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Title: The Art of Musical Diagrams

The Art of Musical Diagrams

From Boethius to Albersheim and Beyond

by Daniel Muzzulini (Volume editor)
©2026 Edited Collection XXVI, 318 Pages
The Arts
Series: Publikationen der Schweizerischen Musikforschenden Gesellschaft. Serie II / Publications de la Société Suisse de Musicologie. Série II, Volume 64

Summary

Creating diagrams is an art—a techne—shared across the several of the seven liberal arts. This richly illustrated volume explores how music has been conceived, taught, and imagined through diagrams from antiquity to the modern era. Musical diagrams can condense theory or guide practice; they may be schematic or pictorial, static or moving, intuitive or paradoxical. Their designs invite non-linear interpretation and mirror changing modes of musical thought. Contributors bring historical and interdisciplinary perspectives to this largely unexplored visual tradition: the measurement of Pythagorean intervals and Nicolaus Oresme's harmony of the polygons; Theinred of Dover's theory of species; circular diagrams in the Islamic world; the evolution from two-dimensional diagrams of musical hands to three-dimensional interactive devices known as volvelles; Baroque volvelles and their dynamic role in musical representation; and visual representations of timbre. This book reveals how visual strategies have long informed musical reasoning, offering new insight into the interplay between sound, image, and thought.
"A fascinating collecting on an important, neglected topic. The world of musical diagrams continually surprises, and the contributors to this volume do excellent work in making connections within and beyond their own disciplines."
— Benjamin Wardhaugh

Table Of Contents

  • Cover
  • Title
  • Copyright
  • Contents
  • Figures
  • Tables
  • Acknowledgements
  • Introduction (Daniel Muzzulini)
  • Chapter 1: Measuring Pythagorean Intervals: From Boethius to Stifel (Daniel Muzzulini)
  • Chapter 2: The Harmony of the Polygons in Nicolaus Oresme’s “Algorismus Proportionum” (Daniel Muzzulini)
  • Chapter 3: Theinred of Dover’s Theory of Species: Revolution, Rotation, and Circularity (John L. Snyder)
  • Chapter 4: Circle Diagrams in the Music Theory of the Islamic World (Judith I. Haug)
  • Chapter 5: Roman de Volvelle: A Story of Visual Aids in Early Modern Musical Texts (Susan Forscher Weiss)
  • Chapter 6: Volvelles in Baroque Music Theory (Michael R. Dodds)
  • Chapter 7: Fantastic Formants and Where to Find Them (Christoph Reuter)
  • Notes on Contributors
  • Publikationen der schweizerischen musikforschenden gesellschaft, Serie II

Contents

  1. List of Figures

  2. List of Tables

  3. Acknowledgements

  4. Introduction

    Daniel Muzzulini

  5. Chapter 1 Measuring Pythagorean Intervals: From Boethius to Stifel

    Daniel Muzzulini

  6. Chapter 2 The Harmony of the Polygons in Nicolaus Oresme’s “Algorismus Proportionum”

    Daniel Muzzulini

  7. Chapter 3 Theinred of Dover’s Theory of Species: Revolution, Rotation, and Circularity

    John L. Snyder

  8. Chapter 4 Circle Diagrams in the Music Theory of the Islamic World

    Judith I. Haug

  9. Chapter 5 Roman de Volvelle: A Story of Visual Aids in Early Modern Musical Texts

    Susan Forscher Weiss

  10. Chapter 6 Volvelles in Baroque Music Theory

    Michael R. Dodds

  11. Chapter 7 Fantastic Formants and Where to Find Them

    Christoph Reuter

  12. Notes on Contributors

Figures

  1. Fig. 1. Definition of the diatonic and chromatic semitone. Boethius (Ms. Harley 5237, 18v)

  2. Fig. 2. Tables for geometric progressions in Nicomachus’ “Introduction to arithmetic.”

  3. Fig. 3. Table for geometric progressions of the ratio 9/8. Boethius (SBG-Hss Msc.Class.9, 91r)

  4. Fig. 4. Transcription of the diagram from Fig. 3.

  5. Fig. 5. Sources: Boethius (BSB Clm 14465, 31v), Boethius (BnF Latin 7200, 40v)

  6. Fig. 6. Transcription of the numbers from Fig. 5 (BnF Latin 7200).

  7. Fig. 7. Table for geometric progressions of the ratio 9/8. Boethius (BnF Latin 7200, 86v)

  8. Fig. 8. Boethius (BnF Latin 7200, 40v, rotated clockwise by 90°)

  9. Fig. 9. Transcription of the numbers from Fig. 8.

  10. Fig. 10. Comparison of six whole tones with the octave. a) Boethius (BSB Clm 14465, 40v); b) Boethius (Ms. Harley 5237, 27r); c) Boethius (BnF Latin 7200, 92v); d) Boethius (BnF Latin 7200, 57v)

  11. Fig. 11. Modified layout for Fig. 10c

  12. Fig. 12. Modified layout for Fig. 10d

  13. Fig. 13. Boethius (BnF Latin 7200, 40v, 57v, 86v, 92v)

  14. Fig. 14. Number triangle with geometric progressions for the non-epimoric common ratio 5/2.

  15. Fig. 15. Triangles for the non-epimoric ratios 5/2 and 7/3. Maurolico (1575, 118)

  16. Fig. 16. Triangles for geometric sequences of length 4 with the common factors b/a (from A to B), c/b (from B to C) and c/a (from A to C).

  17. Fig. 17. Configuration of the base numbers for the grid in Fig. 18.

  18. Fig. 18. Spiderweb of geometric sequences by Jordanus Nemorarius. Sources: Jordanus (BnF Latin 16644, 80v), Jordanus (Basel UBH F II 33, 83v)

  19. Fig. 19. A small number triangle for the Pythagorean comma

  20. Fig. 20. Estimating 19 : 14 by epimoric ratios

  21. Fig. 21. The diagram by Walter Odington proves that the ratio of the Pythagorean comma is between 75 : 74 and 74 : 73. Odington (CCCC MS 410, 9v)

  22. Fig. 22. Transcription of the numbers from Fig. 21.

  23. Fig. 23. Estimation of the Pythagorean comma

  24. Fig. 24. Estimation of the Pythagorean diatonic semitone 256/243 by epimoric ratios

  25. Fig. 25. Estimation of the semitone according to Walter Odington (CCCC MS 410, 10r)

  26. Fig. 26. Estimation of the semitone according to Boethius (BnF Lat. 7200, 50v, 51r)

  27. Fig. 27. Transcription of the numbers from Figs. 25 and 26

  28. Fig. 28. Estimation of the semitone by Pythagorean commas. Boethius (BSB Clm 14465, [35v])

  29. Fig. 29. Interpretation of Fig. 28.

  30. Fig. 30. Decomposition of the tone into two Pythagorean semitones and a Pythagorean comma

  31. Fig. 31. Subtracting multiples of 7,153, the difference of the Pythagorean comma, from 531,441 results in a sequence of increasing intervals.

  32. Fig. 32. Adding multiples of 7,153 to 524,288 results in a sequence of decreasing intervals.

  33. Fig. 33. Comparison of five tones with two fourths. Jacobus (BnF Latin 7207, 117r)

  34. Fig. 34. Transcription of the relevant numbers from Fig. 33

  35. Fig. 35. Estimation of the semitone by commas. Ugolino (I-Rc Ms 2151, 317r)

  36. Fig. 36. Corrected placement of the multiples L and N of the comma difference K from Fig. 35 to visualise the inequality L < M < N

  37. Fig. 37. Estimation of the semitone by commas. Jacobus (BnF Latin 7207, 120r)

  38. Fig. 38. Estimation of the semitone by commas. Boethius (BnF Latin 7200, 52r)

  39. Fig. 39. Interpretation of Fig. 38

  40. Fig. 40. Estimation of the apotome by commas. Boethius (BnF Latin 7200, 52v)

  41. Fig. 41. Interpretation of the main part of Fig. 40

  42. Fig. 42. Estimation of the apotome according to Jacobus (BnF Latin 7207, 121r)

  43. Fig. 43. Linking the arcs from Fig. 42 as in Fig. 37 would result in a change of layout.

  44. Fig. 44. Estimation of the apotome in a Boethius manuscript (BnF Latin 13908, 100v)

  45. Fig. 45. Estimation of the semitone (256/243) by Pythagorean commas

  46. Fig. 46. Ugolino (I-Rc MS 2151, 318r)

  47. Fig. 47. It cannot be inferred from this constellation that nine commas are greater than the tone.

  48. Fig. 48. Comparison of the fourth to the tritonus by Guillermo de Podio (late 15th century). De Podio (I-Bc A 71, 144 r)

  49. Fig. 49. Division of the octave into 53 micro-intervals. Compendium de Musica (B-Br 10162/66, 51r)

  50. Fig. 50. Maurolico’s estimation of the whole tone. Maurolico (BnF Latin 7462, 12v)

  51. Fig. 51. Decomposition of the whole tone by eight successive epimoric ratios

  52. Fig. 52. Symmetrical division of the tone into two diatonic semitones (limma) and a Pythagorean comma. Hothby (BM Add Ms 36986, 38v)

  53. Fig. 53. Division of the tone (9 : 8) into epimoric ratios (18 : 17 and 17 : 16) and into equal intervals. Boethius (SBG-Hss Msc.Class.9, fol. 93r)

