PREFACE This book is based upon my unpublished Cornell dissertation, Equal Temperament: Its History from Ramis (1482) to Rameau (1737), Ithaca, 1932.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, the cent is 21/120° . 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 would beC°-E-1, 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 f2. Euclidean Construction — Euclid's method for finding a mean proportional between two lines, by describing a semicircle upon the sum of the lines taken as a diameter and then erecting a perpendicular at the juncture of the two lines. Fretted Clavichord — See Bonded Clavichord. Fretted Instruments — Such modern instruments as the guitar and banjo, or the earlier lute and viol. Generalized Keyboard — A keyboard arranged conveniently for the performance of multiple divisions. Geometrical Division — The proportional division of two quanti- ties, so that the resultant forms a geometrical progression, as 27:18:12:8. Golden System — A system of tuning based on the ratio of the golden section ( /f5~ - 1):2. Good Temperaments — See Circulating Temperaments. Irregular System — Any tuning system with more than one odd- sized fifth, with the exception of just intonation. Just— Pure: A term applied to intervals, as the just major third. Just Intonation — A system of tuning based on the octave (2:1), the pure fifth (3:2), and the pure major third (5:4). Linear Correction — The arithmetical division of the error in a string-length. Mean-Semitone Temperament — A temperament in which the diatonic notes are in meantone temperament, and the chromatic notes are taken as halves of meantones. Meantone Temperament — Strictly, a system of tuning with flat- tened fifths (y 5:1) and pure major thirds (5:4). See Varieties of Meantone Temperament. x GLOSSARY Meride — Sauveur's tuning unit, 1/43 octave, that is, ^"27 Each meride was divisible into 7 eptamerides, and each of the ep- tamerides into 10 decamerides. Mesolabium — An instrument of the ancients for finding mechan- ically 2 mean proportionals between 2 given lines. See illus- tration, p. 51. Monochord — A string stretched over a wooden base upon which are indicated the string-lengths for some tuning system; a diagram containing these lengths; directions for constructing such a diagram. Monopipe — A variable open pipe, with indicated lengths for a scale in a particular tuning system, thus fulfilling a function similar to that of a monochord. Multiple Division — The division of the octave into more than 12 parts, equal or unequal. Negative System — A regular system whose fifth has a ratio smaller than 3:2. Positive System — A regular system whose fifth has a ratio larger than 3:2. Ptolemaic Comma — See Syntonic Comma. Pythagorean Comma — See Ditonic Comma. Pythagorean Tuning — A system of tuning based on the octave (2:1) and the pure fifth (3:2). Regular Temperament — A temperament in which all the fifths save one are of the same size, such as the Pythagorean tuning or the meantone temperament. (Equal temperament, with all fifths equal, is also a regular temperament, and so are the closed systems of multiple division.) Schisma — The difference between the syntonic and ditonic commas, with ratio 32805:32768, or approximately 2 cents. Semi-Meantone Temperament — See Mean-Semitone Tempera- ment. XI GLOSSARY Sesqui The prefix used to designate a superparticular ratio, as sesquitertia (4:3). Sexagesimal Notation — The use of 60 rather than 10 as a base of numeration, as in the measurement of angles. Split Keys — Separate keys on a keyboard instrument for such a pair of notes as G* and Ab. String- Length — The portion of a string on the monochord that will produce a desired pitch. Subsemitonia — See Split Keys. Superparticular Ratio — A ratio in which the antecedent exceeds the consequent by 1, as 5:4. See Sesqui-. Syntonic Comma — The interval between a just major third (5:4) and a Pythagorean third (81:64). Its ratio is 81:80 and it is conventionally taken as 22 cents. Tabular Differences - The differences between the successive terms in a sequence of numbers, such as a geometrical pro- gression. TemPer - To vary the pitch slightly. A tempered fifth is spe- cifically a flattened fifth. Temperament — A system, some or all of whose intervals can- not be expressed in rational numbers. A Tuning - A system all of whose intervals can be expressed in rational numbers. Tuning Pipe — See Monopipe., Unequal Temperament — Any temperament other than equal temperament, particularly the meantone temperament or some variety thereof. Varieties of Meantone Temperament — Regular temperaments formed on the same principle as the meantone temperament, with flattened fifths and (usually) sharp thirds. Wolf Fifth - The dissonant fifth, usually G#-Eb (notated as a diminished sixth), in any unequal temperament, such as the meantone v/olf fifth of 737 cents. xii CONTENTS Preface Page v Glossary ix I. History of Tuning and Temperament 1 II. Greek Tunings 15 III. Meantone Temperament 25 Other Varieties of Meantone Temperament .... 31 IV. Equal Temperament 45 Geometrical and Mechanical Approximations ... 49 Numerical Approximations 55 V. Just Intonation 89 Theory of Just Intonation 102 VI. Multiple Division 107 Equal Divisions 113 Theory of Multiple Division 128 VII. Irregular Systems 133 Modifications of Regular Temperaments 139 Temperaments Largely Pythagorean 151 Divisions of Ditonic Comma 156 Metius' System 177 "Good" Temperaments 178 VIII. From Theory to Practice 185 Tuning of Keyboard Instruments 188 Just Intonation in Choral Music 196 Present Practice 199 Literature Cited 203 Index 219 xiii CONTENTS LIST OF ILLUSTRATIONS Frontispiece: Fludd's Monochord, with Pythagorean Tuning and Associated Symbolism Fig. A. Schneegass' Division of the Monochord 38 B. The Mesolabium 51 C. Roberval's Method for Finding Two Geo- metric Mean Proportionals 52 D. Nicomedes' Method for Finding Two Geometric Mean Proportionals 53 E. Strahle's Geometrical Approximation for Equal Temperament 66 F. Gibelius' Tuning Pipe 86 G. Mersenne's Keyboard with Thirty-One Notes in the Octave 109 H. Ganassi's Method for Placing Frets on the Lute and Viol 142 I. Bermudo's Method for Placing Frets on the Vihuela 165