Hexachords, solmization, and musica ficta


  1. Christopher Page, "Instruments and Instrumental Music Before 1300," in Richard Crocker and David Hiley, eds., The New Oxford History of Music, Vol. II, 2nd ed.: The Early Middle Ages to 1300, (Oxford University Press, 1990), pp. 455-484; for a facsimile of Hucbald's notation, see p. 458, Fig. 2; for a discussion of the example and its description as a kind of "tablature," see p. 465.

  2. The version of this melody given here appears in Alec Harman, Man & His Music: The Story of Musical Experience in the West, Part One: Medieval and Early Renaissance Music (Schocken, 1969), p. 3, Ex. 1.

  3. Figures such as B3-A3-G3-F3-G3 occur very often, for example, in the 13th-century Cantigas collection of King Alfonso X El Sabio of Spain. As Gordon Anderson comments in his prefatory remarks to his modern edition of Compositions of the Bamberg Manuscript, translations of French texts by Robyn E. Smith, Corpus Mensurabilis Musicae 75 (American Institute of Musicology, 1977), at p. xxxvii, "A G-?-D melodic orbit mostly B-natural to form G-B-D ..., often with a striking use of lower F-natural." More generally, at p. xxxvi, he observes that the tritone either "in a melodic progression" or "in vertical combinations" is "very often not altered, and many cases cannot be altered without involving the performers in new difficulties."

  4. Jacobus of Liege, Jacobus Leodiens Speculum Musicae, ed. Roger Bragard, Corpus Scriptorum de Musica 3 (7 vols), (Rome: American Institute of Musicology. 1955-1973), Vol. 2 (Liber Secundus), Capitulum LXXX, p. 191.

  5. Jacobus of Liege, ibid., Vol. 4 (Liber Quartus), Capitulum XI, p. 22.

  6. Ibid., Capitulum L, p. 123.

  7. See Marchettus of Padua, The Lucidarium of Marchetto of Padua, Jan W. Herlinger, ed. (University of Chicago Press, 1985), pp. 150-151 and Ex. 5 (in facsimile and transcription); and Jan W. Herlinger, "Marchetto's Division of the Whole Tone," Journal of the American Musicological Society 34 (1981), pp. 193-216, at p. 193, Ex. 1. This article offers examples of similar progressions in early 14th-century Italian compositions. Margaret Bent, "Musica Recta and Musica Ficta," Musica Disciplina 26 (1972), 73-100, includes at 82 an example from the opening of a Sanctus in the Old Hall Manuscript (#101) where the outer voices have an 8-M10-12 progression identical to that of Marchettus of Padua, ibid., at pp. 152-153, Ex. 6:

    1   2  | 1 ...
    F4  F#4  G4
    F3  A3   C4
    F3  D3   C3

    Bent remarks that such an inflection is motivated - and the singer guided in its intonation - by a compelling attraction toward "the vertical sonority of a perfect interval."

  8. Andrew Hughes, Manuscript Accidentals: Ficta in Focus 1350-1450, Musical Studies & Documents 27 (American Institute of Musicology, 1972), Latin text at pp. 21-29, English translation and commentary at pp. 29-39; the inflection to obtain a major sixth before an octave is described in sentence 40, p. 23; and paraphrase of sentences 40-46, p. 31 and Ex. 3a.

  9. Ibid., pp. 29-30 and Ex. 1a. Gaston G. Allaire, The Theory of Hexachords, Solmization and the Modal System: A Practical Approach, Musicological Studies and Documents 24 (American Institute of Musicology, 1972), pp. 42-43, likewise notes that "the flat (b) sign did not always require lowering a note by a semitone, nor did the sharp (#) or the B-durum [square-B, later natural] signs always demand raising a note by a semitone."

  10. Richard Hoppin, Medieval Music, Norton Introduction to Music History 1 (Norton, 1978), pp. 70-71.

  11. See Prosdocimo de' Beldomandi, Brevi Summula Proportionum Quantum Ad Musicam Pertinet and Parvus Tractatulus De Modo Monachordum Dividendi: A Short Summary of Ratios Insofar as They Pertain to Music and A Little Treatise on the Method of Dividing the Monachord, Jan Herlinger, ed., Greek and Latin Music Theory (University of Nebraska Press, 1987), facsimile example and transcription with Latin text and English translation at pp. 106-107.

  12. See Hughes (n. 8 above), p. 24, sentence 50 of Latin text; p. 32, paraphrase of sentences 47-51 and Ex. 3b, and commentary at p. 38. Ugolino does explain that this Bb in the tenor is used to obtain a major sixth with the upper part.

  13. See Andrew Hughes and Margaret Bent, eds., The Old Hall Manuscript, Vol. I (Part 1), Corpus Mensurabilis Musicae 46 (1969), prefatory remarks, at pp. XXII-XXIV, and Bent (n. 7 above), pp. 95-96. Bent, ibid., pp. 97-99, and Hughes (n. 8 above), pp. 51-54, address the problem of partial signatures in terms of musica recta and musica ficta, Hughes, p. 52, concluding that the reason for this technique "is simply the preservation of as much recta music as possible."

