Measuring intervals in cents and hearing beats
I can convert a frequency ratio to cents, compare a pure interval with its tempered version, and predict how fast two nearly equal tones beat.
- Round IMeasuring intervals in cents and hearing beats (this page)
- Round IICents in practiceComing
- Round IIIConsonance, beats and roughnessComing
1 Learn
You will be able to
- Convert a frequency ratio r to cents with 1200 × log2(r).
- Add cents instead of multiplying ratios, for example a fifth plus a fourth makes an octave.
- State whether the tempered fifth and major third are wide or narrow of pure, and by how many cents.
- Predict a beat rate as the difference in frequency of two tones.
- LabThe AcademyThe cent as one twelve-hundredth of an octave, the formula 1200 × log2(r), and why cents add where ratios multiply.
- LabPure vs tempered ratiosPure and tempered sizes of every interval in cents, with the signed difference.
- LabBeatsBeat rate equals the difference in frequency, with worked examples at E5 and above A2.
- LabThe harmonic seriesHarmonics read off against the tempered scale in cents, such as the 5th harmonic about 14 cents flat.
- LabTunerThe tuner's Off by readout shows a real string's offset in cents, and the reference tones let you tune by ear on beats.
- LibraryAlgebra: The Skills That Carry Everything · Algebra 2 additionsLogarithms turn products into sums, which is why cents add.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.
-
recall
What frequency ratio defines a pure (just) major 3rd, how many cents is it, and how does the equal-tempered major 3rd compare?
Show answer
A pure major 3rd is the ratio 5:4, about 386.31 cents. The equal-tempered major 3rd is 400 cents, so it is 13.69 cents wide of pure.
-
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.
-
recall
In equal temperament, what is the frequency ratio of every semitone, and which interval within an octave is the only pure one?
Show answer
Every semitone is exactly 2^(1/12), so n semitones give 2^(n/12). That number is irrational, and so never a whole-number ratio, unless n is a multiple of 12. Within an octave, only the octave itself (2:1) is pure.
-
explain
Why does heavy distortion favour power chords (root and fifth) over chords containing a major 3rd?
Show answer
Distortion adds strong upper partials and combination tones. These line up for the nearly pure tempered fifth, which is only 1.96 cents off, but clash for the tempered major 3rd, which is about 14 cents wide of a pure 5:4. This is part of the reason, not the whole story.
-
recall
When two tones of slightly different frequency sound together, how is the beat rate related to the two frequencies?
Show answer
The tones drift in and out of phase, so the loudness pulses at a rate equal to the difference in frequency. For example, 440 Hz against 443 Hz beats 3 times a second.
-
recall
When tuning two guitar strings by ear using beats, what tells you they match?
Show answer
Adjust one string until the beating slows to nothing. When the pulsing stops, the two strings are at the same frequency.
-
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.
-
apply
The pure fifth above A2 (110 Hz) is 165 Hz, while the equal-tempered E3 is about 164.81 Hz. Sounded together, about how fast do these two tones beat, and how does that compare with pure vs tempered E5 (about 0.74 beats a second)?
Show answer
About 0.19 beats a second (165 minus 164.81), roughly a quarter of the E5 rate. Both pairs are off by the same 1.96 cents, but at a lower pitch that error spans fewer hertz, so the beating is slower.
-
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).
-
recall
Where do you touch the open A string (110 Hz) for its 5th harmonic, and how does that harmonic's pitch compare with an equal-tempered C♯5?
Show answer
Touch just behind the 4th fret (about fret 3.9). It sounds 550 Hz, a C♯5 about 14 cents (13.69) flat of equal-tempered C♯5, because it is a pure 5:4 major 3rd above the 4th harmonic, A4 at 440 Hz.
-
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.
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
A cents ledger and a beat prediction
Make a table for the pure intervals 9:8, 6:5, 5:4, 4:3 and 3:2. For each, compute the size in cents with 1200 × log2(r), write the equal-tempered size (200, 300, 400, 500 and 700 cents), and give the difference, saying whether the tempered interval is wide or narrow of pure. Then predict two beat rates: the pure against the tempered fifth above A3 (220 Hz), and the pure against the tempered E5 (660 Hz against 659.26 Hz). In the Beats section of the Science page, play the Pure vs tempered E5 button, then set the slider to 0.4 Hz, the nearest setting to your A3 prediction, and compare the two.
What to hand in: One page, typed or written and photographed: the five-row ledger, one conversion written out in full, both beat predictions with their frequencies, and two sentences, one on what you heard and one on why the same cent error beats at different rates at different pitches. 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.