Quadrivium · the arts of number · art 6 of 7
Music
Musica
Numerical relationships and harmony
- Classical definition
- The science of multitude in relation: ratio and harmony, not performance. (Nicomachus, Introduction to Arithmetic I.3; Boethius, De institutione musica I.34.)
- Proper object
- Numbers in relation, such as double and half, and the ratios of consonant intervals.
- Place in the order
- Quadrivium: multitude in relation · see the whole order
Music is the art of ratio: in Nicomachus's division of the quadrivium, number in relation. It begins with one physical fact, that a string stopped at a simple fraction of its length sounds consonant with the open string, and it ends with a proved impossibility: twelve pure fifths can never close into seven octaves, so no fixed tuning can keep every interval pure. It matters now because every fixed-pitch instrument, from a guitar to a digital piano, carries that compromise, and you can measure it.
The guiding questionWhat makes two notes agree, and what must a tuning give up to keep them agreeing?
The spiral
Three strands, three rounds
Each strand comes back in every round, a level deeper. Finish Round 1 across the arts before Round 2: breadth first, then depth.
Round I Foundations
- RatioIntervals as ratios of string lengthI can state the ratios of the octave, fifth and fourth, and use Mersenne's law to turn a ratio of string lengths into a ratio of frequencies.
- Cents and beatsMeasuring intervals in cents and hearing beatsI can convert a frequency ratio to cents, compare a pure interval with its tempered version, and predict how fast two nearly equal tones beat.
- The commaTwelve fifths and seven octavesI can show by arithmetic that twelve pure fifths overshoot seven octaves, compute the Pythagorean comma in cents, and explain how equal temperament spreads it.
Round II Practice
- RatioBuilding scales from ratiosI can build a seven-note scale from stacked fifths and from simple ratios, and compare the two note by note in cents.
- Cents and beatsCents in practiceI can measure a note in cents with the tuner, convert fret lengths to cents, and tune a fifth and a major third by counting beats between coinciding partials.
- The commaComparing tuning systemsI can compute the fifths and major thirds of Pythagorean, quarter-comma meantone and equal temperament in cents, and say which interval each system favours and which it sacrifices.
Round III Mastery
- RatioRatio, number and the irrationalI can prove that every equal-tempered interval between the unison and the octave has an irrational frequency ratio, and explain what the discovery of incommensurable magnitudes did to the Pythagorean claim that all things are number.
- Cents and beatsConsonance, beats and roughnessI can use beat rates and cents to explain Helmholtz's account of consonance, and state where it fails for real strings.
- The commaChoosing a temperamentI can compare Pythagorean, just, meantone and equal temperaments for one stated purpose and defend a choice using cents for every fifth and major third.
Labs
Core reading in the Library
- LibraryThe Seven Liberal Arts, Hands-On: Logic, Proof and the QuadriviumA practical guide to the trivium and quadrivium, with a practice and a made thing for each art, plus a path through Socratic questioning, Aristotle's logic and Euclid's method.
- LibraryMathematics: Start HereAn orientation to the mathematics wing: why a spine beats a survey, how the trivium frames mathematical work, and which page to read for which need.
- LibraryAlgebra: The Skills That Carry EverythingThe dependency groups of school algebra and the reasons behind each rule, so that errors become diagnosable instead of mysterious.
- LibraryGeometry, Euclid and TrigonometryThe bridge from classical geometry to trigonometry and precalculus, including a track for studying Euclid and Apollonius by reconstruction.
- LibraryProof and Precise Reasoning: From Arguments to TheoremsThe grammar of valid inference: validity and soundness, truth tables, quantifiers, the main proof techniques and counterexamples, framed by the trivium habit of defining, testing and explaining.
Where it is used
- LabThe guitar PathIntervals as ratios on a real fretboard: the tempered fifth is about 2 cents narrow and the major third about 14 cents wide.
- LabGuitar practiceOne string as a ruler: feel whole steps and half steps as distances along the string.
- LabThe guitar LabExperiment 2 holds a drone on low E and plays the notes of C major over it, so each note is heard as an interval against a fixed pitch.
