Informal preparatory notes 2013 rev07harpsichords.weebly.com/uploads/2/5/0/.../monochord... · -...

The monochord as a practical tuning toolInformal notes

Medieval Keyboard Meeting, Utrecht, Tuesday, September 3, 2013

Pierre Verbeek([email protected] - www.clavichords.be)

The monochord as a practical tuning tool - informal notes- Goal of the presentation: to show how the monochord can be used in practice. - The monochord = WYSIWYG- With the monochord you can divide the octave and apply the results to good intonation in accordance with medie-

val practice. - It is a tool not only for theoretical study, but also for tuning the medieval instrumentarium. - Comparisons can be made between the accuracy of monochord-aided tuning and the accuracy of electronic tun-

ing apparatus. - In this session, youʼre invited to practice and to observe:

(a) divide the monochord in accordance with medieval practice(b) identify the intervals on the right part of the monochord and reflect on them(c) identify the intervals on the left part of the monochord(d) imagine other uses of the monochord (eg. with two bridges)(e) feedback and ideas(f) conclusion

The monochord as a practical tuning tool! Verbeek 1

Challenge1. Divide the monochord in accordance with the guidelines of Guido of Arezzo (11th century)2. Check your division with one movable bridge. 3. Verify that the Pythagorean tuning is obtained for all diatonic pitches.4. Identify the intervals at the left of the bridge5. Check your division with two movable bridges.

Enjoy!

_________Hints:- Youʼll find in Appendix 1 numerical tables for division of a few lengths by 9 and by 4.- For checking only: the numerical solution to Guidoʼs monochord division are provided in Appendix 2.


Monochord basics• A single-strung instrument• Said to have been invented by Pythagoras• With string attached across two bridges

• With movable bridge• With resonating box (from 15th c. only)

• Guido of Arezzo 11th c.: “We perceive best, on the monochord, the notes, which are the first elements of our art”. Mi-crologus, Chap. 1.

• Mersenne 1636: “Le Monochorde est lʼinstrument le plus propre pour regler les sons, dʼautant quʼil est le plus iuste de tous, & que la fabrique en est tres facile.” Harm. Univ., Livre premier, ʻdes Instrumentsʼ, Prop. V.

• Mersenne 1636: “il sert à mesurer le grave & lʼaigu des sons, comme la reigle ordinaire des Geometres sert pour me-surer les lignes droites, & le compas pour descrire les Cercles”. Ibid.

Some references• Guidonis Aretini Micrologus, ed. J. Smits van Waesberghe, American Institute of Musicology, 1955 (Corpus scriptorum

de Musica, 4).

• English translation: Huchbald, Guido and John on Music, Three Medieval Treatises, translation W. Babb, ed. and intro-duction C.V. Palisca, New Haven & London, Yale Univ. Press 1978, p. 57-83 (Music Theory Translation Series 3).

• French translation: Gui dʼArezzo, Micrologus, traduction et commentaires de Marie-Noëlle Colette et Christophe Joli-vet, Cité de la Musique, Centre de Ressources Musique et Danse, Paris 1993, 1996.

• Christian Meyer, Mensura Monochordi, la division du monocorde (IXe-XVe siècles), Paris, Publications de la Société française de musicologie, Ed. Klincksieck, 1996. See also Christian Meyer, La tradition du Micrologus de Guy dʼArezzo : une contribution à lʼhistoire du texte, Revue de musicologie, tome 83, n° 1, 1997, Société française de musicologie, p. 5-32.

• Charles M. Atkinson, The Critical Nexus, Tone-System, Mode, and Notation in Early Medieval Music, Oxford Univ. Press, 2009, especially p.202-233.

• Jan Herlinger, Medieval Canonics, (in) The Cambridge History of Western Music Theory, Ed. Thomas Christensen, Cambridge University Press, 2002, p. 168-192.

• David E. Cohen, Notes, scales, and modes in the earlier Middle Ages, (in) The Cambridge History of Western Music Theory, Ed. Thomas Christensen, Cambridge Univ. Press, 2002, p. 307-363, specially p 339-351.

• Nelly van Ree Bernard, The Keyed Monochord, rediscovery of a forgotten instrument, Ed. Muziekcentrum ʻHet Du-intjeʼ, Bennebroek, Holland, 1993.


Basic physics of the monochord• For an “ideal” vibrating string (all other things being equal, such as string tension, string material, position of the sun

relative to the earth, etc.), the vibration frequency f is inversely proportional to the vibrating length of the string L. In other words, the vibrating length of the string L is inversely proportional to the frequency f.

