source file: mills2.txt Subject: temperament spreadsheet (Excel) available From: Bradley P Lehman [snip] I have my own elaborate temperament spreadsheet that I developed while I was working on one of my doctoral projects a few years back. This past New Year's Eve I finally got around to converting it from its original Quattro format over to Excel (Windows 3.1), and then over the next few days put in some new refinements. (But no, it doesn't generate tones.) Features: this spreadsheet is designed to be both objective and subjective in temperament analysis. The objective part: it makes complete beat rate charts for all thirds, fourth, fifths, sixths, and sevenths (including 7/6 minor thirds and several types of minor sevenths) showing how far sharp or flat of pure (simple integer ratio) the interval is in the programmed temperament. It also converts everything to both cents and Hz, and arranges its analyses both chromatically and in a circle of fifths. Then, getting to the subjective part, I've tried to come up with formulas of subjective tolerability of the minor and major thirds and sevenths, and the fourths and fifths. The triads and seventh-chords on all scale degrees get analyzed for usability, tension, etc., and displayed in tables. These subjective formulas are based on my own tuning experience with many temperaments over the years, and reading historical and modern sources about what the ear finds tolerable (so it's not *completely* arbitrary!). The formulas automatically choose the nearest simple ratio (for instance, choosing between the 9/5 and 7/4 minor sevenths, or the 6/5 and 7/6 minor thirds), then look at the beat rates. The upshot of all this is that simply entering a temperament recipe in one place yields beat charts for actually setting it (from either a C or an A fork of any specified frequency), and charts that show which keys are the best/worst to play in, and how the ear might take it on all scale degrees of those keys. For example, in a well temperament, it's easy to see how modulation around the circle of fifths increases/decreases tension in the triads and sevenths. And in 1/4 comma meantone, the Bb-D-F-G# chord shows up sounding like an almost pure dominant seventh (that low pseudo-Ab and all). The spreadsheet makes these features obvious without requiring one to go set the temperament first on an instrument...it could also help in planning an effective overall temperament for a recital, given the keys and modulatory depth of the pieces to be played. Temperaments are specified with superscript comma adjustments plus or minus, the familiar way the old Barbour book did it. My old Quattro version had about 70 temperaments in it, including (I think) all the Barbour ones, some of Jorgensen's, plus others I made up as "what-if" scenarios. I don't remember how many I've converted to the new Excel version, but in any case it's easy to enter new ones...takes only about a minute or two once ye get the hang of it. Twelve notes per octave, and it determines automatically what the enharmonic name of a note is based on the superscript value. The parameters are: start from either an A or C fork; frequency of that fork; temperament name; tuning recipe in fractions of commas; which comma is used; which temperament in the recipe library to display (index number). Everything else is derived. It's also easy to change the subjective formulas that analyze triads and seventh chords. V1.1 has several more temperaments in its library, and displays four more beat-rate intervals (such as the 7/4 minor seventh that a meantone Bb-G# almost is). http://www-personal.umich.edu/~bpl/temper.html ----------------------------------------------------------------------- Bradley Lehman, bpl@umich.edu http://www-personal.umich.edu/~bpl/ Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 12 Apr 1997 23:25 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00411; Sat, 12 Apr 1997 23:25:38 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA00409 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id OAA05668; Sat, 12 Apr 1997 14:23:42 -0700 Date: Sat, 12 Apr 1997 14:23:42 -0700 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu