Tuning theory and measurement

What “in tune” means for an instrument with three partials per note, and how anyone actually checks.

PHY-15What does “in tune” mean for a handpan, and why is it harder than for a guitar?#

intermediate

Short answer. On a guitar, in tune means one number per string and you can fix it in seconds. On a handpan it means three frequencies per note, all in a fixed relationship to each other and to concert pitch, permanently frozen into the shape of the metal. There is no adjustment mechanism at all.

A guitar string has one dominant fundamental and a naturally harmonic overtone series that follows it automatically. Tighten the string and everything moves together. The tuner has one degree of freedom per string and immediate feedback.

A handpan tone field has to satisfy three conditions at once. The fundamental must be at the right absolute pitch. The second partial must be a true octave above it. The third must be a fifth above that octave. And the tuner’s only tools are the shape, thickness and internal stress of the steel, all of which affect all three partials simultaneously and in different proportions. Hammering to flatten the fundamental will move the octave and the compound fifth too, usually by different amounts and sometimes in different directions. Tuning is therefore an iterative search: adjust, measure, adjust again, converge.

Then multiply that by nine or more notes on one continuous piece of metal. Working on one field disturbs its neighbours, so the tuner is not solving nine independent problems but one coupled problem with about thirty simultaneous targets. This is why building a handpan takes days rather than minutes, why makers have waiting lists, and why price correlates so strongly with tuning quality.

The permanence is the other half of it. A guitar goes out of tune constantly and you retune it constantly. A handpan holds its tuning for years and then, when it drifts, requires a skilled person with a hammer to bring it back. There is no user serviceable adjustment. Nothing about the instrument can be corrected by the owner.

One consequence that surprises new owners: a handpan cannot be tuned to another instrument. If you want to play with a piano at A=440, you buy an instrument built at A=440. If you buy one at A=432, you have bought a different instrument, permanently.

See also: PHY-16, PHY-03

PHY-16Cents, A=440 and A=432: what is actually true?#

intermediate

Short answer. A cent is a hundredth of a semitone, the standard unit for talking about small pitch differences. A=440 Hz is the international standard, recommended in 1939 and confirmed by the ISO in 1955. A=432 is about 32 cents lower, roughly a third of a semitone. The historical arguments for 432 are weaker than usually claimed and the health claims have no peer reviewed support, but nothing is wrong with liking how it sounds.

Cents. An octave is 1200 cents and a semitone is 100. The scale is logarithmic, so a cent is the same perceptual size at every pitch even though it corresponds to a different number of hertz. Roughly, most listeners can detect a difference of about 5 to 10 cents on sustained tones; 20 cents is clearly audible; 50 cents is a quarter tone and unmistakable. Makers commonly quote tolerances in single figures of cents.

Where 440 came from. Before the twentieth century, concert pitch varied enormously, from around A=376 Hz on a Lille organ of 1700 to about A=506.9 Hz at Halberstadt in 1361. France legislated the diapason normal at A=435 in 1859, partly in response to complaints from singers about pitch inflation. A=440 was recommended at an international conference in London in 1939 and confirmed by the International Organization for Standardization as ISO 16 in 1955. It is a convention that won, not a discovery.

Where 432 came from. Two strands. Joseph Sauveur’s scientific pitch, around 1700, fixed middle C at exactly 256 Hz for mathematical tidiness, which puts A at about 430.54 Hz, not 432. Separately, Verdi campaigned for standardisation in Italy: he actually advocated matching France at 435, and when the Italian government’s commission proposed 432 for mathematical convenience he accepted it, calling the difference almost imperceptible. Italy decreed 432 in 1884. So “Verdi tuning” is a real historical episode, but Verdi’s own preference was 435 and he treated the difference as trivial.

The modern claims. The frequently repeated link to the Schumann resonance does not survive checking: that resonance is about 7.83 Hz and has no clean arithmetic relationship to 432. There is no peer reviewed evidence that 432 Hz has therapeutic properties that other tunings lack.

What this means for buyers. Both are widely offered. Some makers charge a small premium for 432 because it is less common; many charge the same. The one real consideration is practical: an A=432 instrument will not play with anything tuned to A=440 without retuning the other instrument. Choose on that basis, and on whether you prefer the sound, rather than on physics claims in either direction.