• Examples of f ≈ 1/L on the monochord:(a)Take your monochord(b)Pluck the string and remember the note(c)Divide its vibrating length L by two, using a third bridge(d)Pluck the right-hand side of the string. Listen...(e)Then divide the length by 3, by 4, by 5, etc.(f)Conclusion: f ≈ 1/L

• This explains Mersenneʼs statement already quoted above, that the monochord “... sert à mesurer le grave & lʼaigu des sons, comme la reigle ordinaire des Geometres sert pour mesurer les lignes droites, & le compas pour descrire les Cercles”. (Mersenne)


Divisions of the monochord and canonics- Definition of canonics = “the study of pitch arrays and intervals [on the monochord] and the ratios through which

they are defined” (Jan Herlinger).- Ancient greek music theory with sophisticated canonics.- Boethius (Anicius Manlius Severius Boethius, Rome ca. 480 - ca 524), in his De institutione musica, transmitted

much of the Greek theory to the latin world, including about the division of the monochord. - Enormous influence of Boethius on the Western music theory. - More than 140 different divisions methods of the monochord between 1000 and 1500 AD (Christian Meyer).- For the division of the monochord, Guido of Arezzo depends much on the anonymous Dialogus de musica,

wrongly attributed to Odo of Cluny (ca. 878 - 942) or Odo of Arezzo (10th c.), most probably of north Italian prove-nance. (see Michel Hugloʼs entry ʻOdoʼ in The New Grove, ed. 1980).

- The Dialogus de musica is “the earliest extant treatise to present the letter names of the notes as we still know them today” (Jan Herlinger).

- The monochord division of the Dialogus de musica starts with Γ, which was a change from earlier divisions, which started on A.

- Early methods were complicated and difficult to implement. E.g. the method of the Dialogus requires sixteen divi-sions of the string for sixteen pitches.

- The genius of Guido was to streamline the whole process while extending the compass: in the second division of the Micrologus, only five divisions are required for nineteen pitches from Γ to cc. And the compass can be easily ex-tended further.

- Guido was a master pedagogue; hence his influence has been huge and of great importance for the future of mu-sic practice (not only theory) in Europe. See prologue of the Micrologus: “This is the ranking of notes - which scholars have either kept unknown or have rendered unduly cumbersome - fully explained, even for children, in a complete and concise manner”.

Peculiarities

(a) No use of the sound itself: the theory is enough, no need for experiment.

(b) Use only numerical intervals, no experimentation.

(c) No use of beats (discovered 15t-16th c.).

(d) No use of a second string for comparison.

(e) No use of a second bridge before 16th c.


Disposition of the notes on the monochord (Guidoʼs 2nd method)Note 1 : all notations are Guidoʼs. However, the notation Ω is not found in Guidoʼs work. I suggest, for easy com-munication, to call Ω what Guido calls the “extremity”.

Note 2 : Let the length of the segment between any two points a and b be noted as <a-b>.

One has, by definition, <Γ- Ω> = total sounding length of the free monochord = L.

