Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/tuningtemperamenOObarb Tuning and Temperament 2S A Historical Survey J. Murray Barbour East Lansing Michigan State College Press 1951 Copyright 1951 BY MICHIGAN STATE COLLEGE PRESS East Lansing 2741.9 usic L LITHOPRINTED IN THE UNITED STATES OF AMERICA BY CUSHING-MALLOY, INC., ANN ARBOR, MICHIGAN, 1951 PREFACE This book is based upon my unpublished Cornell dissertation, Equal Temperament: Its History from Ramis (1482) to Rameau (1737), Ithaca, 1932. As the title indicates, the emphasis in the dissertation was upon individual writers. In the present work the emphasis is on the theories rather than on their promulga- tors. Since a great many tuning systems are discussed, a sepa- rate chapter is devoted to each of the principal varieties of tun- ing, with subsidiary divisions wherever necessary. Even so, the whole subject is so complex that it seemed best that these chapters be preceded by a running account (with a minimum of mathematics) of the entire history of tuning and temperament. Chapter I also contains the principal account of the Pythagorean tuning, for it is unnecessary to spend a chapter upon a tuning system that exists in one form only. Most technical terms will be defined when they first occur, as well as in the Glossary, but a few of these terms should be de- fined immediately. Of small intervals arising from tuning, the comma is the most familiar. The ordinary (syntonic or Ptole- maic) comma is the interval between a just major third, with ratio 5:4, and a Pythagorean ditone or major third, with ratio 81:64. The ratio of the comma (the ratio of an interval is ob- tained by dividing the ratio of the higher pitch by that of the lower) is 81:80. The Pythagorean (ditonic) comma is the interval between six tones, with ratio 531441:262144, and the pure octave, with ratio 2:1. Thus its ratio is 531441:524288, which is approximately 74:73. The ditonic comma is about 12/11 as large as the syn- tonic comma. In general, when the word comma is used without qualification, the syntonic comma is meant. There is necessarily some elasticity in the manner in which the different tuning systems are presented in the following chap- ters. Sometimes a writer has described the construction of a monochord, a note at a time. That can be set down easily in the form of ratios. More often he has expressed his monochord as a series of string-lengths, with a convenient length for the fun- damental. (Except in the immediate past, the use of vibration numbers, inversely proportional to the string-lengths, has been so rare that it can be ignored.) Or he may speak of there being so many pure fifths, and other fifths flattened by a fractional v PREFACE part of the comma. Such systems could be transformed into equivalent string -lengths, but this has not been done in this book when the original writer had not done so. Systems with intervals altered by parts of a comma can be shown without difficulty in terms of Ellis' logarithmic unit called the cent, the hundredth part of an equally tempered semitone, or 1/1200 part of an- octave. Since the ratio of the octave is 2:1, 1/120° the cent is 2 . As a matter of fact, such eighteenth century writers on temperament as Neidhardt and Marpurg had a tuning unit very similar to the cent: the twelfth part of the ditonic comma, which they used, is 2 cents, thus making the octave con- tain 600 parts instead of 1200. The systems originally expressed in string- lengths or ratios may be translated into cents also, although with greater difficulty <= They have been so expressed in the tables of this book, in the be- lief that the cents representation is the most convenient way of affording comparisons between systems. In systems where it was thought they would help to clarify the picture, exponents have been attached to the names of the notes. With this method, de- vised by Eitz, all notes joined by pure fifths have the same ex- ponent. Since the fundamental has a zero exponent, all the notes of the Pythagorean tuning have zero exponents. The exponent -1 is attached to notes a comma lower than those with zero expo- nents, i.e., to those forming pure thirds above those in the zero series. Thus in just intonation the notes forming a major third -1 would beC°-E , etc. Similarly, notes that are pure thirds lower than notes already in the system have exponents which are greater by one than those of the higher notes. This use of expo- nents is especially advantageous in comparing various systems of just intonation (see Chapter V). It may be used also, with -fractional exponents, for the different varieties of the meantone temperament. If the fifth C-G, for example, is tempered by 1/4 comma, these notes would be labeled C° and G" . A device related to the use of integral exponents lor the notes in just intonation is the arrangement of such notes to show their *For a discussion of methods of logarithmic representation of intervals see J.Murray Barbour, "Musical Logarithms," Scripta Mathematica , VII (1940), 21-31. VI PREFACE harmonic relationships. Here, all notes that are related by fifths, i.e., that have the same exponent, lie on the same horizontal line, while their pure major thirds lie in a parallel line above them, each forming a 45° angle with the related note below. Since the pure minor thirds below the original notes are lower by a fifth than the major thirds above them, they will lie in the same higher line, but will form 135° angles with the original notes. For ex- ample: A-1 „F-1 „R -1 C G This arrangement is especially good for showing extensions of just intonation with more than twelve notes in the octave, and it is used for that purpose only in this book (see Chapter VI). It is desirable to have some method of evaluating the various tuning systems. Since equal temperament is the ideal system of twelve notes if modulations are to be made freely to every key, the semitone of equal temperament, 100 cents, is taken as the ideal, from which the deviation of each semitone, as C-C , C*-D, D-E", etc., is calculated in cents. These deviations are then added and the sum divided by twelve to find the mean deviation (M.D.) in cents. The standard deviation (S.D.) is found in the usual manner, by taking the root-mean-square. It should be added that there may be criteria for excellence in a tuning system other than its closeness to equal temperament. For example, if no notes beyond E" or G^ are used in the music to be performed and if the greatest consonance is desired for the notes that are used, then probably the 1/5 comma variety of mean- tone temperament would be the ideal, since its fifths and thirds are altered equally, the fifths being 1/5 comma flat and its thirds 1/5 comma sharp. If keys beyond two flats or three sharps are to be touched upon occasionally, but if it is considered desirable to have the greatest consonance in the key of C and the least in the key of G", then our Temperament by Regularly Varied Fifths would be the best. This is a matter that is discussed in detail at the end of Chapter VTI, but it should be mentioned now. My interest in temperament dates from the time in Berlin when Professor Curt Sachs showed me his copy of Mersenne's vii , PREFACE Harmonie universelle . I am indebted to Professor Otto Kinkeldey my major professor at Cornell, and to the Misses Barbara Dun- can and Elizabeth Schmitter of the Sibley Musical Library of the Eastman School of Music, for assistance rendered during my work on the dissertation. Most of my more recent research has been at the Library of Congress. Dr. Harold Spivacke and Mr. Edward N. Waters of the Music Division there deserve especial thanks for encouraging me to write this book. I want also to thank the following men for performing so well the task of read- ing the manuscript: Professor Charles Warren Fox, Eastman School of Music; Professor Bonnie M. Stewart, Michigan State College; Dr. Arnold Small, San Diego Navy Electronics Labora- tory; and Professor Glen Haydon, University of North Carolina. J. Murray Barbour East Lansing, Michigan November, 1950 Vlll GLOSSARY Arithmetical Division — The equal division of the difference be- tween two quantities, so that the resultant forms an arithme- tical progression, as 9:8:7:6. Bonded Clavichord — A clavichord upon which two or more con- secutive semitones were produced upon a single string. Cent — The unit of interval measure. The hundredth part of an 1200i equal semitone, with ratio <y2. Circle of Fifths — The arrangement of the notes of a closed sys- tem by fifths, as C, G, D, A, E, etc. Circulating Temperaments — Temperaments in which all keys are playable, but in which keys with few sharps or flats are favored. Closed System — A regular temperament in which the initial note is eventually reached again. Column of Differences — See Tabular Differences. Comma — A tuning error, such as the interval B^-C in the Py- thagorean tuning. See Ditonic Comma and Syntonic Comma. Ditone — A major third, especially one formed by two equal tones, as L? the Pythagorean tuning (81:64).