  14. To demonstrate the acceptability of the fourth, incidentally, Jacobus (Book VII, Chapter 5) cites an example of three-voice organum from Guido's Micrologus of around 1030, a treatise of about three centuries earlier. See Jacobus of Liege (n. 4 above), Vol. 7 (Liber Septimus), pp. 11-12. In this example of diaphony or organum, the text Spera in Domine et fac bonitatem is set to a series of sonorities with an outer octave, lower fourth, and upper fifth (e.g. C3-F3-C4, D3-G3-D4, etc.). Jacobus comments that in this example "Guido places the fifth above the fourth. This, however, is contrary to a certain modern doctor [likely Johannes de Muris] who posits that the fourth below the fifth is not a consonance, but rather above."

  15. For Boen's passage on the perfect, diminished, or augmented fourth as a consonantia per accidens, see Wolf Frobenius, Johannes Boens Musica und Seine Konsonanzenlehre, Freiburger Schriften zur Musikwissenschaft (Musikwissenschaftliche Verlags-Gesselschaft mbH Stuttgart, 1971), pp. 74-76. Around the early 15th century, the Pythagorean diminished fourth becomes popular on keyboards as a variant tuning of the major third (e.g. written D3-F#3 realized as D3-Gb4, with sharps tuned as Pythagorean flats).

  16. Hughes (n. 8 above), pp. 93-94.

  17. Andrew Hughes, "Solmization," New Grove Dictionary of Music and Musicians 17:458-462, ed. Stanley Sadie. Washington, DC: Grove's Dictionaries of Music (1980), at p. 460. Bent (n. 7 above), pp. 89-90, quotes this passage in fuller context, including a complaint that many singers proceed from re to sol via fa and scarcely ever place a semitone between fa and sol - an apparent objection to the omission of a musica ficta inflection rather than to its commission. From this viewpoint, she interprets the remarks about sol-fa-sol and re-ut-re to protest not the semitonal inflection itself but the licentious solmization. However, the statement that the latter practice confuses the diatonic order and falsifies the chant might suggest that this complaint could be about the inflections themselves, possibly deemed appropriate for secular or polyphonic music but not for plainsong. One tempting approach to reconcile the two complaints might be to guess that the author intended in the first complaint to say that many modern singers scarcely ever ascend from re to sol via fa without singing a semitone between fa and sol. It would be interesting to canvass any more recent scholarly discussions of this passage.

  18. Oliver B. Ellsworth, ed., The Berkeley Manuscript: University of California Music Library, MS. 744 [olim Phillipps 4450], Greek and Latin Music Theory (University of Nebraska Press, 1984), Tractatus Quintus, pp. 240-247.

  19. Oliver B. Ellsworth, "A Fourteenth-Century Proposal for Equal Temperament," Viator 5 (1974), pp. 445-453. For Costeley's tuning, see Kenneth J. Levy, "Costeley's Chromatic Chanson," Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

  20. Ellsworth (n. 18 above), p. 242-243.

  21. Ibid.

  22. This kind of "comprehended" semitone is described by Ramos (Section 3.1 below) about a century later (1482) as a semitonium subintellectum, which one might translate as an "understood" or "mental" semitone.

  23. Act III, Scene I.

  24. Thomas Morley, A Plain & Easy Introduction to Practical Music, Alec Harman, ed., 2nd ed. (W. W. Norton, 1973), p. 13.

  25. Allaire (n. 9 above), pp. 152-153.

  26. Bartolomeo Ramis [or Ramos] de Pareia, Musica Practica, Clement A. Miller, ed. and trans., Musicological Studies & Documents 44 (American Institute of Musicology, Hanssler-Verlag, 1993), p. 93-99.

  27. Mark Lindley, "Fifteenth-Century Evidence for Meantone Temperament," Proceedings of the Royal Musical Association 102 (1976), pp. 37-51.

  28. Ramos (n. 26 above), p. 168; Lindley, ibid., p. 47.

  29. Ramos, ibid., p. 121.

  30. Thus Ramos, ibid. at 168, notes that such an Eb in the middle voice "is not connected to this [B] in a true ratio." In contrast, on a Pythagorean keyboard, a diminished fourth such as B3-Eb4 would actually be smoother than a regular major third.

  31. For a Latin text see Johannes Wolf, ed., Musica Practica Bartolomei Rami de Pareia (Breitkopf & Hartel, 1901), pp. 101-102; and Lindley (n. 27 above) at p. 48 (with English translation).

