Notes to Scherchen's The Nature of Music - First Part, The Foundations of Modern Music

I. The Foundations of Modern Music

Even a deaf man, if he knows the length and thickness of a string, can tune the lute, viol, spinet and all other stringed instruments and find any note on them.

Marin Mersenne, Harmonie Universelle, 1636-37

Joseph Sauveur (1653-1716)

three discoveries:

  • method of determining the pitch of a note by assessing the frequency of its vibrations
  • the investigation of the nature of musical sound
  • the systematic establishment of a tempered scale

life:

  • 1653, born deaf and dumb, never obtained full mastery over his voice, and hearing
  • 1696, had been elected to the French Royal Academy of Sciences
  • Sauveu wanted to: make music the object of scientific research, to ascertain the principal rules of musical composition and to penetrate the metaphysics of pleasing sensation.

More about Sauveur at (Adam Fix 2015).

THE FORERUNNERS

René Descartes (1596—1650)

  • 1618, Compendium Musicae
  • investigating the problems of overtones 60 years before John Wallis's papers in Philosophical Transactions, where he stressed the importance of overtones in the observation of consonance.
  • placed the Major Third after the octave and Fifth in order of consonance, while, denied the concordant value of the Fourth (as negative shadow of the Fifth)
  • Descartes has opposite viewpoint compared with Gioseffo Zarlino (1517–1590), Le Istitutioni Harmoniche (1558),
    • Descartes thought the syncopated suspension is dissonance.
    • while Zarlino, ...
  • From the One includes the Many Axiom, Descartes deduced that, the ear’s preference is for simple sounds even though they may be made up of a composite sound.

More about Gioseffo Zarlino - the most famous music theorist of the 16th century

  • collected from (Judd's CHoWMT, Chap12)
    • 1555-1590, maestro di cappella: taught Venetian school of composers, Claudio Merulo, Girolamo Diruta, Giovanni Croce, Vincenzo Galilei, Giovanni Artusi.
    • ... (need further research)
  • compositions

Marin Mersenne (1588—1648)

  • inspiration came from his contaction with artists and craftsmen
  • Mersenne and his friends held scientific conferences, Académie des Sciences de Paris, 1666
  • friend of Descartes

Mersenne & Descartes

  • 1618, Opera Mathematica, vol. 1. (same year with Descartes's Compendium Musicae)
    • when the fundamental of a sounding string ceases, certain higher notes, the twelfth and the seventeenth, are left sounding, i.e. the notes which result from the division of the string into three and ​five parts respectively. (need further research)
    • Descartes commented in a 1629 letter: these sympathetic notes were probably set up by divisions of the vibrating string; Latterly Descartes wrongly assumed that the partition of a string into vibrating subdivisions, and the consonance of the different kinds of vibration resulting from it, only occurred in faulty strings.
  • 1613, El melopeo y maestro
    • anthology of music theorists from Fuclid and Glarean to the treatise of Cerone
  • 1636, Harmonie universelle, four experiments on the frequency of vibration of strings of equal dimen​sions.

Mersenne & Galilei

  • 1638, Galilei, Discourses and Mathematical Demonstrations Relating to Two New Sciences
  • Mersenne and Galilei came independently to some conclusions ... (see below)

Mersenne's Laws:

Mersenne's First Law: The fundamental frequency $f_{0}$ of a stretched string is inversely proportional to its length $L$. (A.K.A Pythagoras's Law)

$$ f_{0} \propto \frac{1}{L} $$

Mersenne's Second Law: The fundamental frequency $f_{0}$ of a stretched string is proportional to the square root of its tension $F$.

$$ f_{0} \propto \sqrt{ F } $$

Mersenne's Third Law: The fundamental frequency $f_{0}$ of a stretched string is Inversely proportional to the square root of its mass per length $\mu$.

$$ f_{0}\propto \frac{1}{\sqrt{ \mu }} $$

Three laws are derived from Mersenne's equation 22:

$$ f_{0}=\frac{v}{\lambda}=\frac{1}{2L}\sqrt{ \frac{F}{\mu} } $$

Additional Sources Used

  • H. F. Cohen, Quantifying Music The Science of Music at the First Stage of Scientific Revolution 1580–1650 (Consonance Problem, Octave Division, Singer's Dilemma, Coincidence Theory; Kepler, Stevin, Galilei, Mersenne, Beeckman, Descartes, Benedetti, Huygens, Pythagoras etc.)