V (the leading tone) is smaller that way. In the quarter-comma > meantone case, the mediant's error is 0 and the leading tone's is > 1/4 comma, whereas in fifth-comma meantone a total error of 1/4 > comma, the mediant's is 1/5 comma, and the leading tone's is 0, a > total of 1/5 comma. I've been trying to catch up with a lot of list messages, and among them I noticed the above passage; I've looked for a subsequent retraction/correction but can't find any; I've also tried squinting at it in various ways in the hope that the words will somehow make sense, but with no success. First, an ad hoc definition: let's call any key (or mode) that is in keeping with the chosen set of pitch classes in a meantone system an "available" key (or mode). In this context, I am excluding the notion of enharmonic equivalence from "pitch class". On a keyboard of twelve notes to the octave, for example, the most common choice of pitch classes (in the 15th/16th centuries) was: C, C#, D, Eb, E, F, F#, G, G#, A Bb, B Given this choice, the major keys of Bb, F, C, G, D and A are all "available", in the given sense (I am speaking only of the availability of pitch classes for precisely these keys -- of course even a simple piece in Bb is likely at some stage to require an Ab, and similarly for the rest). No other major keys are available, since they must utilise wolf intervals, e.g. the dominant triad of E major will consist of a diminished fourth plus an augmented second in the given meantone tuning, rather than a major third plus minor third. I know perfectly well that this is all common knowledge, but I have to get the issue of enharmonic equivalence out of the way to avoid confusion. Now in any available key, thus defined, a given interval type is uniform in size, not only in this key, but also in all other available keys. Thus, in a given meantone tuning, the third from C to E will be the same as the third from F to A and the third from G to B. In 1/4-comma meantone, all these major thirds will be just, as will any major third in any available key. In any other meantone tuning, the major thirds in available keys will not, of course, be just; for any meantone tuning based on a rational power of the syntonic comma, there will, of course, be at least one just interval class, but aside from 1/4-comma, where maj. thirds and min. sixths are just, the only other meantone which was chosen (in the period from 15th - 17th centuries) to produce just intervals of musical interest was 1/3-comma, with just min. thirds and maj. sixths. What on earth, then, is the meaning of the passage quoted at the top? The fact that 1/5-comma meantone has just, i.e. 15/8, major sevenths in no way distinguishes the tuning of the dominant chord in an available key from the tuning of the tonic chord; likewise, 1/3-comma doesn't distinguish between the tuning of subdominants and tonics in available keys. In any available key, a given meantone system will no more favour one triad over another than it will favour one presidential candidate over another. -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 27 Feb 1997 21:06 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA32146; Thu, 27 Feb 1997 21:06:38 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA32241 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id MAA26696; Thu, 27 Feb 1997 12:04:28 -0800 Date: Thu, 27 Feb 1997 12:04:28 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu