II. Counterpoint and Galant Schemas
Gradus ad Parnassum Exercises
Mark Gotham
The previous chapters have introduced species counterpoint and the iconic pedagogical treatise on cantus firmus composition, Johann Joseph Fux’s Gradus ad Parnassum (1725). This page provides the full cantus firmus exercises of that treatise so you can try your hand at species counterpoint à la Fux.
Gradus ad Parnassum is in three sections:
- Two Voices (46 exercises)
- Three Voices (44 exercises)
- Four Voices (32 exercises)
For each of these sections, we provide four files:
- Exercises: all of the cantus firmus exercises with only the cantus firmus present and the other part(s) left blank for you to complete.
- Solutions: all of Fux’s solutions to those exercises—i.e., both the cantus firmus and the additional part(s) that Fux wrote as answers.
- Annotations: Those solutions annotated with the interval that every note in each additional part forms with the cantus firmus part.
- Distinct: Every distinct cantus firmus in all parts. All of the exercises are based on this format, so these simple files distill all possibilities: you can use this file to do any species exercise, on any cantus firmus, in any part arrangement.
Everything is provided in an editable format so teachers can adjust and combine exercises freely for their own classes’ needs. Specifically, the files are provided in the following formats:
- .mxl: open these in any music notation package like MuseScore, Sibelius, or Finale
- .mscz: for MuseScore specifically, preserving all original formatting
View Online and/or Download
View these files on MuseScore.com (and download in multiple formats if you have an account):
- The “Annotations” files on OMT’s MuseScore page
- All of the files on Four Score and More’s MuseScore page
Click on the links to download any particular type. Please note that the .mxl files are hosted externally on http://fourscoreandmore.org/species/, so you will be redirected and may prefer to open in a new tab.
| Part | Exercises | Solutions | Annotated | Distinct |
|---|---|---|---|---|
| I | .mxl, .mscz, |
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| II | .mxl, .mscz, |
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| III | .mxl, .mscz, |
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Gradus ad Parnassum Data
Each exercise includes the following information:
- Figure number (in the modern Norton/Mann edition, 1965)
- Species type
- Modal final
- Cantus firmus part
For ease of reference, the following subsections and tables set out that information in full.
Finally, a note on clefs. These files use the clefs in the Norton/Mann edition (1965) for parts I and II. Part III keeps the same clefs throughout to be consistent both internally and with the modern choral score layout (treble, treble, treble 8vb, bass). Teachers may wish to choose their own clefs as part of adapting these files for class.
Part I (Two Voices)
| Figure | Species | Mode (final) | Cantus firmus part |
|---|---|---|---|
| 5 | 1 | d | Lower |
| 6 | 1 | d | Upper |
| 11 | 1 | e | Lower |
| 12 | 1 | e | Upper |
| 13 | 1 | f | Lower |
| 14 | 1 | f | Upper |
| 15 | 1 | g | Lower |
| 21 | 1 | g | Upper |
| 22 | 1 | a | Lower |
| 23 | 1 | a | Upper |
| 33 | 2 | d | Lower |
| 35 | 2 | d | Upper |
| 36 | 2 | e | Lower |
| 37 | 2 | e | Upper |
| 38 | 2 | f | Lower |
| 39 | 2 | f | Upper |
| 40 | 2 | g | Lower |
| 41 | 2 | g | Upper |
| 42 | 2 | a | Lower |
| 43 | 2 | a | Upper |
| 44 | 2 | c | Lower |
| 45 | 2 | c | Upper |
| 55 | 3 | d | Lower |
| 56 | 3 | d | Upper |
| 57 | 3 | e | Lower |
| 58 | 3 | e | Upper |
| 59 | 3 | f | Lower |
| 59 | 3 | f | Upper |
| 73 | 4 | d | Lower |
| 74 | 4 | d | Upper |
| 75 | 4 | e | Lower |
| 76 | 4 | e | Upper |
| 77 | 4 | f | Lower |
| 78 | 4 | f | Upper |
| 82 | 5 | d | Lower |
| 83 | 5 | d | Upper |
| 84a | 5 | e | Lower |
