1 Although not formally incorporated until 1662, the Society's minutes date from November 1660.Google Scholar

2 For a listing of items on music in the archives of the Royal Society, with descriptions of each item and an introductory essay, see Leta Miller and Albert Cohen, Music in the Royal Society of London, 1660–1806, Detroit Studies in Music Bibliography, 56 (Detroit, 1987).Google Scholar

3 Royal Society, Boyle Papers XLI, no. 1.Google Scholar

4 Birchensha, John, ‘An Extract of a Letter Written to the Royall Society … concerning Musick’ (26 April 1664), Royal Society, Letter Book Copy i, 171–2.Google Scholar

5 From the Society's minutes of 27 April 1664 as printed in Thomas Birch, History of the Royal Society of London (London, 1756–7; repr. ed. A. Rupert and Marie Boas Hall, New York and London, 1968), i, 418. The minutes are preserved in a Journal Book, which exists in both an original and a copy. However, up to 1686, they were published as part of Birch's History. Comparison of Birch's minutes with the original shows him to have been very faithful to the source. His rendition differs from the Journal Book only in the clarification of some details and by his inclusion of several complete papers. The committee appointed by the Society to examine Birchensha's theories had been named on 20 April upon the recommendation of Silas Taylor. See note 24 below.Google Scholar

6 Birchensha, ‘An Extract of a Letter Written to the Royall Society’, Royal Society, Letter Book Copy i, 166–73.Google Scholar

7 ‘Two Advertisements; one, concerning a Body of Algebra …; the other, a System of Musick, ready to be publish't by John Birchencha’, Philosophical Transactions of the Royal Society of London, 7, no. 90 (20 January 1672/3), 5153–4. The Transactions were founded in 1665 by the Society's first secretary, Henry Oldenburg, who issued the journal approximately once a month until his death in 1677. Thereafter, subsequent secretaries took responsibility for the publication until 1741, when a committee of the Society was established to oversee the journal.Google Scholar

8 Birchensha, John, ‘An Account of Divers Particulars, Remarkable in my Book; In which I will Write of Musick Philosophically, Mathematically, and Practically …’, Royal Society Classified Papers 22(1), no. 7 [February 1676]. The Classified Papers consist of 39 volumes of documents presented to the Society during the period 1660–1740. The papers are arranged primarily by subject. The secretary has noted in the margin of Birchensha's paper that the document was read at the meeting of 10 February 1675[/6].Google Scholar

9 Templum musicum: or the Musical Synopsis of the Learned and Famous Johannes-Henricus Alstedius being a Compendium of the Rudiments both of the Mathematical and Practical Part of Musick, trans. and ed. John Birchensha (London, 1664).Google Scholar

10 ‘A Collection of Rules in Musicke from the Most Knowing Masters in that Science with Mr. Birchensha's 6 Rules of Composition; and his Enlargements thereon to the Right Honorable William Lord Viscount Brouncker’, British Library, Add. MS 4910, ff. 39–60.Google Scholar

11 Royal Society, Boyle Papers XLI, no. 1.Google Scholar

12 Ibid., 20.Google Scholar

13 Penelope Mary Gouk, ‘Music in the Natural Philosophy of the Early Royal Society’ (Ph.D. dissertation, University of London, 1982), 202–10. Gouk (pp. 210–17) includes a thorough discussion of Birchensha's system and describes, in addition, two commentaries on Birchensha's work, the first an involved set of calculations by John Pell (1611–85), the second, a refutation of the Pythagorean system in favour of equal temperament possibly written by Nicolas Mercator (Niklaus Kauffman, c.1619–87); see also Louis Chenette, ‘Music Theory in the British Isles’ (Ph.D dissertation, Ohio State University, 1967), 132–40.Google Scholar

14 Birchensha, ‘An Extract of a Letter Written to the Royall Society’, 170.Google Scholar

15 Birch, History of the Royal Society, iii, 295.Google Scholar

16 Quotation and example from Birchensha, Compendious Discourse, 17.Google Scholar

17 Ibid., 14.Google Scholar

18 Ibid.Google Scholar

19 Birchensha's use of the word ‘scale’ to characterize his diagram of intervallic ratios and relationships is consistent with the Oxford English Dictionary's definition of the word as ‘a set or series of graduations (marked along a straight line or curve) used for measuring distance’; and for music in particular as ‘any of the graduated series of sounds into which the octave is divided’.Google Scholar

20 ‘Concerning the mathematicall part of musick, first I shall lay downe divers certain and easy rules for the finding out of these intervals, which are required to the compleating of a diapason and all other simple diastems, and the placing of them in their proper seats and cells in a disdiapasonick system. … Every interval must be placed in the scale in its proper place, and have its proper term of difference assigned to him, that the scale may be perfect, i.e. so far as the eye and ear can judge it so to be. I will [also] deliver certain rules for the finding out of the numbers and termes of difference which ought to be adscribed to every perticular [sic] interval from a comma to an octave.’ (Birchensha, ‘An Extract of a Letter Written to the Royall Society’, 168–9).Google Scholar

21 An arithmetic progression is one in which the magnitudes of two intervals (i.e. the difference between any two terms in the series) is equal. Thus an octave (2:1) divided with the fourth (4:3) below the fifth (3:2) would result in the following arithmetic progression in which 4–3 = 3–2.Google Scholar

A	d	a
4:	3:	2

By necessity, the ratios of the two intervals will be unequal (i.e. 4/3 ≠ 3/2).Google Scholar

In the geometric progression, the ratios of any two adjacent terms are equal, and this ratio is equivalent to the ratio of their magnitudes, as in the seriesGoogle Scholar

A	e	b
9:	6:	4

where 9/6 = 6/4 = 3/2 (3 representing the magnitude of 9:6, and 2, the magnitude of 6:4).Google Scholar

In the harmonic division, the ratio of the highest term to the lowest equals the ratio of the magnitudes of the terms. When the octave (6:3) is divided with the fifth (6:4) on the bottom and the fourth (4:3) above -Google Scholar

E	B	e
6:	4:	3

- the ratio of outer tones (6:3) will be equal to 2:1, representing the ratio of the magnitudes of the individual intervals (6–4 = 2, 4–3 = 1).Google Scholar

22 ‘Two Advertisements’, 5154.Google Scholar

23 Taylor (1624–78) published works on antiquities, such as the History and Antiquities of Harwich and Dovercourt (1730), and composed sacred vocal music and dance pieces.Google Scholar

24 In 1664, Taylor's recommendation of Birchensha as ‘a gentleman, who pretended to discover some musical errors, generally committed by all modern masters of music, touching the scales, and the proportions of notes’ aroused the Society's interest in Birchensha and prompted the appointment of a committee to review his theories. See note 5 above. (Quotation from the Society's minutes of 20 April 1664 as printed in Birch, History of the Royal Society, i, 416.)Google Scholar

25 British Library, Add. MS 4910, f. 47. Similarly, on f. 49, Taylor writes about rule 2, ‘This from Mr. Birchensha himselfe’, and on f. 53 he has noted regarding rule 4, ‘This collected out of Mr. Birchensha's works by Mr. Silas Domvill als Taylor’.Google Scholar

26 Unless otherwise noted, the quotations in the section on monophonic composition are taken from Birchensha, Compendious Discourse, last four (unnumbered) pages.Google Scholar

27 Ibid., 6.Google Scholar

28 The formal close occurs at the end of a section, while the final close appears at the end of the piece.Google Scholar

29 That the manuscript was rebound and the pages reordered at some point is clear from the double set of numbers on many of the pages; the original numbering has been crossed out and replaced by new pagination.Google Scholar

30 The five rules are found on ff. 47, 49, 51, 53 and 55 respectively. Examples for each rule are on the facing versos.Google Scholar

31 ‘A Collection of Rules in Musicke’, f. 49.Google Scholar

32 Ibid., f. 40.Google Scholar

33 On 24 February, for example, Pepys spent the morning with Birchensha finishing his song ‘Gaze not on swans’ and then returned to the latter's house that afternoon as well. See The Diary of Samuel Pepys, ed. Robert Latham and William Matthews (Berkeley and Los Angeles, 1970–83), iii, 34–5. Pepys's lessons with Birchensha ran from 14 January (ibid., 8–9) to 27 February (ibid., 36–7).Google Scholar