- LibraryQuantitative Reasoning with Stated Assumptions · The same discipline elsewhereEngineering measurement held to the same standard: exact ratios, stated units and what was held constant.2 min glance · 6 min
Seminars
Read the sources
One primary text for each round, with a question to think through. All 21 seminars →
Music · Round 1
- Boethius, De institutione musica, Book I, chapters 1-2 and 10-11 (Friedlein's chapter numbering)Latin only at this link (Friedlein's text); the standard English translation (Calvin Bower, Fundamentals of Music, Yale, 1989) is in print and not freely online. I.1: music is joined to us by nature and can ennoble or corrupt character. I.2: three kinds of music, of the world (mundana), of the human being (humana, the joining of soul and body) and of instruments. I.10: the smithy story, with hammers weighing 12, 9, 8 and 6. I.11: the tests Pythagoras is said to have made at home, with weights on strings, lengths of pipes and cups struck with a rod, ending with the length and thickness of strings and the 'regula', the measuring rule. Read I.10-11 as legend, not physics. Hammer pitch does not follow hammer weight in the ratios claimed. Where weights in these ratios are hung on equal strings, as in the parallel story in Nicomachus (Manual of Harmonics 6), frequency goes as the square root of the stretching weight, so a 2:1 weight gives a frequency ratio of about 1.41:1 (600 cents), not an octave. The ratios 2:1, 3:2 and 4:3 are right for string lengths, which the monochord shows.
- Plato, Republic, Book VII, 530d-531cShorey's English translation at Perseus, with a link to the Greek. Jowett's translation is free at Project Gutenberg (ebook 1497), without Stephanus numbers. At 530d Socrates says that as the eyes are fitted for astronomy, the ears are fitted for harmonic motion, and that the two are sister sciences, as the Pythagoreans say (Shorey has 'kindred sciences'). At 530e-531a he says that students of harmony measure heard concords and sounds against one another and labour in vain. Glaucon mocks those who put their ears to the strings hunting for the smallest interval, and Socrates sets aside those who 'torture the strings' on the pegs (531b). His real target is the Pythagoreans, who seek numbers in heard concords but do not rise to problems, asking which numbers are concordant and which are not, and why (531c). Harmonics is paired with astronomy here, not rhythm.
At 531c Socrates faults the Pythagoreans: they look for the numbers in the concords they hear, but never rise to problems, asking which numbers are concordant and which are not, and why. What is the difference between those two activities, and which of them does a monochord let you do?
- Boethius says music is joined to us by nature and can ennoble or corrupt character (I.1). What would count as evidence for or against that, and is it a claim about numbers at all?
- The smithy story is false as physics, yet the ratios it reports are right for string lengths. What should a reader do with a source whose story fails but whose result holds?
- Boethius's three kinds of music include the music of the world and the music of the human being, the joining of soul and body (I.2). Is that an analogy, a hypothesis or a claim, and how would you decide?
Music · Round 2
- Plato, Timaeus, 35b-36bLamb's English translation at Perseus, with a link to the Greek; Jowett's translation is at Project Gutenberg (ebook 1572), without Stephanus numbers. At 35b-c the maker marks off portions 1, 2, 3, 4, 9, 8 and 27, a series of doubles and a series of triples. At 35c-36a he fills the double and triple intervals with two kinds of mean, which yields intervals of 3:2, 4:3 and 9:8. At 36a-b he fills each 4:3 with 9:8 tones, leaving a remainder of 256:243, about 90.2 cents. The music is exact arithmetic using only the factors 2 and 3. The cosmic setting is cosmology offered as a 'likely story' (29d), not a measurement.
The maker of the world builds the soul from portions 1, 2, 3, 4, 9, 8 and 27, fills the gaps with means to get intervals of 3:2, 4:3 and 9:8, then fills each 4:3 with 9:8 tones until only a small remainder, 256:243, is left. Why might a maker who wants perfect order choose these numbers, and what does the remainder say about the order that can be achieved?
- The numbers use only the factors 2 and 3. What would be gained or lost if the factor 5 were allowed?
- Is the 256:243 remainder a flaw in the plan or part of it?
- Timaeus calls his account a likely story (29d). Which parts of this passage could be tested, and which could not?
Music · Round 3
- Hermann von Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music (Alexander J. Ellis, translator), Part II, Chapter VIII, On the Beats of Simple Tones (pp. 159-173), and Chapter X, Beats of the Upper Partial Tones (pp. 179-197)Ellis's English translation is public domain and scanned at the Internet Archive. This scan is the 1895 printing, called the third edition but reprinted from the 1885 second edition, and the page numbers above are taken from its table of contents. Helmholtz explains the smoothness or roughness of an interval by whether the partials of the two notes coincide, beat slowly or beat fast. It is a physical account of roughness in sustained tones. Whether roughness is the same thing as dissonance in music is a question for the seminar.
Helmholtz accounts for the smoothness of some intervals by the way the partials of two notes coincide or beat. Is that an explanation of consonance in music, or only of roughness in sustained tones?
- Which of his steps would fail for a note whose partials are not whole-number multiples of the fundamental, as in a stiff string?
- The tempered fifth beats slowly and the tempered major third beats faster. What does the tolerance of players and listeners for those beats tell us about what consonance is?
- What would a counterexample to his account look like, and how would you look for one?