| 84b | 5 | e | Upper |
| 85a | 5 | f | Lower |
| 85b | 5 | f | Upper |
| 86a | 5 | g | Lower |
| 86b | 5 | g | Upper |
| 87a | 5 | a | Lower |
| 87a | 5 | a | Upper |
| 88a | 5 | c | Lower |
| 88b | 5 | c | Upper |
Fux basically uses one cantus firmus for each modal final throughout. The note sequences are as follows, along with their usage counts (46 total) and the minor exceptions:
| Modal final | Pitches | Number of times used, and exceptions |
|---|---|---|
| D | D4–F4–E4–D4–G4–F4–A4–G4–F4–E4–D4 | 10 counts |
| E | E4–C4–D4–C4–A3–A4–G4–E4–F4–E4 | 10 counts |
| F | F3–G3–A3–F3–D3–E3–F3–C4–A3–F3–G3–F3 | 10 counts including one at the octave above, (fig. 13: starting F4) |
| G | G3–C4–B3–G3–C4–E4–D4–G4–E4–C4–D4–B3–A3–G3 | 6 counts |
| A | A3–C4–B3–D4–C4–E4–F4–E4–D4–C4–B3–A3 | 6 counts including one lacking the first D (fig. 42: A3–C4–B3–C4–E4 …) |
| C | C4–E4–F4–G4–E4–A4–G4–E4–F4–E4–D4–C4 | 2 counts; the most anomalous case |
| C | C4–D4–F4–E4–G4–E4–F4–E4–D4–C4 | 2 counts; more divergent than for the other modes |
Part II (Three Voices)
| Figure | Species | Mode (final) | Cantus firmus part |
|---|---|---|---|
| 101 | 1 | d | 1 |
| 104 | 1 | d | 2 |
| 105 | 1 | d | 3 |
| 106 | 1 | e | 1 |
| 108 | 1 | e | 2 |
| 109 | 1 | e | 3 |
| 110 | 1 | f | 1 |
| 111 | 1 | f | 2 |
| 112 | 1 | f | 3 |
| 113 | 1 | g | 1 |
| 114 | 1 | g | 2 |
| 115 | 1 | a | 1 |
| 116 | 1 | a | 2 |
| 117 | 1 | a | 3 |
| 118 | 1 | c | 1 |
| 119 | 1 | c | 2 |
| 121 | 2 | d | 1 |
| 122 | 2 | d | 2 |
| 123 | 2 | d | 2 |
| 124 | 2 | e | 2 |
| 125 | 2 | e | 3 |
| 126 | 2 | e | 1 |
| 127 | 2 | f | 1 |
| 128 | 2 | f | 2 |
| 129 | 2 | f | 2 |
| 130 | 3 | d | 2 |
| 131 | 3 | d | 3 |
| 132 | 3 | d | 2 |
| 134 | 2, 3 | d | 3 |
| 141 | 4 | d | 2 |
| 143 | 4 | d | 3 |
| 144 | 4 | d | 1 |
| 146 | 4 | e | 2 |
| 147 | 4 | e | 1 |
| 148 | 4 | e | 2 |
| 149 | 4 | f | 2 |
| 150 | 4 | f | 3 |
| 151 | 4 | f | 1 |
| 154 | 5 | d | 2 |
| 155 | 5 | d | 1 |
| 156 | 5 | d | 2 |
| 157 | 5 | e | 2 |
| 158 | 5 | e | 1 |
| 159 | 5 | e | 2 |
The table below summarizes the cantus firmi used in Part II:
| Modal final | Pitches | Number of times used, and exceptions |
|---|---|---|
| D | D4–F4–E4–D4–G4–F4–A4–G4–F4–E4–D4 | 16 counts at two octaves (D3 and D4) |
| E | E4–C4–D4–C4–A3–A4–G4–E4–F4–E4 | 12 counts at two octaves (E3 and E4) |
| F | F3–G3–A3–F3–D3–E3–F3–C4–A3–F3–G3–F3 | 9 counts at two octaves (F3 and F4) |
| G | G3–C4–B3–G3–C4–E4–D4–G4–E4–C4–D4–B3–A3–G3 | 2 counts |
| A | A3–C4–B3–D4–C4–E4–F4–E4–D4–C4–B3–A3 | 3 counts at three octaves (A2, A3 and A4) |
| C | C4–E4–F4–G4–E4–A4–G4–E4–F4–E4–D4–C4 | 2 counts; corresponds to the first case for Part I |
Part III (Four Voices)
| Figure | Species | Mode (final) | Cantus firmus part |
|---|---|---|---|
| 160 | 1 | d | 2 |
| 163 | 1 | d | 1 |
| 164 | 1 | d | 3 |
| 165 | 1 | d | 4 |
| 166 | 1 | e | 1 |
| 167 | 1 | e | 2 |
| 168 | 1 | e | 3 |
| 169 | 1 | f | 2 |
| 170 | 1 | f | 1 |
| 171 | 1 | f | 3 |
| 172 | 1 | f | 4 |
| 173 | 2 | d | 1 |
| 174 | 2 | d | 2 |
| 175 | 2 | d | 3 |
| 176 | 2 | d | 4 |
| 177 | 3 | d | 2 |
| 180 | 3 | d | 4 |
| 181 | 3 | d | 2 |
| 182 | 3 | d | 2 |
| 183 | 3 | e | 2 |
| 184 | 3 | e | 1 |
| 185 | 3 | e | 2 |
| 186 | 3 | e | 3 |
| 193 | 4 | d | 2 |
| 195 | 4 | d | 4 |
| 196 | 4 | d | 1 |
| 197 | 4 | d | 2 |
| 200 | 5 | d | 2 |
| 201 | 5 | d | 1 |
| 202 | 5 | d | 4 |
| 203 | 5 | d | 2 |
| 204 | 2, 3, 4 | d | 4 |
The table below summarizes the cantus firmi used in Part III:
| Modal final | Pitches | Number of times used, and exceptions |
|---|---|---|
| D | D4–F4–E4–D4–G4–F4–A4–G4–F4–E4–D4 | 21 counts at two octaves (D3 and D4) |
| E | E4–C4–D4–C4–A3–A4–G4–E4–F4–E4 | 7 counts, one at E5 (fig. 184) |
| F | F3–G3–A3–F3–D3–E3–F3–C4–A3–F3–G3–F3 | 4 counts at two octaves (F3 and F4) |
- Fux, Johann Joseph. (1725) 1965. Gradus ad Parnassum. Translated and edited by Alfred Mann. New York: W. W. Norton.