In brief
Tuning establishes pitch relationships and a reference. Temperament adjusts selected intervals to reconcile competing demands within a tuning system, often on instruments with fixed pitches.
What you’ll learn
- Distinguish a reference pitch from a tuning system.
- Compare a just major third with its equal-tempered counterpart.
- Explain a temperament as a musical compromise rather than a mistake.
Useful first: Pitch and frequency · Intervals · Timbre and overtones
Two questions before tuning
Ask first which reference pitch the musicians will share, and second how the other pitches relate to it. Agreeing on A4 = 440 Hz answers only the first question. An ensemble could use that reference with different interval systems. A singer can adjust a note while listening to another player; a keyboard usually offers a set of pitches determined before the performance begins.
The practical problem depends on the music. A sustained chord exposes relationships that a rapid passage may not foreground. A piece that moves through many tonal centres asks something different from one built around a stable drone. Before calling a pitch inaccurate, identify the intended reference, the instrument's possibilities and the surrounding musical practice. A measurement is useful only after those assumptions are clear.
References: [1]
Simple ratios and their consequences
In just intonation, selected intervals are expressed through simple frequency ratios. A 3:2 ratio gives a just fifth, and 5:4 gives a just major third. With suitable harmonic sounds, these relationships align some upper components and can reduce beating. Such alignments help explain an acoustic property of the interval; they do not establish a universal ranking of musical styles or expressive effects.
Problems arise when a fixed pitch has to fulfil several roles. Three successive 5:4 thirds produce a ratio of 125:64, which is approximately 1.953, rather than the 2:1 octave that the equivalent chain reaches on an equal-tempered keyboard. You cannot keep all those thirds exactly pure and also make the chain close as that octave. The conflicting requirements are mathematical, not evidence of poor musicianship.
References: [1]
Twelve equal steps are one solution
Twelve-tone equal temperament divides the octave into twelve identical frequency ratios. Each step multiplies frequency by 2^(1/12); four steps give an equal-tempered major third of 2^(4/12), approximately 1.2599. A just major third is 1.25. These intervals are close, but not the same. Equal temperament makes corresponding interval sizes consistent when a passage moves to another pitch level.
That consistency is valuable for many instruments and repertoires, but it is not a definition of music itself. Other equal divisions, unequal temperaments and flexible performance practices answer different needs. The word equal concerns ratios between steps, not equal numbers of hertz. When learning a historical or traditional repertoire, investigate its actual performance conventions instead of assuming that a modern piano supplies every intended pitch.
References: [2]
Listen before reaching for a rule
When two nearby steady frequencies sound together, their interference can produce a fluctuation called beating. Musicians can use changing beat rates while tuning. Listen carefully to which components interact, however: complex instruments contain more than one frequency, and room acoustics or vibrato can complicate the impression. A simple two-tone exercise isolates the phenomenon more clearly than a full ensemble.
At rehearsal, document the agreed reference and test a representative passage. A keyboard, voice and fretless instrument may have different room to adjust. Listen to a sustained sonority, then to its motion into the next one. A tuning choice that seems attractive in isolation may require a different compromise in context. Treat the discussion as musical coordination rather than a contest between supposedly natural and artificial systems.
References: [2]
A worked example
Two major thirds above 240 Hz
Take a base frequency of 240 Hz. Compare two theoretical major thirds: a 5:4 just third and a four-semitone equal-tempered third. This calculation uses idealised frequencies, not a claim about every instrument's behaviour.
- The just third is 240 × 5/4 = 300 Hz.
- The equal-tempered third is 240 × 2^(4/12), approximately 302.38 Hz.
- The two upper notes differ by about 2.38 Hz. They serve similarly named interval roles but occupy different frequencies.
- For harmonic tones, compare the base tone's fifth harmonic, 1,200 Hz, with the upper note's fourth harmonic.
- At 300 Hz the fourth harmonic is 1,200 Hz. At 302.38 Hz it is about 1,209.52 Hz, so the components no longer coincide.
Notice: The example makes one acoustic consequence visible. It does not tell you which version a whole composition, ensemble or musical tradition requires.
Try it / Listening experiment
Hear a relationship.
Two short, pure tones: a starting pitch followed by a higher pitch. This model uses twelve-tone equal temperament. It does not model a raga’s intonation or ornamentation.
Frequency ratio: 27/12 ≈ 1.4983
Choose an interval, then listen to the two tones.
Start with a comfortable device volume. Audio plays only when you press Listen.
Try it yourself
Test why three pure thirds do not close an octave.
- Begin with the abstract ratio 1.
- Multiply by 5/4 three times.
- Compare the result with 2 and explain which assumption would need to change if both were required to match.
Compare your thinking
The product is 125/64 = 1.953125, smaller than 2. To close exactly at the octave, at least one third must change, or the final pitch must remain distinct from the octave-equivalent starting pitch. A temperament is one way to organise such adjustments across a fixed system.
What to remember
- Reference pitch and tuning system are different decisions.
- Simple ratios can conflict when combined into a fixed pitch network.
- Temperaments balance musical requirements; none describes every tradition.
References & further study
Use these sources to check terminology and continue studying. The worked examples and exercises are written for this collection.
- Tartini tones and temperament: an introduction for musicians
Joe Wolfe, UNSW Music Acoustics
Explains just intervals, equal temperament and the conflict between chains of pure thirds and octave equivalence.
- Tartini tones and temperament
UNSW Physclips
Provides demonstrations and acoustic explanations of beating, consonance and tuning choices.
Educational writing by Burnt Chapati. Send a correction or suggest a source.