<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd"> <html> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <meta http-equiv="Content-Style-Type" content="text/css"> <title></title> <meta name="Generator" content="Cocoa HTML Writer"> <meta name="CocoaVersion" content="824.41"> <style type="text/css"> p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica} p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica; min-height: 14.0px} </style> </head> <body> This spreadsheet was created following instructions outlined in detail in A Guide to Musical Temperament by Thomas Donahue, published in 2005 by the Scarecrow Press. It contains three worksheets, which are identical in format. Each worksheet is designed so that values may be entered into one column, and all the rest of the values will be calculated automatically. This makes it possible to predict beat rates for intervals based on cents offset tables, among other things. Sheet 1 allows you to input cents offset for each note of the temperament. This information is very commonly available today for many temperaments, both on line, in books and articles, and pre-loaded in tuning devices. Note that the temperament octave is C3 to C4. This follows the precedent of Helmholtz' monumental book, On the Sensations of Tone, though it is not commonly used today by many tuners. Note that the spreadsheet is set up for A440. Hence, if the cents offset chart you are working from is based on C (which will be obvious, as the value for C will be 0.0), you will need to convert to A if you want to rely on some of the data in the spreadsheet, like frequency of each note. To convert, take the value for A and subtract it from each input number. (If A has a negative value, subtraction means changing the sign to positive and adding). Sheet 2 allows for input in the column "Variable Accumulated Semitones," again a pattern deriving from Helmholtz, and used by many writers on temperament. Starting with C3 as zero, each note is measured in cents above C3, with C3 always serving as the single reference pitch. Sheet 3 allows for input of "Variable 5th commas." The value of the comma must be entered in cents. Briefly, the entire Pythagorean comma (PC) is +24 cents (rounded). To come up with an octave, one needs to subtract a total of 24 cents from the among the 12 5ths. Hence, equal temperament, which can be described as 1/12 PC, divides the PC equally among the 12 5ths, so that one would enter -2.0 for each of the 5ths. 1/6 PC = -4 cents, 1/4 PC = 6 cents. There is also a comma known as the syntonic comma (SC). It is the difference between a series of four just (~beatless) 5ths and a just (~beatless) major third (M3). Eg., tune the 5ths, or their inversion 4ths, FC, CG, GD, DA. The result is a major third. If the 5ths are all tuned "just" the M3 will be wide by 22 cents. The SC is +22 cents. Temperaments, especially meantone, are often described in terms of syntonic comma, as one often wants to end up with a beatless M3, as in the common 1/4 SC meantone. So the 1/4 comma will be 22/4, or 5.5 cents, a little smaller than 1/4 PC. This may or may not be significant for a given tuner or tuning. For further information, please consult A Guide to Musical Temperament. Should you find any errors, please email fssturm@unm.edu. </body> </html>