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Note	Approximate frequency (Hz)	Audible?
C₋₄	1
C₋₃	2
C₋₂	4
C₋₁	8	Y
C₀	16	Y
C₁	32	Y
C₂	64	Y
C₃	128	Y
C₄	256	Y
C₅	512	Y
C₆	1024	Y
C₇	2048	Y
C₈	4096	Y
C₉	8192	Y
C₁₀	16384	Y
C₁₁	32768
C₁₂	65536

Scientific pitch, also known as philosophical pitch, Sauveur pitch or Verdi tuning, is an absolute concert pitch standard which is based on middle C (C₄) being set to 256 Hz rather than approximately 261.63 Hz,^[1] making it approximately 31.77 cents lower than the common A440 pitch standard. It was first proposed in 1713 by French physicist Joseph Sauveur, promoted briefly by Italian composer Giuseppe Verdi in the 19th century, then advocated by the Schiller Institute beginning in the 1980s with reference to the composer, but naming a pitch slightly lower than Verdi's preferred 432 Hz for A, and making controversial claims regarding the effects of this pitch.

Scientific pitch is not used by concert orchestras but is still sometimes favored in scientific writings for the convenience of all the octaves of C being an exact power of 2 when expressed in hertz (symbol Hz).^[2]^[3] The octaves of C remain a whole number in Hz all the way down to 1 Hz in both binary and decimal counting systems.^[4]^[5] Instead of A above middle C (A₄) being set to the widely used standard of 440 Hz, scientific pitch assigns it a frequency of 430.54 Hz.^[6]

Since 256 is a power of 2, only octaves (factor 2:1) and, in just tuning, higher-pitched perfect fifths (factor 3:2) of the scientific pitch standard will have a frequency of a convenient integer value. With a Verdi pitch standard of A₄ = 432 Hz = 2⁴ × 3³, in just tuning all octaves (factor 2), perfect fourths (factor 4:3) and fifths (factor 3:2) will have pitch frequencies of integer numbers, but not the major thirds (factor 5:4) nor major sixths (factor 5:3) which have a prime factor 5 in their ratios. However scientific tuning implies an equal temperament tuning where the frequency ratio between each half tone in the scale is the same, being the 12th root of 2 (a factor of approximately 1.059463), which is not a rational number: therefore in scientific pitch only the octaves of C have a frequency of a whole number in hertz.

History

Concert tuning pitches tended to vary from group to group^{[clarification needed]}, and by the 17th century the pitches had been generally creeping upward (i.e., becoming "sharper"). The French acoustic physicist Joseph Sauveur, a non-musician, researched musical pitches and determined their frequencies. He found several frequency values for A₄ as presented to him by musicians and their instruments, with A₄ ranging from 405 to 421 Hz. (Other contemporary researchers such as Christiaan Huygens, Vittorio Francesco Stancari and Brook Taylor were finding similar and lower values for A₄, as low as 383 Hz.) In 1701, Sauveur proposed that all musical pitches should be based on a son fixe (fixed sound), that is, one unspecified note set to 100 Hz, from which all others would be derived. In 1713, Sauveur changed his proposal to one based on C₄ set to 256 Hz; this was later called "philosophical pitch" or "Sauveur pitch". Sauveur's push to standardize a concert pitch was strongly resisted by the musicians with whom he was working, and the proposed standard was not adopted.^[7] The notion was revived periodically, including by mathematician Sir John Herschel and composer John Pyke Hullah in the mid-19th century, but never established as a standard.^[8]

In the 19th century, Italian composer Giuseppe Verdi tried to stop the increase in the pitch to which orchestras were tuned. In 1874 he wrote his Requiem using the official French standard diapason normal pitch of A₄ tuned to 435 Hz. Later, he indicated that 432 Hz would be slightly better for orchestras.^[9] One solution he proposed was scientific pitch, but he had little success.^[9]^[10]

In 1988, Lyndon LaRouche's Schiller Institute initiated a campaign to establish scientific pitch as the classical music concert pitch standard. The Institute called this pitch "Verdi tuning" because of the connection to the famous composer.^[11] Even though Verdi tuning uses 432 Hz for A₄ and not 430.54, it is said by the Schiller Institute to be derived from the same mathematical basis: 256 Hz for middle C.^[12] The Institute's arguments for the notation included points about historical accuracy and references to Johannes Kepler's treatise on the movement of planetary masses.^[13] The Schiller Institute initiative was opposed by opera singer Stefan Zucker. According to Zucker, the Institute offered a bill in Italy to impose scientific notation on state-sponsored musicians that included provisions for fines and confiscation of all other tuning forks. Zucker has written that he believes the Schiller Institute claims about Verdi tuning are historically inaccurate. Institute followers are reported by Tim Page of Newsday to have stood outside concert halls with petitions to ban the music of Antonio Vivaldi and even to have disrupted a concert conducted by Leonard Slatkin in order to pass out pamphlets titled "Leonard Slatkin Serves Satan".^[14]

References

^ ${\frac {440}{2^{\!^{\tfrac {9}{12}}}}}$
^ Marshall Long, Architectural acoustics, p.81, Elsevier, 2006 ISBN 0-12-455551-9.
^ Clarence Grant Hamilton, Sound and Its Relation to Music, p.56, Read Books, 2009 ISBN 1-4446-7429-3.
^ Eli Maor, Trigonometric delights, p.210, Princeton University Press, 2002 ISBN 0-691-09541-8. "Scientific pitch...has the advantage that all octaves of C correspond to powers of two."
^ Herbert Stanley Allen, Harry Moore, A text-book of practical physics, p.202, Macmillan, 1916. "The reason for the choice of 256 as middle C in scientific work is in order that the number of vibrations corresponding with any C shall be a whole number."
^ Turtur, Claus Wilhelm (2011). Prüfungstrainer Physik: Klausur- und Übungsaufgaben mit vollständigen Musterlösungen (in German) (3 ed.). Springer. p. 151. ISBN 978-3834809407.
^ Haynes, Bruce (2002). A History of Performing Pitch: The Story of 'A'. Scarecrow Press. p. 224. ISBN 1461664152.
^ Pole, William (January 29, 1869). "Musical Pitch". Journal of the Society of Arts. 17 (845). London: Bell and Daldy: 165–166.
^ ^a ^b Rosen, David (September 14, 1995). Verdi: Requiem. Cambridge University Press. ISBN 9780521397674 – via Google Books.
^ Letter from Verdi to Giulio Ricordi, Verdi's Aida, Giuseppe Verdi, Hans Busch
^ Johnston, Ian (2009). Measured Tones: The Interplay of Physics and Music, Second Edition (3 ed.). CRC Press. p. 36. ISBN 978-1420093476.
^ "For a Verdi Opera in the Verdi Tuning in 2001". Schiller Institute. 2001. Retrieved April 21, 2013.
^ "The Science of Music". The Schiller Institute. Retrieved 2009-07-28.
^ "Opera Fanatic Magazine". Bel Canto Society. Archived from the original on 2008-08-19. Retrieved 2008-10-23.