  32. Ramos (n. 26 above), p. 106.

  33. Pietro Aaron, Toscanello in Musica, trans. Peter Bergquist (3 vols.), Colorado College Music Press Translations 4 (Colorado College Music Press, 1970), Vol. 2 (Book II, Chapters I-XXXVI), Chapter 20, at p. 34, gives an example of a written sonority E3-E4-G4 where he notes that the singer of the highest part would add an inflection of G#4 in order to obtain a major tenth above the bass, see Section 3.2 below. In his discussion of the modes in polyphonic music, Trattato della natura e cognizione di tutti gli toni di canto figurato (1525), Aaron gives examples of four-voice cadences in the first and second modes on D with an F# inflection to provide a major third in the final sonority; and likewise such cadences in the third and fourth modes on E with a G# inflection to provide such a closing major third. See excerpts from this treatise in Oliver Strunk, ed. and trans., The Renaissance, Source Readings in Music History 2 (W. W. Norton, 1965), pp. 22, 24.

  34. Lindley (n. 27 above), p. 48, and p. 47, Ex. 1 (iv)-(vi).

  35. Nicola Vicentino, Ancient Music Adapted to Modern Practice, tr. Maria Rika Maniates, ed. Claude V. Palisca (Yale University Press, 1996), Book III on Music Practice, Chapter 32, p. 176, at the conclusion of Example 32.3 showing some four-voice cadences in the third mode "in the Hard Hexachord" - that is, E Phrygian, characteristically using B-natural.

  36. Ramos (n. 26 above), p. 120; Wolf (n. 31 above), p. 66.

  37. See also nn. 17-22 above.

  38. Ramos (n. 26 above), pp. 93-94.

  39. Ibid., p. 94 n. 63. Here the melodic figure is A3-G3-A3-D3, or la-sol-la-re in the natural hexachord C3-A3; Gaffurius thus suggests that a singer might actually intone A3-G[#]3-A3-D3.

  40. See Aaron, n. 33 above, for Peter Bergquist's complete translation including a Supplement on questions of accidentals added to the 1529 and later editions.

  41. Ibid., Vol. 2, Book II, Chapter XVI, pp. 27-28.

  42. Ibid., Vol. 2, Book 2, Chapter XIV, p. 26.

  43. Ibid., Vol. 2, Book 2, Chapter XVIII, pp. 30-31.

  44. Ibid., Vol. 3 (Book II, Chapters XXXVII-XXXX; Supplement), "Supplement to the Toscanello Made for the Satisfaction of my Friends," pp. 12-35; in addition to the discussion of accidentals and accidental signs, pp. 12-24, Aaron discusses the modality of Gregorian chants used for the Ordinary of the Mass and for the Te Deum, pp. 24-33. He opens the first portion of his discourse by explaining that "[d]oubts and disputations are circulating among some lovers of music," about the use of the b molle (flat) and diesis (sharp) signs, and "whether composers are constrained by necessity to show them in their compositions, or whether the singer should be held to understand and recognize the hidden secret of all the places where these figures or signs are needed," ibid. at p. 12.

  45. See, e.g., Lowinsky, Edward E., The Secret Chromatic Art in the Netherlands Motet, trans. Carl Buchman, Columbia University Studies in Musicology 6 (Russell & Russell, 1967); Margaret Bent, "Diatonic Ficta," Early Music History 4 (1984), 1-48. For a more recent dialogue regarding some issues of Bent's approach, see Roger Wibberley, "Josquin's Ave Maria: Musica Ficta versus Mode," Music Theory Online 2.5 (1996); Margaret Bent, "Diatonic ficta revisited: Josquin's Ave Maria in context, Music Theory Online 2.6 (1996); and Roger Wibberley, "'Mode versus Ficta' in Context," Music Theory Online 2.7 (1996).

  46. On the evidence provided by intabulations, see Robert Toft, Aural Images of Lost Traditions: Sharps and Flats in the Sixteenth Century (University of Toronto Press, 1992), especially pp. 106-111 on a motet by Clemens non Papa, Fremuit spiritu Jesus (a piece proposed by Lowinsky for a "secret chromatic art" reading), and pp. 161-162 n. 5.

  47. Aaron (n. 33 above), vol. 3, Supplement, p. 15.

  48. Ibid., p. 16.

  49. Ibid., p. 17.

  50. Ibid., p. 17.

  51. Ibid., vol. 3, Book II, Chapter XX, pp. 33-34; see also Toft (n. 46 above), pp. 23-24. Vicentino (n. 35 above), p. 253, likewise advises: "When a composer requires repose on a consonance in the middle of a composition, it is infinitely better to stop on a major than on a minor third."

  52. Aaron, n. 33 above, vol. 3, Supplement, p. 23.

  53. Ibid., p. 20.

  54. Ibid., p. 20.

  55. Toft (n. 46 above), p. 65, Ex. 2.30 (b); Pater Noster I, mm. 96-97.

  56. Ibid.

  57. Ibid., p. 97.

  58. Morley (n. 24 above), p. 147.

  59. Toft (n. 46 above), p. 45.

  60. Ibid., p. 45.

  61. Bernard Thomas, ed., Philippe Verdelot: Twenty-two madrigals for four voices or instruments, The Italian Madrigal 3, LPM MA3 (London Pro Musica Edition, c. 1980), p. 3, "Editorial Note."