See also: PHY-15

PHY-17How do temperature and humidity shift a handpan’s pitch?#

intermediate

Short answer. Less than people expect, and mostly indirectly. The steel’s own pitch moves only slightly with temperature. The bigger and more audible effect is on the air: temperature changes the speed of sound, which moves the Helmholtz resonance and shifts the crossover bands, so certain notes become wobbly in the cold and settle in the warm.

Take the two mechanisms separately.

The steel. Heating steel expands it slightly and reduces its elastic modulus slightly. Both effects lower the natural frequencies of a tone field, but for the temperature swings you encounter in normal life the movement is small compared with the 5 to 10 cents a listener can detect. Makers are consistent that ordinary everyday temperature change is not something to worry about. What does matter is repeated exposure to extremes, which stresses the metal and, importantly, the adhesive seam over time. A hot car is the standard cautionary example, and the seam is usually the least heat tolerant part of the instrument.

The air. This is where the audible change lives. The speed of sound in air rises with temperature by roughly 0.6 m/s per degree Celsius, so a cold instrument has a lower Helmholtz frequency and lower cavity standing wave frequencies than a warm one. Since the crossover bands are set by those standing waves, they move with temperature. Saraz give a specific example: A4 on their 21 inch chamber sounds good at around 27 degrees Celsius but becomes problematic below about 16. The note itself has not gone out of tune. The interference pattern has moved onto it.

Humidity changes the speed of sound too, because water vapour is lighter than dry air, but the effect is small compared with temperature at ordinary conditions. Humidity’s real significance for a handpan is corrosion, not tuning: moisture on nitrided or raw steel is what causes rust. See ANA-15.

The practical upshot for players is to let a cold instrument come up to room temperature before judging it, and never to evaluate an instrument you have just carried in from the cold. A quarter of an hour is usually enough. Avoid forcing it with heaters or hot water, which stresses the seam for no benefit.

See also: PHY-14, CAR section

PHY-18Equal temperament or just intonation: which does a handpan use?#

advanced

Short answer. Almost every handpan is built in twelve tone equal temperament, with each note’s fundamental placed at its standard equal tempered frequency. But each individual note’s internal partials are tuned to pure just intervals: a true 2 : 1 octave and a true 3 : 1 twelfth. So a handpan is equal tempered horizontally and justly intoned vertically, at the same time.

This sounds contradictory until you separate the two things being tuned.

Between notes, makers use equal temperament because that is what everything else uses. It lets the instrument play with pianos, guitars and recordings, and it makes scale names meaningful. In equal temperament every semitone is exactly 100 cents, which means most intervals are slightly impure: an equal tempered perfect fifth is about 2 cents narrow of a pure 3 : 2, and an equal tempered major third is about 14 cents wide of a pure 5 : 4. Those compromises are the price of being able to play in any key.

Within a note, the tuner aims at pure ratios. The octave partial should be exactly twice the fundamental and the compound fifth exactly three times it. There is no reason to compromise here, because these are partials of a single sound rather than separate notes, and any deviation produces beating within the note itself. See PHY-03.

The interesting consequence is that a handpan is subtly more consonant than the same notes on a piano. Because a handpan scale is usually a single mode with seven to nine notes and no chromatic obligations, and because each note carries a strong pure twelfth, the intervals you actually play tend to reinforce each other. Play a fifth: the compound fifth partial of the lower note lands very close to a partial of the upper note, and they lock.

Genuinely just intoned handpans, where the notes themselves are placed at pure ratios from the tonic rather than at equal tempered frequencies, do exist. They are rare, almost always commissioned, and they trade away compatibility with other instruments. A few makers experiment with other tunings entirely, including microtonal and non Western systems, and PANArt eventually abandoned an external reference altogether: from 2010 the Free Integral Hang was freed from the 440 Hz standard and each instrument was attuned to itself.

Ask before assuming. If it matters to you, the maker will tell you what reference they use.

