When you tune an instrument using electronic tuners, you're using equal temperament — a deliberate mathematical compromise that made every key sound equally in-tune (and equally slightly out-of-tune) by spacing the 12 semitones exactly 100 cents apart rather than the pure frequency ratios that would make one key perfect and every other key wrong
The previous articles on this site covered frequency unit basics, the electromagnetic spectrum, digital audio and the Nyquist theorem, and CPU clock speeds. This article addresses musical tuning systems — specifically why the frequency ratios between musical notes aren't "natural" numbers, what temperament means, and why a piano can't be tuned perfectly in all keys simultaneously.
Pure intervals: the physics of harmonics and why they don't fit in 12 notes
Vibrating strings and air columns produce harmonics — whole-number multiples of a fundamental frequency. A string vibrating at 440 Hz (concert A) also vibrates at 880 Hz (octave), 1320 Hz (perfect fifth above that octave), 1760 Hz (two octaves), and so on.
The "pure" perfect fifth is a frequency ratio of 3:2 — if A = 440 Hz, the E above it is "purely" 660 Hz (440 × 3/2). This sounds consonant because the harmonics align well: 660's second harmonic (1320 Hz) matches 440's third harmonic (1320 Hz).
The Pythagorean comma problem: if you build a scale by stacking pure perfect fifths — starting at C and going up: C → G → D → A → E → B → F# → C# → G# → Eb → Bb → F → and then back to C — the mathematics says you should return to exactly the starting note after 12 fifths. You don't. You return to a note that's slightly sharp — by a ratio of approximately 1.0136, called the Pythagorean comma (about 23.46 cents).
This tiny but audible error is the fundamental tuning problem: pure intervals and 12 evenly-spaced notes are mathematically incompatible.
Equal temperament: the mathematical compromise
Equal temperament solves the comma problem by distributing the error evenly — instead of pure 3:2 perfect fifths, every fifth is tuned slightly flat (by about 1.955 cents — a tiny, nearly inaudible amount). After 12 such fifths, you arrive back at exactly the octave, with no leftover comma.
The frequency ratios in equal temperament: each semitone is a ratio of 2^(1/12) ≈ 1.05946. This is an irrational number — none of the intervals (except the octave) have simple integer ratios. The equal-tempered perfect fifth is 2^(7/12) ≈ 1.49831 — close to 3:2 (1.5), but not exactly.
The result: every key sounds equally in-tune — and equally slightly "off" compared to pure intervals. A chord in C sounds as consonant as a chord in F# or Db. Before equal temperament, some keys were nearly unplayable on keyboard instruments tuned in other systems.
Why this matters in Hz: the frequency differences are measurable
The equal-tempered A4 is defined as exactly 440 Hz (the modern standard, established in 1939 by the International Organization for Standardization).
E5, a perfect fifth above A4:
- Pure (Pythagorean): 440 × 3/2 = 660.000 Hz
- Equal temperament: 440 × 2^(7/12) = 659.255 Hz
The difference: 0.745 Hz — small but producing audible beats when the two notes are sounded together. A beat frequency of 0.745 Hz means the sound "pulses" about 0.745 times per second — just under once per second. A trained ear can hear this.
String players and singers can adjust in real-time — they're not constrained by fixed pitch (unlike a piano), and often play pure intervals when playing long sustained chords, "bending" toward mathematically pure ratios. A string quartet playing without piano sounds slightly different from the same music played with piano accompaniment, because the string players naturally adjust toward pure intervals while the piano holds to equal temperament.
Alternative temperaments: well temperament and meantone
Before equal temperament became universal (roughly the 18th-19th century), various alternative tuning systems were used:
Meantone temperament: tuned pure major thirds (5:4 ratio) at the expense of worse fourths and some unusable "wolf" fifths. Works beautifully in a few keys; other keys sound harsh.
Well temperament: not the same as equal temperament. Various historical "well temperament" systems (Werckmeister, Kirnberger, and others) made all 12 keys playable while preserving more purity in common keys and less in remote keys. The "character" of different keys — one key feeling brighter, another more melancholic — may partially reflect the different interval purities across keys in these systems.
The question of Bach: whether Bach's "Well-Tempered Clavier" was written for equal temperament or for a specific historical well temperament is actively debated among musicologists. Modern performances on equal temperament pianos sound different from recordings on historically-informed tuning systems.
How to use the Frequency Converter on sadiqbd.com
- For calculating note frequencies: any note in equal temperament can be calculated as
440 × 2^((n-69)/12)where n is the MIDI note number (A4 = 69). Converting the resulting Hz value to a different frequency unit is what this tool handles - For comparing pure vs tempered intervals: calculate the pure frequency (using integer ratios) and the equal-tempered frequency — the difference in Hz shows the size of the tuning compromise
- For audio engineering work: sample rates (44,100 Hz, 48,000 Hz, 96,000 Hz) are frequencies too — converting between sample rates and understanding the Nyquist frequency (half the sample rate) involves frequency unit conversions this tool handles directly
Frequently Asked Questions
If equal temperament is a compromise, why don't we use pure tuning instead? Because pure tuning only works for a single key at a time on a fixed-pitch instrument. A piano tuned with pure intervals in C major would have a nearly perfect-sounding C major chord — and would be unplayably out-of-tune in Gb major. The benefit of equal temperament is that all 12 major and minor keys are usable, equally impure in the same way. For instruments that can continuously adjust pitch (voice, strings, trombone), pure intervals are used in practice — but for fixed-pitch instruments that must serve multiple keys in the same composition, equal temperament's uniform compromise is the enabling technology.
Is the Frequency Converter free? Yes — completely free, no sign-up required.
Try the Frequency Converter free at sadiqbd.com — convert between Hz, kHz, MHz, GHz, and RPM instantly.