34 Ibid., 46.Google Scholar

35 Ibid., 34–5.Google Scholar

36 Emslie, Macdonald, ‘Pepys's Songs and Songbooks in the Diary Period’, The Library, 12 (1957), 242–3.Google Scholar

37 Pepys Library (Magdalene College, Cambridge), MS 2591, ff. 166b–167b.Google Scholar

38 There are repeated references to ‘musique practice’ in the Diary during this time.Google Scholar

39 The Diary of Samuel Pepys, iii, 36–7.Google Scholar

40 Ibid. Pepys's anger may have been motivated not only by his own frustration but by his concern with the financial arrangement for the lessons as well. On 24 February he noted that ‘I did give [Birchensha] 5l for this month or five weeks that he hath taught me, which is a great deal of money and troubled me to part with it’ (ibid., 35). When the lessons ended three days later, Pepys wrote, ‘It is not for me to continue with him at 5l. per mensem’ (ibid., 37). He then records that he spent much of that day writing Birchensha's rules ‘in fair order in a book’, but such a document is as yet untraced (ibid., 37 and note 1).Google Scholar

41 The Diary of Samuel Pepys, iii, 35.Google Scholar

42 Birchensha, Templum musicum, Preface.Google Scholar

43 Birchensha, John, ‘An Account of divers particulars, remarkable in my Book …’, last (unnumbered) page.Google Scholar

44 Birchensha, ‘An Extract of a Letter Written to the Royall Society’, 168.Google Scholar

45 Birchensha, Compendious Discourse, 6.Google Scholar

46 Birchensha, ‘An Extract of a Letter Written to the Royall Society’, 170–1.Google Scholar

47 ‘The proportions on which almost all the sciences are founded prove sufficiently that there ought to be a great analogy among these sciences; I am only knowledgeable in music and consequently am not able to venture beyond it, but I dare to regard it rightly as the mirror of nature in the scientific part of the arts, because I doubt that there are other areas where nature permits us to develop her secrets with the same certainty: here reason and the ear agree in every respect.’ Rameau, letter to the Royal Society, 26 February 1750 (Royal Society, Letters and Papers, II, no. 125). Complete letter transcribed in Leta Miller, ‘Rameau and the Royal Society of London: New Letters and Documents’, Music and Letters, 66 (1985), 27.Google Scholar

48 In Salmon's system, the bottom line of the staff would always be G, and the letters T (treble), M (mean), or B (bass) would appear in place of the clef to indicate the octave range.Google Scholar

49 ‘An Accompt of some Books: III, An Essay to the Advancement of Musick, by Tho. Salmon’, Philosophical Transactions of the Royal Society of London, 6, no. 80 (February 1671/2), 3095.Google Scholar

50 The frets were placed independently on each string so as to produce both major and minor tones (9:8 and 10:9) and five sizes of semitone (16:15, 17:16; 18:17, 19:18 and 20:19). Interchangeable fingerboards, which could easily be snapped off and on, accommodated the demands of the various keys. For a complete account of the system, see Salmon's second treatise, A Proposal to Perform Musick in Perfect and Mathematical Proportions (1688), the commentary on this proposal by John Wallis included with the treatise, and Salmon's revised system as described in an article published by the Royal Society: Thomas Salmon, ‘The Theory of Musick reduced to Arithmetical and Geometrical Proportions’, Philosophical Transactions of the Royal Society of London, 24, no. 302 (August 1705), seven irregularly numbered pages (2072, 2069, 2041, 2080, 2076, 2073, 2077). Salmon demonstrated his viol at a meeting of the Society on 27 June 1705 and the following week the Fellows heard a performance by the brothers Frederick and Christian Steffkin using instruments equipped with his new fingerboard. After playing a ‘lesson’ for two viols, they were joined by the violinist Gasparo Visconti in a Corelli sonata. See Society, Royal, Journal Book Copy x, 110–11.Google Scholar