Intervals as ratios of string length
I 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.
- Round IIntervals as ratios of string length (this page)
- Round IIBuilding scales from ratiosComing
- Round IIIRatio, number and the irrationalComing
1 Learn
You will be able to
- Write the octave, fifth and fourth as frequency ratios (2:1, 3:2, 4:3) and as sounding-length ratios (1/2, 2/3, 3/4).
- Use frequency inversely proportional to length to find the pitch of a shortened string.
- Say how tension and mass per unit length change the pitch of a string, using Mersenne's law.
- Separate the fact (string-length ratios) from the legend (Pythagoras and the hammers).
- LabMonochordHear the octave, fifth and fourth as stopped-length ratios against the open string.
- LabString physicsMersenne's law: halving the length doubles the frequency, and each fret removes the same fraction of the string.
- LabMersenne's string lawThe same law with its status (Established, not Demonstrated), Helmholtz's coincident-partials explanation, and its limits for real strings.
- LabThe harmonic seriesThe first harmonics of the A string sound at 110, 220, 330, 440 and 550 Hz, so neighbouring harmonics stand in the ratios 2:1, 3:2, 4:3 and 5:4.
- LabPure vs tempered ratiosThe ratio column for every interval in one table.
- LibraryThe Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium · The seven arts and their orderWhy music is called number in relation (Nicomachus), and where it sits among the four arts.2 min glance · 7 min
- LibraryMathematics: Start Here · Music as applied numberA short statement of the string ratios, with the tradition about Pythagoras kept apart from verified fact.2 min glance · 6 min
2 Practise
Answer each card from memory, then grade yourself honestly. The cards join your review deck and come back just before you would forget them.
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recall
What is a monochord, and how do you use it to compare two notes?
Show answer
A monochord is one string with a movable bridge. You slide the bridge, then compare the sound of the stopped part of the string with the sound of the open string.
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recall
On a monochord, what do you hear when the stopped part of the string and the open string form a simple length ratio, and who is this discovery attributed to?
Show answer
The two notes blend. The discovery is attributed to Pythagoras, and the effect is real physics.
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explain
Why is the story of Pythagoras hearing musical ratios in a blacksmith's hammers treated as legend, while the monochord's string-length ratios are treated as fact?
Show answer
Hammer pitch does not scale with hammer weight the way the story says, so the tale fails as physics. The string ratios hold up: on a monochord, where the stopped and open lengths form a simple ratio, you can hear the two notes blend.
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recall
According to Helmholtz (1863), why do two notes whose frequencies stand in a simple whole-number ratio sound consonant?
Show answer
Because of coincident partials: when the ratio is simple, the upper harmonics of the two notes line up instead of beating against one another.
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recall
In the stretched-string formula f = (1 / 2L)·√(T/μ), how does frequency depend on length L, tension T and mass per unit length μ?
Show answer
Frequency is inversely proportional to length L, proportional to the square root of tension T, and inversely proportional to the square root of mass per unit length μ.
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apply
By Mersenne's law for a stretched string, an ideal string sounding at 300 Hz is shortened to 2/3 of its original sounding length, with tension and mass per unit length unchanged. What is the new frequency?
Show answer
Frequency varies inversely with length, so it is multiplied by 1/(2/3) = 3/2: 300 × 3/2 = 450 Hz. That is the 3:2 ratio of the monochord.
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recall
What does Mersenne's law say about how a string's frequency depends on its length, tension and mass per unit length?
Show answer
Frequency is inversely proportional to the vibrating length L, and proportional to the square root of tension T divided by mass per unit length μ: f = (1/2L)·√(T/μ).
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explain
Why do guitar frets get closer together as you go up the neck?
Show answer
Each fret shortens the remaining vibrating length by the same fraction, 1 − 2^(−1/12), about 5.6%. As the string gets shorter, that fixed fraction is a smaller distance, so the frets crowd closer together.
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recall
Since a string's frequency is proportional to the square root of its tension, how must tension change as a bend raises the pitch?
Show answer
Tension must rise with the square of the frequency, so each extra semitone of bend costs more effort than the last. On an ideal string, a two-semitone bend needs about 26% more tension.
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recall
At the same length and tension, how does a string with 4 times the mass per unit length sound compared with the lighter string?
Show answer
It sounds an octave lower. Frequency scales with 1 over the square root of mass per unit length, and √4 = 2, so the frequency halves.
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recall
What are the partials of a plucked string, and what do they give a note?
Show answer
A plucked string vibrates at its full length and at halves, thirds, quarters and so on, all at once. These partials, at 1×, 2×, 3× and so on of the fundamental, give a note its tone.
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apply
An open A string has a 110 Hz fundamental. What frequency is its 3rd harmonic, which note is that, and where do you touch the string to play it?
Show answer
It is 330 Hz (3 × 110), an E4, an octave plus a fifth above the open string. Touch the string lightly over the 7th fret, or the 19th.
3 Prove it: the mastery check
5 questions drawn from a pool of 11. The pass mark is 80%. There is no time limit, and you can retake it with new questions; your best result counts.
4 Prove it: the performance task
Build a monochord table
Use a string of 648 mm (the guitar scale length on the Science page) with an open pitch of 220 Hz. For the octave (2:1), perfect fifth (3:2), perfect fourth (4:3) and whole tone (9:8), work out the sounding length as a fraction of the open length, the stopping point in millimetres measured from the nut, and the resulting frequency. Then play two of these ratios with the presets on the monochord tool and note whether the stopped part blends with the open string. Finish with a short note on what is fact and what is legend in the story of Pythagoras and the ratios.
What to hand in: One page, typed or written and photographed: a table with four rows (one per interval) and columns for frequency ratio, length fraction, stopping point in millimetres and frequency in hertz, then one sentence for each of the two intervals you played, then a note of two to four sentences. Save it on the page or download it.
Saved only in this browser. To keep a copy, download it or export your progress.
5 Discuss: the seminar
Read the text, then think, write or talk through the question with someone. There is no answer key: the aim is a better question.
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?
Where music 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
Self-administered checks and self-assessed tasks: no credential is awarded. The design borrows from Khan Academy (mastery levels), WGU (competencies proved by assessment) and St. John's College (seminars on primary texts); this site is not affiliated with any of them.