  62. Ibid., Madonna, il tuo bel viso (#13), at p. 34, mm. 47-48.

  63. Toft (n. 46 above), p. 56.

  64. Ibid., p. 56.

  65. See ibid., pp. 42-45, for a discussion of the relevance of the theory of vocal polyphony to instrumentalists, and on the intabulation technique of building the texture line by line, revising accidentals as the process proceeds; on the use of imperfect octaves, see pp. 80-82, and also pp. 110-111 and 130 for some instances which Toft considers more problematic as judged by "normal Renaissance procedures." For a recent canvassing of the views of Peter Urquhart and others, see the thread on the Usenet newsgroup rec.music.early, "To ficta or not to ficta" (December 1999-January 2000).

  66. Toft (n. 46 above), pp. 30-32.

  67. Ibid., p. 97.

  68. See ibid., e.g. pp. 45-47 and 62-71 on cadential progressions, and pp. 73-79 on vertical or linear tritones or diminished fifths.

  69. Ibid., p. 44.

  70. Allaire (n. 9 above), p. 45.

  71. Ibid., p. 51.

  72. See, e.g., Hughes, "Solmization" (n. 17 above), p. 460.

  73. Thus Allaire (n. 9 above), p. 97 and p. 99, suggests that "technical deficiencies on the part of either the singer or copyist" might account for some discrepancies between sources such as choirbooks, citing the example of Montpellier Codex H. 159 "in which the rule of fa supra la is often not applied." Ibid. at p. 46, he asserts that "[t]he rule of fa supra la is confirmed in the writings of the medieval theorists," citing an example from Anonymous XI (early 15th century?); ibid. at p. 26, he proposes a basis for this rule in the nature of hexachord orders. For remarks querying the earliest statement of the "fa supra la" rule, and questioning its systematic application to medieval practice, see Hughes and Bent (n. 13 above), p. XXIII and n. 19; and Bent, "Musica Recta and Musica Ficta" (n. 7 above), p. 92 and n. 21, citing a statement from the Enchiridion (Rhaw or Rhau, 1518), that if a melody in the first or second mode proceeds beyond la [only] as far as a second, one should always sing fa; and likewise, if the melody descends again to F fa ut; otherwise, mi should be sung, "as you see in the hymn Ave Maris Stella." Hughes, "Solmization" (n. 17 above) suggests that while such a "semitone excursion" to a single step above or below a hexachord without benefit of a proper mutation arose as a "license" at "a much earlier date" than its endorsement by some Renaissance theorists, it should not be taken as standard medieval practice. See also Anderson's statement, n. 3 above, on the fluid use of B-natural and Bb in 13th-century polyphony.

  74. Toft (n. 46 above), p. 68, Ex. 1 (a), from Memor esto I, mm. 64-66, (Newsidler 1536).

  75. Ibid., p. 68; for both sides of this intonational question regarding figures such as A3-B[b?]3-A3-D3, or D3-E[b?]3-D3-G2 in pieces with a Bb signature (G Dorian), see discussion and examples at pp. 65-70.

  76. Hughes, "Solmization" (n. 17 above), p. 460, refers to "the sharpening of the note below the hexachord which one sings as mi."

  77. Toft (n. 46 above), pp. 24-25.

  78. See Hughes, "Solmization" (n. 17 above), p. 462; and Allaire (n. 9 above), pp. 51-52, on the solmization of such larger intervals, quoting Yssandon's rule (1582) that the same hexachord syllabe be used for both notes, but also noting that leaps remaining within the bounds of a single hexachord would not require the kind of mutation for which Yssandon's rule might serve as a shortcut.

  79. For this example I am indebted to Blanco y Negro: Hispanic Songs of the Renaissance from the Old and New World, Ancient Consort Singers directed by John Alexander, Ancient Instrumental Ensemble directed by Ron Purcell (Patrician, Klavier Records KS-540, 1975), liner notes, Side II, Band 2 (version of La mi sola Laureola a4 by Juan Ponce, c. 1510): "His graceful setting borrows the first words to form solmization syllables, `La-mi-sol-la,' and thus the first melodic pitches, which enter imitatively."

  80. Donald Jay Grout, A History of Western Music, 1st ed. (Norton, 1960), p. 182.

  81. Gioseffo Zarlino, The Art of Counterpoint: Part Three of Le Istitutioni harmoniche, 1558, trans. Guy A. Marco and Claude Palisca (W. W. Norton, 1976). In Chapter 38, Zarlino presents this rule: "One should proceed from one consonance by means of the nearest. By this should be understood that the composer moving from an imperfect consonance to a perfect consonance must be certain that the imperfect one is actually the nearest." As examples he includes the traditional progressions by stepwise contrary motion M6-8 and m3-1, and also the typical 16th-century progression by contrary motion from the major third to the octave (which he notes elsewhere commonly occurs in multi-voice cadences where the bass descends a fifth and one of the upper voices rises by a semitone, e.g. D3-F#3 to G2-G3, Chapter 53, pp. 147-148). Zarlino emphasizes that the rule applies to oblique as well as contrary motion, thus m6-5 rather than M6-5. While similarly prescribing an oblique progression of M3-5, he adds that in stepwise contrary motion, "especially in two-voice writing," m3-5 is preferred "although more distant," in order to avoid a nonharmonic relation of the tritone.

  82. Ibid., p. 82.

  83. Ibid., p. 82.

  84. Ibid., p. 83.

  85. Ibid., Chapter 42, p. 103. Here Zarlino is addressing the situation where a minim or semibreve is followed by two semiminims, concluding that it is the first semiminim which may be dissonant, while the second must be consonant. He adds that it would not do for the "ordered science" of music "to have anything at all disorderly," which would be the case "if these semiminims were written one way by some composers and another way by the rest."

  86. Toft (n. 46 above), pp. 95-102.

  87. "The German Custom" is the title of Toft's Chapter 3, Ibid., pp. 95-102. As Toft notes in this chapter, however, German intabulators do frequently use inflected semitones of the usual variety; and he notes some example of cadences involving m6-8 or M3-1, etc., in non-German intabulations (see pp. 45-64). Toft describes such cadences as "subtonal" (both voices moving by whole-tone), in contrast to "subsemitonal" cadences (ascending semitone) and "suprasemitonal" cadences (descending semitone).

  88. Ibid., pp. 95-96.

  89. See n. 35 above.

  90. Ibid., Book II on Music Practice, Chapter 19, p. 113.

  91. Ibid., Chapter 18, p. 112, Vicentino finds the minor sixth "somewhat sonorous and rather sad. It inclines readily toward the fifth. Indeed it is so close to the latter that when these two intervals wish to be joined together, they require but one step, that of the major semitone, to pass from one to the other." (On the major semitone, the usual diatonic semitone in 16th-century tunings for voices or keyboards, see n. 100 below.) He explains that the fifth "has so much intrinsic harmony" that it bestows this harmony upon the minor sixth "owing to the proximity of the two," comparing their relation to that of the sun and the moon. Zarlino (n. 81 above), Chapter 38, p. 81, likewise notes a "kinship and consensus" between the minor sixth and the fifth: "Therefore we go to the fifth from the minor sixth, for it shares a consensus with it and is nearest to it."

  92. Vicentino, ibid., Book III on Music Practice, Chapter 15, p. 150; here Vicentino is discussing the practice of shifting from one modal order to another in order to express the words, which he finds appropriate for genres where the music is not obliged to "sustain the mode" (as it is in "sacred works that anticipate the response of choir or organ, such as masses, psalms, hymns, or other responses expecting a reply"). In one of his chapters on the minor sixth, Book II on Music Practice, Chapter 18, p. 112, he observes that some progressions involving this interval are more pleasing than others. "But all of them may be used, the well and the badly composed, for a composer selects them according to the subject of the words."

  93. Oliver Strunk, ed. and trans., The Baroque Era, Source Readings in Music History 3 (W. W. Norton, 1965), pp. 45-52; this manifesto, appearing at the end of Claudio Monteverdi's Scherzo musicale (1607) is largely a gloss of Giulio Cesare Monteverdi upon his brother Claudio's brief remarks offered in his Fifth Book of Madrigals (1605) in response to his critic Giovanni Maria Artusi.

  94. Zarlino (n. 81 above), Chapter 53, pp. 144-145.

  95. Vicentino (n. 35 above), Book III on Music Practice, Chapter 27, p. 168.

  96. Ibid., p. 168.

  97. In usual vocal and keyboard tunings of the 16th century, the augmented sixth would actually be about 1/5-tone smaller than a usual minor seventh; on a lute with equal semitones, the two intervals would be identical.

  98. This approach may reflect the experimental nature of Vicentino's music, using the ancient Greek genera (diatonic, chromatic, enharmonic) as the basis for a polyphonic practice. Thus ibid., in Book III on Music Practice, Chapter 52, p. 211, Vicentino introduces the first part of an otherwise unknown madrigal Dolce mio ben (transcribed pp. 211-213) which can be sung in the "diatonic" genus (disregarding all inflection marks, "flats, naturals, sharps, or enharmonic dots"); then in the "chromatic" (with flats, naturals, and sharps); and in the "enharmonic," including also the dots above notes which indicate their raising by a diesis of around 1/5-tone. For a discussion of this kind of performance option in Vicentino, and complications in practice, see the editorial introduction, pp. liii-lviii. Ibid. at 211, Vicentino adds that "you can benefit contemporary compositions by adding sharps and enharmonic dots between their whole tones and semitones," apparently referring to pieces in the usual style.

