source file: mills2.txt Date: Thu, 13 Mar 1997 08:44:48 -0800 Subject: Linear Temperaments cont. From: John Chalmers < A slightly lower unweighted sum of sq'ed errors, duh, by definition. I think the "duh" is unnecessary. The mathematics are obvious; my point was about the appropriateness of the weightings which is a psychoacoustic question whose answer I don't really know. You mentioned (ironically) "sharp or flat unisons." I was pointing out that octaves, fifths, and unisons modified by a 1/4 tone were used by Wyschnegradsky for expressive purposes, so the statement is not as humorously vacuous as it might appear at first sight. I too agree that the 7/4 is not always the best tuning for the 7th of a dom7th chord. Bosanquet thought that it was disturbing in melodic passages, though beautiful in full chords (in meantone systems). I'm using Bosanquet's nomenclature in which a positive tuning is one in which the fifth is sharper than 700 cents (some contemporary writers use the term for fifths sharper than 3/2). The degree to which a tuning is positive or negative is the number of steps 12 of its fifths exceeds or falls short of 7 octaves. For untempered systems, this definition may be modified by approximating their fifths to those of a tempered system. Thus 1/4-meantone may be treated as a -1 system by approximating it to 31-tet (others are 7,19,43,55,67). Doubly negative systems are 14, 26,38,50,62, 74... Primary positive systems are 5,17,29,41,53... and doubly positive systems are 10,22,34,46.... Triply negative systems are 9, 21, 33, 45, 57,69,81 .... and triply positive, 3, 15, 27, 39... As the chain length for major thirds and harmonic sevenths is so long in positive systems, more than 12 tones is mandatory for harmonic music. I wouldn't put too much faith in the precision of the tunings I posted. I calculated them in double precision, but I'm not sure the accuracy is anywhere near the number of decimal places shown. I simply didn't have the time to round them down to 8 places as I did for the negative ones prior to posting them. I am aware that errors may cancel when the ABS(Sum) is taken rather than the Sum(ABS) and I thought the resulting tunings might be interesting for exactly this reason. Thus in the 1/3-comma tuning the 6/5 is just, in the 1/5 comma, the 15/8 is. I stress these tunings, for the most part, were generated as a theoretical study many years ago. I am not proposing them seriously today as I think the corresponding equal temperaments would be perceptually equivalent and easier to implement and use, though Eduardo Sabat-Garibaldi has successfully embodied the 1/9th skhisma tuning on a specially fretted guitar. The negative tunings might have some value in realizing early music, but I'm not sure one could really tell the difference from 1/4-comma meantone or 31-tet within a 12-tone gamut. For a larger series of notes, perhaps.. Bosanquet himself decided that tuning his organ to the 1/7 skhisma system was not worth the extra trouble and henceforth used 3/2's. XH17 will appear when I get all the promised articles. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 13 Mar 1997 18:46 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA10373; Thu, 13 Mar 1997 18:46:40 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA10361 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id JAA20345; Thu, 13 Mar 1997 09:40:35 -0800 Date: Thu, 13 Mar 1997 09:40:35 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu