Most of us were taught that Western music contains twelve notes:
C, C♯, D, E♭, E, F, F♯, G, A♭, A, B♭, and B—after which the pattern repeats at the next octave.
On a modern piano, that is essentially true. Press middle C and the piano produces one predetermined frequency. Press the black key between G and A and you get one predetermined pitch, whether the composer calls it G♯ or A♭.
But that is not a law of nature. It is a compromise humans imposed on musical instruments.
In naturally tuned music, a note does not always have one permanent frequency. Its ideal pitch can change according to the key, chord, and harmonic function in which it appears. The difference can be small, but it is not imaginary. In one particularly striking case, two versions of what a piano treats as the same note can differ by approximately 41 cents—about 41 percent of a half-step, or nearly half of the distance between two neighboring piano keys.
That means a “single note” can move almost halfway toward the next note depending upon what it is doing musically.
Nature Does Not Divide the Octave Into Twelve Equal Pieces
Musical harmony begins with physical relationships between frequencies.
If one string vibrates at 220 cycles per second and another vibrates at 440 cycles per second, the second is vibrating exactly twice as fast. We hear those two pitches as versions of the same note, one octave apart. Their frequency relationship is:
2:1
Other simple numerical relationships also produce intervals that human beings generally perceive as especially consonant:
- An octave is 2:1
- A perfect fifth is 3:2
- A perfect fourth is 4:3
- A major third is 5:4
- A minor third is 6:5
- A major sixth is 5:3
This method of tuning intervals according to simple whole-number frequency ratios is called just intonation.
The intervals sound exceptionally smooth because their vibrations regularly coincide. A note vibrating three times during the same period in which another vibrates twice forms a 3:2 perfect fifth. Their wave patterns repeatedly meet in an orderly way. The notes seem to lock together.
The difficulty appears when we try to use all of those pure relationships at once.
The mathematics does not close perfectly.
The Piano’s Solution: Make Every Note Slightly Wrong
Modern pianos are normally tuned using twelve-tone equal temperament. The octave is divided into twelve mathematically equal half-steps. Each note has a frequency equal to the previous note multiplied by the twelfth root of two:
2¹⁄¹² ≈ 1.059463
After multiplying by that number twelve times, the frequency has exactly doubled, completing the octave.
This system gives musicians an enormous practical advantage: every key works. A pianist can begin in C major, modulate to E major, wander into A♭ major, and eventually return to C without retuning the instrument. The same black key can function as G♯ in one passage and A♭ in another.
The price is that, except for the octave, the intervals are not perfectly pure.
For example:
- A pure 3:2 perfect fifth is approximately 701.96 cents
- An equal-tempered perfect fifth is 700 cents
- A pure 5:4 major third is approximately 386.31 cents
- An equal-tempered major third is 400 cents
A cent is one one-hundredth of an equal-tempered half-step. There are 100 cents between adjacent piano keys and 1,200 cents in an octave.
The equal-tempered major third is therefore almost 14 cents wider than the natural 5:4 third. It is close enough to work, but it is not acoustically pure. On a sustained instrument, one can hear the slight beating produced by the mismatch.
Equal temperament does not make every interval perfect. It distributes the imperfection so evenly that no ordinary key is unusable.
It is less a discovery than a diplomatic agreement among the notes.
The Same D Can Be Two Different Pitches
Consider the note D within the key of C major.
If C has a frequency value of 1, D can be tuned as a major whole-step above it using the ratio:
D = 9:8
But suppose D becomes the root of a D-minor chord containing D, F, and A. In a justly tuned C-major scale:
- F can be tuned at 4:3 above C
- A can be tuned at 5:3 above C
For D–F to be a pure 6:5 minor third, and D–A to be a pure 3:2 perfect fifth, D needs to be:
D = 10:9
We now have two mathematically reasonable versions of D:
- D as 9:8
- D as 10:9
The relationship between them is:
(9/8) ÷ (10/9) = 81/80
The ratio 81:80 is called the syntonic comma. It is approximately:
21.51 cents
Thus, the D that works perfectly as the fifth of a G chord can be approximately 21.5 cents higher than the D that works perfectly as the root of a D-minor chord.
The written note has not changed. It is still D. But its harmonic job has changed, and nature gives us two slightly different answers for where D should be.
A fixed-pitch piano cannot move the note while it is being played. A capable singer, violinist, trombonist, or string ensemble can. Musicians often adjust notes instinctively, listening for the point at which the chord settles and the audible beating diminishes. They may never calculate the ratio 81:80, but their ears can lead them toward it.
This is one reason a fine unaccompanied choir or string quartet can produce chords that seem to become unusually still, luminous, or resonant. The musicians are not necessarily adhering rigidly to twelve immovable piano pitches. They are tuning the notes to one another.
The Nearly Half-Step Difference
The more dramatic example involves enharmonic notes such as G♯ and A♭.
On a modern piano, G♯ and A♭ are the same key. In musical notation, however, they do not necessarily mean the same thing.
A G♯ may function as the major third above E. If C is our reference pitch, and E is tuned as a pure 5:4 major third above C, then another pure 5:4 major third above E produces G♯:
G♯ = 5/4 × 5/4 = 25/16
A♭, meanwhile, can be tuned as a pure minor sixth above C:
A♭ = 8/5
Compare the two:
(8/5) ÷ (25/16) = 128/125
The ratio 128:125 is known as the lesser diesis, or enharmonic diesis. Its size is approximately:
41.06 cents
That is not quite half of a 100-cent half-step, but it is remarkably close.
