^{page 88 note 1} Macr., p. 180 (Mb. 21); cf. 9. 199 (Mb. 46), p. 207 (Mb. 57), p. 211 (Mb. 62).

^{page 88 note 2} Thus one tone=204 cents, the half-tone=102 cents, the quarter-tone=51 cents, the third-tone=68 cents, the sixth-tone=34 cents, the eighth-tone=25·5 cents, the twelfth-tone=17 cents, the twenty-fourth tone=8·5 cents.

^{page 89 note 1} In relation of tempering, Erich M. von Hornbostel states ‘that the most efficient of our piano tuners, making use of beats for the determination of correct tempering [an aid to the ear due to sympathetic resonance of the strings and their unisons, which is very powerful on the piano but very weak on the Kithara, if not altogether negligible.–K. S.], are wont to make errors of as much as four vibrations per second in the middle octave’–Notiz übcr die Musik d. Einwohner v. Süd-Neu-Mecklenburg. Abh. z. vergl. Musik-wissenschaft. München, 1922, bd. I., pp. 352–353. A. J. Ellis has made similar statements, giving exact results of tests.

^{page 89 note 2} Tempering in relation to Aristoxenus is a theory advanced by the foremost authorities of the French School, viz. A. J. Vincent, Théodore Reinach, Louis Laloy, etc., all of whom have been led to adopt tempering as a solution of the difficulties raised by the error of a comma involved in the method suggested by Aristoxenus (Macr., pp. 207–208; Mb., pp. 56–58) for verifying his assumption that the Fourth consists of two and a half tones.

^{page 90 note 1} When equal temperament was adopted in the eighteenth century there was a strong inducement or necessity as driving power: it was in order to satisfy the desire for modulation into various tonalities in the fact of the prohibitive technical exigencies of musical instruments.

^{page 90 note 2} The system of the Harmonists and the ratios of the Aulos-scales have been identified and established by well-authenticated evidence in a work which I have in preparation for the Press.

^{page 90 note 3} P. 193 (37 Mb.). The translation by Macran of δίεσις as ‘quarter-tone’ is not a happy one here, for the diesis had no definite magnitude; it was valued by Aristoxenus merely as something less than a semitone.

^{page 91 note 1} It is recognized that an acceptance of this statement involves a leap in the dark, but this paper anticipates the publication of a detailed work on the subject.

^{page 92 note 1} Which the scope of this little paper does not allow the writer to produce.

^{page 93 note 1} Harm., Lib. II., cap 5.

^{page 94 note 1} A translation into French of Al-Farabi's treatise (Grand Traité de la Musique) Kitabū L-Mūsīqī Al-Kabīr by Baron Rodolphe d'Erlanger, as the first volume of a projected series to be published under the general title of ‘La Musique Arabe.’ Paris, Librairie Orientaliste, Paul Geuthner, 1930. See section on Flutes, with diagrams, pp. 263 sqq. Those who are versed in the acoustic properties of reed-blown pipes and flutes will be able to distinguish erroneous from true statements in this section. For the Tanbur of Bagdad see pp. 218 sqq. Al-Farabi, born A.D. 872, died A.D. 950.

^{page 94 note 2} The flute was found in a Roman dump during the excavation of the Bucheum by Dr. Robert Mond at Armant, with Mr. Oliver Myers as Director of the Expedition sent out by the Egypt Exploration Society.

^{page 94 note 3} Ptolemy thus uses the ratios of the Harmonic Series while demonstrating the practice by means of lengths of string, a contradiction which would account for certain difficulties in interpretation encountered throughout his treatise.

^{page 95 note 1} Über die altgriech. Musik in der griech. Kirche, by Dr. Joh. Tzetzes, München, 1874. See pp. 30, 52, 77, 83, 93, etc.

^{page 95 note 2} Ascertained from an Arabian Professor of Music in Cairo by means of monochord tests by M. F. Grant.

^{page 95 note 3} De Musica, ed. Weil and Reinach, C. II, p. 1135, pp. 42–51, §§ 108, 114–117; and C. 19, pp. 72–77, §§ 168–177, Cf. Aristoxenus, Mb., p. 37, and Macr., p. 193.

^{page 95 note 4} De Musica, Lib. I., p. 28 Mb.

^{page 96 note 1} De Musica, pp. 112–113, C. 29, § 287.

^{page 96 note 2} Macr., pp. 188–189 (Mb., p. 32).

^{page 96 note 3} Ibid., p. 194 sq. (Mb., pp. 39–41).

^{page 96 note 4} Arist. Quint., Lib. III., p. 116 Mb.

^{page 96 note 5} Cf. Aristoxenus, Macr., p. 206 (Mb., p. 55), in which the process is given for the Ditone, and by implication for the scale.

^{page 96 note 6} Macr., p. 192 (Mb., p. 36)