Step Division of segment Interval of interest Note obtained Sounding length

1.1 divide L = <Γ- Ω> in 9 parts

first divider 8/9 A <A-Ω> = 8/9 L

1.2

divide L = <Γ- Ω> in 9 parts

second divider 7/9 -

1.3


third divider 6/9 D <D-Ω> = 6/9 L

1.4


fourth divider 5/9 -

1.5


fifth divider 4/9 a <a-Ω> = 4/9 L

1.6


sixth divider 3/9 d <d-Ω> = 3/9 L = 1/3 L

1.7


seventh divider 2/9 aa <aa-Ω> = 2/9 L

1.8


eighth divider 1/9 -

2.1 divide <A- Ω> in 9 parts first divider 8/9 B [b square] <B-Ω> = 8/9 <A-Ω>

2.2

divide <A- Ω> in 9 parts

second divider 7/9 -

2.3


third divider 6/9 E <E-Ω> = 6/9 <A-Ω>

2.4


fourth divider 5/9 -

2.5


fifth divider 4/9 ե [b square] <ե-Ω> = 4/9 <A-Ω>

2.6


sixth divider 3/9 e <e-Ω> = 3/9 <A-Ω>

2.7


seventh divider 2/9 եե [bb square] <եե-Ω> = 2/9 <A-Ω>

2.8


eighth divider 1/9 -

3.1 divide <Γ- Ω> in 4 parts first divider 3/4 C <C-Ω> = 3/4 L

3.2

divide <Γ- Ω> in 4 parts

second divider 2/4 G <G-Ω> = 2/4 L = 1/2 L

3.3

divide <Γ- Ω> in 4 parts

third divider 1/4 g <g-Ω> = 1/4 L

4.1 divide <C- Ω> in 4 parts first divider 3/4 F <F-Ω> = 3/4 <C-Ω>

4.2

divide <C- Ω> in 4 parts

second divider 2/4 c <c-Ω> = 2/4 <C-Ω>

4.3

divide <C- Ω> in 4 parts

third divider 1/4 cc <cc-Ω> = 1/4 <C-Ω>

5.1 divide <F- Ω> in 4 parts first divider 3/4 b [b flat] = b rotun-dum = b mol

<b-Ω> = 3/4 <F-Ω>

5.2

divide <F- Ω> in 4 parts

second divider 2/4 f <f-Ω> = 2/4 <F-Ω>

Note 3: Guido stops the division at cc but doesnʼt exclude the possibility to continue. Some later manuscripts con-tinue until dd.


Modern view of Guidoʼs monochord division

Step Note Ratio to preced-ing division

note

Ratio to preced-ing division

note

Ratio to ΩRatio to ΩRatio to Ω CommentsStep Note

Abso-lute

Cents Absolute frac-tion

Abso-lute al-gebraic

Cents (c)

Comments

Γ 1 1.000 0

1.1 A 8/9 204 8/9 0.889 204 tone above Γ [= + 204 c]

1.3 D 6/9 702 6/9 0.667 702 pure fifth of Γ [= + 702 c]pure fourth of A [= + 498 c]

1.5 a 4/9 1,404 4/9 0.444 1,404 octave of A [= +1200 c]pure fifth of D [= + 702 c]

1.6 d 3/9 1,902 3/9 0.333 1,902 octave of D [= +1200 c]pure fourth of a [= + 498 c]

1.7 aa 2/9 2,604 2/9 0.222 2,604 octave of a [= +1200 c]pure fifth of d [= + 702 c]

2.1 B 8/9 204 8/9 * 8/9 0.790 408 tone above A [= + 204 c]

2.3 E 6/9 702 6/9 * 8/9 0.593 906pure fifth of A [= + 702 c]

pure fourth of B [= + 498 c]tone above D [= + 204 c]

2.5 ե 4/9 1,404 4/9 * 8/9 0.395 1,608octave of B [= + 1200 c] pure fifth of E [= + 702 c]tone above a [= + 204 c]

2.6 e 3/9 1,902 3/9 * 8/9 0.296 2,106octave of E [= + 1200 c]pure fifth of a [= + 702 c]tone above d [= + 204 c]

2.7 եե 2/9 2,604 2/9 * 8/9 0.198 2,808octave of ե [= + 1200 c]pure fifth of e [= + 702 c]tone above aa [= + 204 c]

3.1 C 3/4 498 3/4 0.750 498pure fourth of Γ [= 498 c]tone below D [= - 204 c]

semitone above B [= +90 c]

3.2 G 2/4 1,200 2/4 0.500 1,200octave of Γ [= + 1200 c]pure fifth of C [= + 702 c]pure fourth of d [= - 498 c]

tone below a [= -204 c]

3.3 g 1/4 2,400 1/4 0.250 2,400octave of G [= + 1200 c]

pure fourth of d [= + 498 c]tone below aa [= -204 c]

4.1 F 3/4 498 3/4*3/4 0.563 996pure fourth of C [= + 498 c]

tone below G [= - 204 c]semitone above E [= + 90 c]


Step Note Ratio to preced-ing division

note

Ratio to preced-ing division

note

Ratio to ΩRatio to ΩRatio to Ω CommentsStep Note

Abso-lute

Cents Absolute frac-tion


Cents (c)

Comments

4.2 c 2/4 1,200 2/4*3/4 0.375 1,698

octave of C [= + 1200 c]pure fifth of F [= + 702 c]

pure fourth of G [= + 498 c]tone below d [= - 204 c]

semitone above ե [= + 90 c]

4.3 cc 1/4 2,400 1/4*3/4 0.188 2,898octave of c [= + 1200 c]

pure fourth of g [= + 498 c]semitone above եե [= -+ 90 c]