PHY-19How do makers and players measure tuning?#

advanced

Short answer. With a chromatic tuner for a quick check of fundamentals, and with a real time spectrum analyser or a strobe style tuner for anything serious, because a single number tuner cannot see the octave and compound fifth partials separately.

Why an ordinary tuner is not enough. A guitar tuner takes an input, picks out what it thinks the fundamental is, and shows one number. Feed it a handpan note and two problems appear. First, it tells you nothing about the two partials that determine whether the note actually sounds right. Second, it can lock onto the wrong partial, especially on notes where the octave is strong, and report a pitch an octave up. It is fine for confirming that a note is roughly where the maker said, and useless for diagnosing why a note sounds wrong.

What is actually used. Tuners work with real time FFT spectrum analysis. The signal from a microphone is decomposed into its frequency components and displayed as a spectrum, so all three partials are visible simultaneously along with everything above them. Software commonly mentioned in the community includes general purpose analysers on laptops and phone apps built for the job; the specific tool matters much less than having enough frequency resolution and a display that updates fast enough to watch a note decay. Strobe tuners are also used, since they resolve individual partials precisely.

Practicalities that catch people out.

  • Microphone placement. Close to the field for isolation, further back for a representative overall sound. They give different answers, and both are legitimate.
  • Strike consistency. Nonlinearity means a hard strike momentarily sharpens the note. Measure at a moderate, repeatable dynamic and read the pitch after the initial transient has passed, not during it.
  • Which part of the decay. Partials do not decay at the same rate. A note measured immediately after the strike and the same note measured two seconds later can read differently, particularly if crossover is affecting it.
  • Temperature. Let the instrument settle at room temperature first. See PHY-17.
  • Neighbours. Other fields ring sympathetically and put their own peaks in your spectrum. Learn to recognise which peaks belong to the note you struck.

For an owner, the honest advice is that measurement is diagnostic, not prescriptive. Knowing a note is 8 cents flat does not help you fix it, and small readings are easy to over interpret. Measure when something sounds wrong, to find out what.

See also: PHY-20

PHY-20How do I read a tuning report or a spectrum?#

advanced

Short answer. A tuning report lists, for each note, the target and measured frequencies of the fundamental, the octave and the compound fifth, usually with the deviation in cents. Read down the deviations first: you are looking for small numbers and, more importantly, for consistency across the instrument. In a spectrum, find the three tall peaks in a roughly 1 : 2 : 3 ratio, and check the middle and upper peak sit exactly where the fundamental predicts.

Not every maker supplies a report and there is no standard format. A typical one gives per note: the note name, the intended fundamental in hertz, the measured fundamental, and then the same for the two tuned partials, with deviations in cents.

How to read it.

  • Deviation on the fundamental tells you whether the instrument is at the pitch standard it claims. A consistent offset across all notes usually means the whole instrument is a little sharp or flat rather than mistuned, which is not a fault.
  • Deviation on the octave and compound fifth is the interesting column. This is where fusion lives. A note whose fundamental is spot on but whose octave is 20 cents out will sound worse than one whose fundamental is 8 cents off but whose partials are perfectly aligned.
  • Consistency across notes matters more than any single figure. An instrument where every note sits within a few cents is the work of someone who converged properly. One with two immaculate notes and one wild one usually means the tuner ran out of patience or out of metal.
  • Do not expect zeros. Steel is not a spreadsheet. Small deviations are normal on excellent instruments.

Reading a live spectrum. Strike a note and look for the pattern. The fundamental is normally, though not always, the tallest peak. Multiply its frequency by 2 and by 3 and check that peaks sit at those points. If the peak near 2f is visibly offset from exactly 2f, that is the drift you can hear. Above the third peak you will see a scatter of unaligned peaks, which is normal and is the bell-like layer of the sound. Down at the bottom, often below 100 Hz, there is usually a broad low peak that appears on every note: that is the Helmholtz resonance of the cavity, and PANArt noted it is detectable in every spectrum the instrument produces.

A caution. A report describes the instrument on the day it was measured, in the conditions it was measured in, by the person who built it. It is evidence of care, and it is a useful baseline for spotting drift years later. It is not a guarantee, and it is not a substitute for playing the instrument.

See also: PHY-02, BUY section