  99. Ibid., Book II on Music Practice, Chapter 9, pp. 101-102 and Ex. 9.2.

  100. Ibid., p. 102. Vicentino in this passage more precisely specifies the "two major semitones" as having this virtue, as does Zarlino in other passages on various standard progressions where one of the voices moves by a semitone (e.g. m3-1, M6-8, oblique m6-5). Renaissance keyboard tunings in meantone generally divide a whole-tone into a larger or "major" diatonic semitone (e.g. A-Bb) and a smaller or "minor" chromatic semitone (e.g. Bb-B), and 16th-century vocal tunings gravitating toward pure thirds and sixths share this property. In medieval Pythagorean tuning (based on pure fifth and fourths), in contrast, the diatonic semitone (e.g. A-Bb) is smaller than the chromatic (e.g. Bb-B). While this intonational shift reflects a musical change in the role of thirds and sixths, it has no dramatic effect on the basic theory of solmization or musica ficta. The hexachord step mi-fa remains a diatonic semitone of whatever size, and "closest approach" progressions such as M6-8 likewise involve the progression of a whole-tone in one voice and a diatonic semitone in the other.

  101. Allaire (n. 25 above), pp. 57-58 (quotation given in French).

  102. Tomás de Santa María (or Thomas de Sancta Maria), Libro llamado Arte de tañer Fantasia, assi para Tecla como para Vihuela, y todo instrumento, en que se pudiere tañer a tres, y a quatro vozes, y a mas ("The Book called the Art of playing Fantasia, for Keyboard as for Vihuela, and every instrument on which it is possible to play in three, four, or more voices"), facsimile edition (Gregg International Publishers Limited, 1972) of 1565 edition (Francisco Fernandez de Cordova). For the solmization steps of the transposed or "accidental" modes, see Part I, Chapter 25, pp. 71r-73v. A modern English edition is also available: Almonte C. Howell, Jr. and Warren E. Hultberg, eds., The art of playing the fantasia by Fray Thomas de Sancta Maria (Latin American Literary Review Press, 1991). For a discussion of Santa María's technique of four-voice composition or improvisation using mostly progressions in note-against-note motion, see Miguel A. Roig-Francoli, "Playing in consonances: A Spanish Renaissance technique of chordal improvisation," Early Music (August 1995), pp. 461-471.

  103. Ibid., Chapter 26, p. 76r.

  104. Ibid., p. 78v.

  105. Another way of stating Santa María's rule to avoid such a result would be that a cadential suspension of the eleventh must resolve to a major tenth, again requiring a subsemitonal motion (here A4-G#4-A4).

  106. See Morley, n. 24 above, pp. 280-281 n. 4 for the actual example from Vox in Rama, based on Lowinsky (n. 46 above), pp. 11-13; given the ambiguous conventions of this example with the tenor progression Bb3-A3-G#3 placed in brackets, but both accidentals set before the notes in the usual manner, it may be helpful to explain that while the Bb is a manuscript accidental, the G# is an editorial addition regarded as obligatory by Lowinsky and evidently concurred on by Harman.

  107. Ibid., pp. 261-262.

  108. Giovanni Paolo Cima, Partito de Ricercari & Canzoni Alla Francese (1606), ed. Clare G. Rayner, Corpus of Early Keyboard Music 20, gen. ed. Willi Apel (American Institute of Musicology, 1969). See pp. 63-71 for the original piece and its 11 transpositions, each transposition prefaced by Cima's instructions for retuning, pp. 62 for his general remarks on such transpositions, and pp. 89-90 for English translations.

  109. Fabio Colonna, La Sambuca Lincea, overo Dell'Istromento Musico Perfetto, con annotazioni critiche manoscritte di Scipione Stella (1618-1622), ed. Patrizio Barbieri, Musurgiana 24 (Lucca: Libreria Musicale Italiana, 1991), pp. 63-65, demonstrating such transpositions as Ionian, Mixolydian, Dorian, Aeolian, and Phrygian on finals of Gb and Db. In Colonna's likely tuning, 1/4-comma meantone with pure major thirds, a diatonic or major semitone is equal to about 3/5-tone. Thus every mi-fa semitone requires three steps in his five-part division of the tone, and every hexachordal whole-tone five steps. From C to D, for example, the steps on his keyboard are notated C-Cx-C#-Db-Dh-D, where "h" stands for a square-B sign (like the natural sign). From E to F, a diatonic semitone, the steps are E-Ex-E#/Fh-F. Thus the steps for Gb Dorian are written:

        T    S    T     T     T    S    T
    Gb4   Ab4 Ax4   Bx4   Db5   Eb5 Ex5   Gb5
    re    mi  fa    sol   re    mi  fa    sol
        5    3    5     5     5    3    5
  110. Ramos (n. 26 above), pp. 64-65.