To put real frequencies to it, suppose C is approximately 261.63 hertz:
- G♯ at 25:16 would be approximately 408.79 Hz
- A♭ at 8:5 would be approximately 418.60 Hz
- The equal-tempered piano key used for both would be approximately 415.30 Hz
The natural G♯ and natural A♭ in this example differ by almost 10 vibrations per second.
The piano compromises by placing its single black key between them.
So when we say that G♯ and A♭ are “the same note,” what we really mean is that equal temperament assigns them to the same key. They are equivalent for the mechanical convenience of a twelve-note keyboard. They are not necessarily identical in their harmonic origin or ideal natural pitch.
Why the Key Matters
A musical key establishes a network of relationships.
In C major, C is the tonal center. G tends to be understood in relation to C; E forms the major third above C; B functions as a leading tone pointing toward C.
Move into E major and those same written or sounding pitches acquire different jobs. Now E is home. G♯ is the major third of the tonic chord. B is its perfect fifth. D♯ pulls upward toward E.
If every important interval is to remain acoustically pure, some notes must move slightly when the harmonic center changes. A note tuned perfectly for one chord may be imperfect for the next. Correcting it for the second chord can create a discrepancy elsewhere.
This is not evidence that the mathematics is faulty. It is evidence that a closed twelve-note system cannot simultaneously preserve every simple frequency relationship in every key.
The problem is sometimes called comma drift. A progression can travel through a sequence of perfectly tuned intervals and arrive at what is nominally the original note—but at a slightly different frequency. The musical spelling says that we have come home; the multiplication of the natural ratios says that home has moved.
That tiny difference is one of the reasons tuning systems and temperaments exist.
Comma, Diesis, and Half-Step Are Not the Same Thing
Several related measurements can easily become confused:
- The syntonic comma, ratio 81:80, is approximately 21.51 cents
- The Pythagorean comma, produced by the mismatch between twelve pure fifths and seven octaves, is approximately 23.46 cents
- The lesser diesis, ratio 128:125, is approximately 41.06 cents
- An equal-tempered half-step is exactly 100 cents
Therefore, the statement “the same note can differ by nearly half a half-step depending on its key or harmonic meaning” refers most directly to the approximately 41-cent lesser diesis.
The more common adjustment of the same named note between two harmonic roles is often the smaller 21.5-cent syntonic comma.
Both illustrate the same underlying truth: musical notation groups together pitches that nature does not always place at precisely the same frequency.
Sharps and Flats Are Not Merely Different Spellings
Beginning musicians are often told that G♯ and A♭ are two names for the same note. That explanation is useful at a piano keyboard, but it conceals some of the deeper logic of music.
A sharp generally means that a note has been raised in relation to a particular scale or harmony. A flat means that a note has been lowered. The two spellings tell us where the note came from, where it is going, and what job it performs.
G♯ in an E-major chord is not conceptually interchangeable with A♭ in an A♭-major or F-minor chord, even when equal temperament forces both onto the same piano key.
The notation preserves a distinction that the keyboard erases.
Historically, this was not merely theoretical. Some keyboard instruments were built with split black keys so that performers could play separate sharp and flat pitches. Nicola Vicentino’s sixteenth-century archicembalo went much further, dividing the octave into many more than twelve available pitches. Meantone temperaments likewise produced noticeably different enharmonic notes, although they solved and distributed the tuning discrepancies differently from strict just intonation.
Equal temperament eventually prevailed largely because it made unrestricted modulation and standardized fixed-pitch instruments practical—not because it revealed that G♯ and A♭ had always been physically identical.
The Note Is a Relationship
We tend to imagine a musical note as a fixed object: C is C, D is D, and A is 440 hertz.
But even A = 440 is only a modern tuning convention. Orchestras can tune somewhat higher or lower. Earlier periods and different regions used other reference pitches. More importantly, after selecting a reference pitch, musicians must still decide how all the remaining notes relate to it.
A note is therefore not merely a frequency. It is also a relationship:
- a relationship to the tonal center,
- a relationship to the other notes in the chord,
- a relationship to the notes immediately before and after it,
- and a relationship to the tuning system being used.
The piano gives us the enormously useful illusion that music is constructed from twelve permanent sonic bricks. Natural harmony reveals something more fluid. The notes behave less like fixed points and more like members of a living family, subtly adjusting themselves according to whom they are standing beside.
That adjustment can be 21.5 cents.
In certain enharmonic situations, it can exceed 41 cents—nearly half the distance to the next piano key.
The name of the note may remain the same. The mark on the page may remain the same. Our piano may offer only one key for it.
But the mathematically pure note nature asks for can change with the harmony.
And somehow, the human ear knows.
References and Further Reading
The mathematical distinction between the just 5:4 major third, the Pythagorean 81:64 major third, and their 81:80 syntonic comma is discussed by the Society for Music Theory in “Scholtz, Footnotes”.
A detailed academic treatment of the unavoidable discrepancies within just tuning appears in Just Tuning and the Unavoidable Discrepancies, published in the Indiana Theory Review.
For the historical use of enharmonic divisions, including Nicola Vicentino’s approximately 40-cent diesis, see The 31-Tone Tuning System of Nicola Vicentino.
John Baez’s mathematical treatment, The Mathematics of Tuning Systems, illustrates both the 81:80 syntonic comma and the 128:125 lesser diesis.