5.1 b flat 3/4 498 3/4*3/4*3/4 0.422 1,494

pure fourth of F [= + 498 c]tone below c [= - 204 c]

semitone above A [= + 90 c]

5.2 f 2/4 1,200 2/4*3/4*3/4 0.281 2,196

octave of F [= + 1200 c]pure fifth of b flat [= + 702 c]

pure fifth of cc [= - 702 c]tone below g [= - 204 c]

semitone above e [= 90 c]


Intervals according to Guidoʼs Micrologus Chapter IVThere exist six ways of joining notes (consonantias) = intervals (clausulis):

Name Definition Guidoʼs examples

tone (tonus) large space between two adjacent notes on the monochord Γ-AA-B

semitone (semitonium) small space between two adjacent notes on the monochord B-C

major third (ditonus) tone + tone C-E

minor third (semiditonus) tone + semitonesemitone + tone

D-F

fourth (diatessaron) tone + tone + semitonetone+semitone + tonesemitone+tone+tone

A-DB-E

fifth (diapente) tone + tone + tone + semitonetone + tone + semitone + tonetone +semitone + tone + tonesemitone + tone + tone + tone

C-GA-E

Note 4: The octave (diapason) is the interval joining together a fourth and a fifth; e.g. A-D and D-a give the diapa-son A-a (Guidoʼs Micrologus Chapter V). Same letter used for a note and its diapason, e.g. A-a-aa, B-ե, C-c, D-d, etc. These notes have a natural correspondence (concordia): “[when two or three singers sing them together,] you will notice with astonishment that one obtains the same notes, which are at different levels, but very much look alike, and that the same song at the bass, at the treble and at the super-treble, nevertheless sounds uniform”.

The exist three consonant intervals (symphonias), i.e. pleasing combinations of sounds (Chapter VI):- octave (diapason)- fifth (diapente)- fourth (diatessaron)


Lodovico Fogliani 1529 - the monochord in 16th c. practice


Appendix 1 : table of numerical values useful for dividing a 900 mm monochord

Length mm

8/9 7/9 3/4 6/9 5/9 1/2 4/9 3/9 1/4 2/9 1/9

900 800.0 700.0 675.0 600.0 500.0 450.0 400.0 300.0 225.0 200.0 100.0

800 711.1 622.2 600.0 533.3 444.4 400.0 355.6 266.7 200.0 177.8 88.9

675 600.0 525.0 506.3 450.0 375.0 337.5 300.0 225.0 168.8 150.0 75.0

506.3 450.0 393.8 379.7 337.5 281.3 253.1 225.0 168.8 126.6 112.5 56.3


Appendix 2 : numeric solution to Guidoʼs diatonic monochord division in modern terms The table below provides numerical values of distances on Guidoʼs diatonic monochord, for a 900 mm sounding length

monochord. These data are provided only for reference. It is highly recommended to use the division method of the Micrologus instead.

Step Note Ratio to ΩRatio to ΩRatio to Ω Length to Ω(mm)

Length from Γ(mm)

Step Note

Absolute frac-tion


Cents (c)

Length to Ω(mm)

Length from Γ(mm)

Γ 1 1.000 0 900.0 0.0

1.1 A 8/9 0.889 204 800.0 100.0

2.1 B 8/9 * 8/9 0.790 408 711.1 188.9

3.1 C 3/4 0.750 498 675.0 225.0

1.3 D 6/9 0.667 702 600.0 300.0

2.3 E 6/9 * 8/9 0.593 906 533.3 366.7

4.1 F 3/4*3/4 0.563 996 506.3 393.8

3.2 G 2/4 0.500 1,200 450.0 450.0

1.5 a 4/9 0.444 1,404 400.0 500.0

5.1 b flat 3/4*3/4*3/4 0.422 1,494 379.7 520.3

2.5 ե 4/9 * 8/9 0.395 1,608 355.6 544.4

4.2 c 2/4*3/4 0.375 1,698 337.5 562.5

1.6 d 3/9 0.333 1,902 300.0 600.0

2.6 e 3/9 * 8/9 0.296 2,106 266.7 633.3

5.2 f 2/4*3/4*3/4 0.281 2,196 253.1 646.9

3.3 g 1/4 0.250 2,400 225.0 675.0

1.7 aa 2/9 0.222 2,604 200.0 700.0

2.7 եե 2/9 * 8/9 0.198 2,808 177.8 722.2

4.3 cc 1/4*3/4 0.188 2,898 168.8 731.3


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