  111. Ibid., p. 65.

  112. Ibid., p. 64.

  113. Ibid., p. 67.

  114. Ibid., pp. 94-95.

  115. Ibid.

  116. Ibid.

  117. Ibid., p. 95.

  118. See ibid., pp. 96-98 and facsimile of Ramos' hand at p. 98.

  119. See Hughes (n. 17 above), p. 462.

  120. Ramos (n. 26 above), p. 95.

  121. See, e.g., Johannes Lippius, Synopsis of new music; Synopsis musicae novae, Benito V. Rivera, trans., Colorado College Music Press Translations 8 (Colorado College Music Press, 1977); and Benito V. Rivera, German Music Theory in the Early 17th Century: The Treatises of Johannes Lippius (George Buelow, ed., Studies in Musicology 17), UMI Research Press, Ann Arbor, 1980.

  122. Lippius, ibid., pp. 31-33; and Rivera, ibid., pp. 63-64 and 73 n. 101. Rivera, and also an entry on "Bocedization" in the New Grove Dictionary of Music and Musicians (n. 17 above), 2:831, discuss some possible origins for this and related systems, one reputed inventor being the composer Hubert Waelrunt (c. 1515-1595) who was said to have introduced this technique at the music school he founded in Antwerp in 1547, and another Belgic candidate being David Mostart (c. 1550-1615). Other German theorists drawn to such systems around 1600 include Burmeister and Calvisius. A feature of bocedization is that it retains the familiar Guidonian series of vowels for the second through sixth syllables: [bo]-ce-di-ga-lo-ma-[ni], [ut]-re-mi-fa-sol-la.

  123. Lippius, ibid., p. 131.

  124. However, some other theorists such as Calvisius proposed that the additional heptachord step should be fluid, B/Bb in the C-C heptachord and E/Eb in the F-F hexachord: In addition to the usual ni for B in C-C and E in F-F, he suggests pa for Bb in C-C or Eb in F-F. See "Bocedization," n. 122 above. Others in this epoch advocate heptachord systems adding an extra syllable to the usual Guidonian series, e.g. ut-re-mi-fa-sol-la-sa/si (with a fluid seventh degree, e.g. in the C-C heptachord, sa for Bb and si for B-natural). Note that either the ba/ni or sa/si scheme has a vowel analogy to the familiar Guidonian fa/mi: ba or sa (like fa) has semitone immediately below, while ni or si (like mi) has a semitone immediately above. If one adds this fluid seventh step to each hexachord of the usual musica recta and musica ficta systems, the result is a scheme which avoids the need for some mutations while supporting all of the traditional mutations and inflections. It would be interesting to see whether and how theorists around 1600 advocating heptachord systems approached the matter of musica ficta.

  125. Lippius (n. 121 above), p. 31; Rivera (n. 121 above), p. 63.

  126. Lippius, ibid. at p. 33, offers a diagram showing the intervals and ratios of a sample two-octave segment of his ideal gamut and tuning system (favoring, like Zarlino, Ptolemy's syntonic diatonic, which gives pure ratios of 5:4 and 6:5 for major and minor thirds). He uses an extended system of lines and spaces like those of a usual musical staff, but in place of the customary clefs uses the letters B-L-b-l-bb to show the first and fifth degrees B(o) and L(o) ascending through two octaves. These clefs could be equivalent, for example, to F2-C3-F3-C4-F4 in cantus mollis (F-F heptachords, Bb signature), or C2-G2-C3-G3-C4 in cantus durus (C-C heptachords, no signature). Ibid., p. 32, he proposes to replace the usual letter names of the notes with "the initials of the seven syllables," b-c-d-g-l-m-n, "which will be valid for both cantus durus and mollis."

  127. See "Bebization," New Grove Dictionary of Music and Musicians (n. 17 above), 2:328-329; and Martin Ruhnke, "Solmisation," Die Musik in Geschichte und Gegenwart: Allgemeine Enzyklopadie der Musik (Barenreiter Kassel, 1965), 12:843-852 (page numbers are actually column numbers, with two columns per page), with Hitzler's system shown at p. 848, Tab. 1.

  128. See "Bebization," ibid., at p. 329.

  129. See Ruhnke, "Solmization" (n. 127 above), pp. 848-849 and Tab. 2.

  130. See Lindley (n. 27 above), pp. 48-49.

  131. Morley (n. 24 above), pp. 10-18 on solmization, and p. 18 for a series of "plainsong" exercises solmized using only mi-fa-sol-la.

  132. See, e.g., Warren Steel, Sacred Harp Singing FAQ (Frequently Asked Questions). As Steel notes, the "Rudiments of Music" sections in the shape-note music books such as the Sacred Harp and Southern Harmony give rules such as: "If B be flat, mi is in E."

