Ptolemy's intense diatonic scale

Ptolemy's intense diatonic scale, also known as the Ptolemaic sequence,^[1] justly tuned major scale,^[2]^[3]^[4] Ptolemy's tense diatonic scale, or the syntonous (or syntonic) diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy,^[5] and corresponding with modern 5-limit just intonation.^[6] While Ptolemy is famous for this version of just intonation, it is important to realize this was only one of several genera of just, diatonic intonations he describes. He also describes 7-limit "soft" diatonics and an 11-limit "even" diatonic.

This tuning was declared by Zarlino to be the only tuning that could be reasonably sung, it was also supported by Giuseppe Tartini,^[7] and is equivalent to Indian Gandhar tuning which features exactly the same intervals.

It is produced through a tetrachord consisting of a greater tone (9:8), lesser tone (10:9), and just diatonic semitone (16:15).^[6] This is called Ptolemy's intense diatonic tetrachord (or "tense"), as opposed to Ptolemy's soft diatonic tetrachord (or "relaxed"), which is formed by 21:20, 10:9 and 8:7 intervals.^[8]

^ Partch, Harry (1979). Genesis of a Music, pp. 165, 173. ISBN 978-0-306-80106-8.
^ Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, pp. 172–73. ISBN 978-0-19-816505-7.
^ Wright, David (2009). Mathematics and Music, pp. 140–41. ISBN 978-0-8218-4873-9.
^ Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), "Maximum clarity" and Other Writings on Music, p. 78. ISBN 978-0-252-03098-7.
^ see Wallis, John (1699). Opera Mathematica, Vol. III. Oxford. p. 39. (Contains Harmonics by Claudius Ptolemy.)
^ ^a ^b Chisholm, Hugh (1911). The Encyclopædia Britannica, Vol.28, p. 961. The Encyclopædia Britannica Company.
^ Dr. Crotch (October 1, 1861). "On the Derivation of the Scale, Tuning, Temperament, the Monochord, etc.", The Musical Times, p. 115.
^ Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 2, Page 9

[1] Partch, Harry (1979). Genesis of a Music, pp. 165, 173. ISBN 978-0-306-80106-8.

[2] Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, pp. 172–73. ISBN 978-0-19-816505-7.

[3] Wright, David (2009). Mathematics and Music, pp. 140–41. ISBN 978-0-8218-4873-9.

[4] Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), "Maximum clarity" and Other Writings on Music, p. 78. ISBN 978-0-252-03098-7.

[5] see Wallis, John (1699). Opera Mathematica, Vol. III. Oxford. p. 39. (Contains Harmonics by Claudius Ptolemy.)

[EB-6] Chisholm, Hugh (1911). The Encyclopædia Britannica, Vol.28, p. 961. The Encyclopædia Britannica Company.

[7] Dr. Crotch (October 1, 1861). "On the Derivation of the Scale, Tuning, Temperament, the Monochord, etc.", The Musical Times, p. 115.

[8] Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 2, Page 9

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