Twelve fifths and seven octaves
I 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 ITwelve fifths and seven octaves (this page)
- Round IIComparing tuning systemsComing
- Round IIIChoosing a temperamentComing
1 Learn
You will be able to
- Compute (3/2)^12 and 2^7 and the ratio between them, 531441/524288.
- Convert the comma to cents and get 23.46.
- Explain why 3^12 = 2^19 is impossible, using odd and even numbers.
- State what equal temperament does to each fifth, and name the cost of two other tuning choices.
- LabTwelve fifths ≠ seven octavesThe full demonstration, what it costs, and where it stops being true.
- LabThe Pythagorean commaThe spiral of fifths: watch the stack overshoot, and see equal temperament shave 1.955 cents from each fifth.
- LabThe AcademyIts test: twelve fifth errors of 1.955 cents add up to the comma.
- LabThe ElenchusA worked claim about the tempered fifth, showing the mathematical verdict and the musical verdict kept separate.
- LabBeatsWhat a 1.96-cent error actually sounds like: slow beating.
- LibraryProof and Precise Reasoning: From Arguments to Theorems · The main proof formsProof by contradiction, the form behind the odd versus even argument.2 min glance · 6 min
- LibraryThe Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium · The quadrivium in practiceThe music exercise compares three major thirds in cents and places the comma at about 23.5 cents.2 min glance · 7 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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explain
Why can twelve perfect fifths never exactly equal seven octaves?
Show answer
A fifth multiplies frequency by 3/2 and an octave by 2, so equality would need (3/2)¹² = 2⁷, that is 3¹² = 2¹⁹. But 3¹² is odd and 2¹⁹ is even; by unique factorisation no power of three equals a power of two.
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recall
What is the Pythagorean comma, and how large is it as a frequency ratio and in cents?
Show answer
The amount by which twelve perfect fifths overshoot seven octaves: 3¹²/2¹⁹ = 531441/524288 ≈ 1.01364, or about 23.46 cents.
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recall
How does equal temperament deal with the Pythagorean comma, and by how much does it narrow each fifth?
Show answer
It spreads the 23.46-cent comma evenly across all twelve fifths, narrowing each by about 1.955 cents, so no key is favoured over another.
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recall
What is the Pythagorean comma, and how large is it in cents?
Show answer
It is the amount by which twelve stacked pure 3:2 fifths overshoot seven octaves. It measures 23.46 cents.
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recall
Stacking twelve pure 3:2 fifths from C should in theory return to C, seven octaves up. Which two numbers show that it does not?
Show answer
(3/2)^12 is about 129.746, but seven octaves is 2^7 = 128. The two values differ, so the circle of fifths does not close.
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explain
Why can a stack of twelve pure 3:2 fifths never land exactly seven octaves up?
Show answer
Equality would need (3/2)^12 = 2^7, which means 3^12 = 2^19. But 3^12 is odd and 2^19 is even, so the two can never be equal.
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recall
How does equal temperament handle the 23.46-cent Pythagorean comma, and what does that do to a guitar's fifths?
Show answer
It spreads the comma evenly, shaving about 1.955 cents off each of the twelve fifths (23.46 divided by 12). That is why a guitar's fifths are very slightly narrow: 700 cents instead of the pure 701.96.
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recall
Why can twelve-tone equal temperament never produce a pure perfect fifth, and by how many cents does its fifth miss?
Show answer
The pure fifth is exactly 3:2, a rational number, while the tempered fifth is exactly 2^(7/12) ≈ 1.498307, an irrational number, so they cannot be equal. The tempered fifth falls short by 1.955 cents.
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explain
The claim that twelve-tone equal temperament cannot produce a pure perfect fifth is given two verdicts, Demonstrated and Provisional. Why two?
Show answer
The mathematics is Demonstrated: an irrational ratio cannot equal 3:2. Whether a 1.955-cent gap matters musically stays Provisional, because two cents is near the limit of pitch discrimination, yet the two fifths beat audibly when sustained. Both verdicts are recorded and neither is hidden.
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recall
How does the equal-tempered perfect 5th on a guitar compare with the pure 3:2 fifth?
Show answer
The tempered fifth is 700 cents and the pure 3:2 fifth is 701.96 cents, so the tempered fifth is 1.96 cents narrow.
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recall
A pure E5 (660 Hz) and an equal-tempered E5 (659.26 Hz) sound together. About how fast do they beat, and what does that show about the tempered fifth?
Show answer
They beat about 0.74 times a second. That shows the equal-tempered fifth is slightly impure, but the beating is slow enough that most players never notice it.
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recall
How many cents make up an octave, and how do you convert a frequency ratio r into cents?
Show answer
An octave is 1200 cents, because one cent is the interval with frequency ratio 2^(1/1200). The cents of a ratio r are 1200·log₂(r).
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
Prove it, measure it, repeat it
Write the proof that twelve pure fifths cannot equal seven octaves in five numbered steps or fewer, then compute the comma in cents. Repeat the method for three pure major thirds (5:4) against one octave: find the gap and its size in cents. Finish with a short paragraph on what equal temperament does to the fifth and one cost of a different tuning, taken from the Academy page.
What to hand in: One page, typed or written and photographed: the numbered proof, a calculation table for both cases, and a closing paragraph of three 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
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