  133. George Pullen Jackson, "The Story of the Sacred Harp," in B. F. White and E. J. King, The Sacred Harp, facsimile of the third edition, 1859 (Broadman Press, 1968), at p. xiv, describes the Dorian mode used in practice: "It is that scale which has the lowered third and seventh and perfect sixth." (Emphasis in original.) The use of "perfect sixth" to describe a major sixth recalls such theorists of the Ars Nova as Johannes Boen (1357, see n. 15 above) and Prosdocimus (1413, see n. 11 above). Jackson adds regarding the performance of Sacred Harp pieces notated in minor keys: "And the seventh is nearly always sung as a lowered or natural tone, even though it may not be printed as such," ibid. (Emphasis in original.) In other words, written sharps on this degree are as a rule disregarded.

  134. For an example, see Charles Seeger, Studies in Musicology 1935-1975 (University of California Press, 1977), "Contrapuntal Style in the Three-Voice Shape-Note Hymns of the United States," pp. 237-251 at pp. 239 and 241 Ex. 3, a facsimile of the song What Wondrous Love. Although this version from the Harp of Columbia (1849) uses seven-shape notation based on the common heptachordal do-re-mi system rather than fasola, the first three notes in the tenor (middle) part at m. 6, F5-E[b]5-D5, would have the same syllables and shapes in either system (following a signature of Bb and Eb): sol-fa-mi, or round-triangle-diamond noteheads. The notehead sequence for the actually sung F5-E5-D5 (mi-la-sol) in four-shape fasola would be diamond-square-round; in seven-note notation (do-ti-la), it would be x-arrow-square (with do as an x-shaped notehead and ti as an arrowlike shape centered on the stem). For this piece, as Seeger observes (p. 239), the signature accidentals are not actually written, but the note-shape of G as la (square head) implies two flats.

  135. Ibid., p. 239. Strictly speaking, singing an inflection of B-natural (written or unwritten) in a piece with a signature of Bb and Eb could qualify as musica ficta in the original medieval sense also, if we apply the concept of gamut transpositions (Section 1.6). Such a signature would transpose the regular gamut down a whole-tone, so that we would have recta hexachords of Bb-G, F-D, and Eb-C (corresponding to C-A, G-E, and F-D). In this transposed gamut, there is no recta note of E-natural (Emi), which therefore involves musica ficta.

  136. Pal Jardanyi, "The Determining of Scales and Solmization in Hungarian Musical Folklore," in Studia Memoriae Belae Bartok Sacra (Boosey and Hawkes, 1959), pp. 305-310.

  137. Ibid., at p. 307; the title of the song is Erre kakas, erre tyu/k.

  138. For the term "melodic orbit," apparently synonymous with "melodic segment," see Anderson (n. 3 above), pp. xxxvi-xxvii. "A melodic segment centres around the interval of a fifth, to which may be added as important elements the third and octave (upper and lower, depending on position" (p. xxxvi n. 11). Examples of such orbits within a typical 13th-century motet range (say C3-A4) include "an F-A-C (lower) melodic orbit" often calling for Bb; "a higher A-C-E orbit" retaining B-natural; a "C-E-G (upper or lower) orbit" generally avoiding Bb; a "lower D-F-A melodic orbit" with "many striking uses of B-natural..., but also a number of normal uses of Bb"; and a "G-?-D melodic orbit" which "mostly has B-natural to form G-B-D."

  139. Allaire (n. 9 above), p. 63.

  140. Ruhnke (n. 129 above) mentions a proposal of the theorist Joachim Burmeister (1601) for a heptachord system using Bb-se/B-si.

  141. Three-voice conducti suggesting an F/F# contrast include Gedeonis area and Crucificat omnes. Anderson (n. 3 above), p. xxxvii, finds that F# "is almost wholly confined" to the triplum, at least in the Bamberg repertory.

  142. Rivera (n. 118 above), p. 103.

  143. Guido's Micrologus, translated in Warren Babb, trans., Claude V. Palisca, ed., Hucbald, Guido, and John on Music: Three Medieval Treatises (Yale University Press, 1978), pp. 61-62.

  144. Vicentino (n. 35 above), e.g. Book III on Music Practice, Chapter 32, p. 173: "The principal limits of a mode fall on the first and last notes of its fifth and fourth." While the derivation of a mode from an octave divided into fifth-plus-fourth (authentic) or fourth-plus-fifth (plagal) is standard in medieval theory, Vicentino's emphasis on the importance of these intervals in the bass part of a polyphonic composition has a new 16th-century flavor. See ibid., Chapter 15, at p. 151, on determining the mode of a piece: "Let students first rely on the bass, for in that part there appear the fourths and fifths that form all the modes." Chapter 16, p. 153, gives an example (Ex. 16.1) in the second mode (Hypodorian) to demonstrate how "the modes ... are formed with their fourths and fifths" in usual polyphonic music; a brief bass line (eight measures in transcription) is shown with numerous prominent leaps such as D3-A2, A3-D3, D3-A3, and a concluding D3-A2-D3, as well as other melodic outlines of these intervals.

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Margo Schulter