  54. Fig. 54. Interpretation of Philolaus’ division of the whole tone

  55. Fig. 55. Zarlino’s illustration of the division of the whole tone according to Philolaus. Zarlino (1571, 168)

  56. Fig. 56. Halving the tone 36 : 32 = 9 : 8 approximately by “averaging” the semitones 18 : 17 and 17 : 16 by 35 : 33

  57. Fig. 57. Ratios of the semitone minus multiples of the Pythagorean comma. Stifel (1544, 75v)

  58. Fig. 58. Stifel’s chromatic scale. Stifel (1544, 74r)

  59. Fig. 59. Each of the chromatic semitones from the scale in Fig. 58 is replaced by a diatonic semitone and a Pythagorean comma. Stifel (1544, 75r)

  60. Fig. 60. In the “Compendium de Musica” (14th century), each of the five whole tones of the diatonic scale is divided into two diatonic semitones (limmae) and a comma, with the comma at the top. Compendium de Musica (B-Br 10162/66, 51r, 50v)

  61. Fig. 61. John Hothby divides each tone symmetrically into two diatonic semitones and a central Pythagorean comma, unlike Stifel with the comma on top. Source: Hothby (BM Add Ms 36986, 39r) (late 15th century)

  62. Fig. 62. The fourth is equal to 5 diatonic semitones plus 2 Pythagorean commas. Algebraically and graphically. Stifel (1544, 76v)

  63. Fig. 63. Multiplication of the diminished fifth by ¾. Stifel (1544, 78r)

  64. Fig. 64. Bisection of the semitone (256/243) and the fourth (4/3). Stifel (1544, 79r)

  65. Fig. 65. Mathematical notation for fractions and fractional powers of ratios. Sources: Oresme (1868, n. p.), Oresme (Basel F II 33, 95v)

  66. Fig. 66. Complete arc diagrams for the heavenly aspects “oppositus,” “trinus,” “quartus,” “sextilis.” Sources: Oresme (Basel F II 33, 98v); Oresme (Erf. CA 4° 348, 45v)

  67. Fig. 67. The tetraktys {1, 2, 3, 4} as a twofold complete graph with four nodes. Source: Jacobus Leodiensis (BnF Latin 7207, 216v)

  68. Fig. 68. Traditional tetraktys diagrams. Left: The ratios between the numbers 1, 2, 3 and 4 define the primary Pythagorean consonances. Right: Symmetrical division of the octave defining the whole tone (9/8) as the interval difference of the fifth (diapente: 3/2) and the fourth (diatessaron: 4/3).

  69. Fig. 69. The interpretation of Oresme’s diagram on the right is obtained by taking square roots from the numbers and ratios in the tetraktys diagram on the left.

  70. Fig. 70. Two correlated diagrams, on different pages of the same manuscript. Source: Oresme (Basel F II 33, 98r/98v)

  71. Fig. 71. The ratios of the lengths of the inscribed diameter, the sides of the equilateral triangle, square and regular hexagon from the diagram on the left are expressed as a complete arc diagram on the right.

  72. Fig. 72. In “De caelo et mundo” the complete arc diagram from Figs. 70 and 71 is replaced by a legend which indicates all six side ratios. Sources: Oresme (BnF Français 24278, 87r; BnF Français 565, 117r; BnF Français 1082, 127v)

  73. Fig. 73. Diagrams from Chalcidius’ comment on Timaeus. Left: Lambda diagram for powers of 2 and 3. Right: Assignment of the seven numbers of the lambda diagram to celestial bodies. Source: Chalcidius (BnF Latin 6280, 13r, 26r)

  74. Fig. 74. The first two stages in the creation of the World Soul as a musical structure. Source: Lambert of Saint-Omer, Liber Floridus, Universiteitsbibliotheek Ghent, MS 92, 222r

Details

Pages
XXVI, 318
Publication Year
2026
ISBN (PDF)
9783034363563
ISBN (ePUB)
9783034363570
ISBN (Softcover)
9783034363556
DOI
10.3726/b23428
Language
English
Publication date
2026 (June)
Keywords
diagram diagrammatology history of music theory circular diagrams volvelles complete graphs medieval diagrams Guidonian hand timbre space
Published
Lausanne, Berlin, Bruxelles, Chennai, New York, Oxford, 2026. xxvi, 318 pp., 185 fig. col., 99 fig. b/w, 9 tables.
Product Safety
Peter Lang Group AG

Biographical notes

Daniel Muzzulini (Volume editor)

Daniel Muzzulini, musciologist and mathematician, received his PhD with Genealogie der Klangfarbe (2006) at Zurich University. Since 2015 he has been the manager of the project Sound Colour Space - A Virtual Museum at the Zurich University of the Arts. His main research topics are the history of mathematical approaches to music theory and diagrams.

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Title: The Art of Musical Diagrams