What transposition is, and why instruments need it
Transposition moves every pitch in a piece by the same interval. Do it to a whole score and nothing about the music changes except its absolute pitch: the intervals between notes, the chord qualities and the shape of every melody survive intact. That is why it works at all, and why it reduces to one number.
Two entirely different jobs use the same arithmetic. The first is musical: a singer cannot reach the top of a song in Eb, so you move it down to C. The second is notational, and it is the one that trips people up. A large family of wind instruments is transposing - the note written on the page is not the note that comes out. Press the fingering a trumpeter calls C on a Bb trumpet and a concert Bb sounds, a major second lower. Nobody did this to be difficult. It means a player who learns one set of fingerings can move between the Bb, Eb and C members of an instrument family and keep reading the same shapes on the page.
The convention is stated as an interval: "Bb instruments sound a major second below written pitch." Turn that round and you get the rule you actually use when writing a part: written = sounding + 2 semitones. Every transposing instrument has such a number. Once you have it, transposing a part is addition modulo twelve, and the only remaining craft is choosing how to spell the answer.
The formula, and the sign that everyone gets backwards
Assign each instrument a transposition T, defined as the number of semitones the written note lies above the sounding note. C instruments have T = 0. A Bb instrument has T = 2, because you write a C to get a Bb. An Eb alto saxophone has T = 9: write A, hear C. A horn in F has T = 7: write C, hear F a fifth below. A clarinet in A has T = 3. An alto flute in G has T = 5. A trumpet in D has T = -2, because it sounds above what is written.
Going from concert pitch to a written part, you add T. Going from a written part back to concert pitch, you subtract it. Going directly from one transposing instrument to another - a tenor sax part handed to an alto player - the shift is the difference: Δ = Ttarget − Tsource = 9 − 2 = 7 semitones up.
The sign error is so common it has a diagnostic: if your transposed part sounds a whole tone away from right in the wrong direction, you added where you should have subtracted. The safe habit is to route everything through concert pitch. Convert the source part to concert first, then convert concert to the target. Two subtractions and additions you can check separately beat one composite step you cannot.
Pitch classes are then reduced modulo twelve, which is why the calculator returns a key name rather than a specific octave. Octave placement is a separate decision driven by range: a part that sits comfortably for a tenor sax may need to be written an octave up for an alto, and no amount of modular arithmetic will tell you that. Check the target instrument's written range before you commit.
Worked example: a concert Bb chart for a Bb trumpet and an Eb alto sax
You have a piano and vocal chart in Bb major - two flats - and you need parts for a trumpet and an alto saxophone.
- Identify the starting pitch class. Bb is pitch class 10 (C = 0, so C 0, C# 1, D 2, Eb 3, E 4, F 5, F# 6, G 7, Ab 8, A 9, Bb 10, B 11).
- Trumpet: apply T = 2. The source is concert pitch, T = 0, so Δ = 2 − 0 = 2. New pitch class = (10 + 2) mod 12 = 12 mod 12 = 0, which is C. The trumpet part is in C major, no sharps or flats. That is exactly why so much brass-band and jazz writing lands in flat concert keys - it puts the Bb instruments in easy keys.
- Transpose the chords. Every symbol moves by the same two semitones. Bbmaj7 becomes Cmaj7, Gm7 becomes Am7, Cm7 becomes Dm7, F7 becomes G7. The qualities never change, only the roots.
- Alto sax: apply T = 9. Δ = 9 − 0 = 9. New pitch class = (10 + 9) mod 12 = 19 mod 12 = 7, which is G. The alto part is in G major, one sharp. Bbmaj7 becomes Gmaj7.
- Cross-check by interval. Nine semitones up is a major sixth. Bb up a major sixth is G. It agrees.
- Trumpet part to alto part directly. Δ = 9 − 2 = 7 semitones, a perfect fifth up. C major becomes G major, which matches the answer already derived through concert pitch. The two routes agreeing is the check worth doing every time.
A guitarist who wants to play along with the trumpet part without relearning it uses a capo. Two semitones up is fret 2: hold the Bb-major shapes, capo at 2, and the sounding key is C.
Choosing the right spelling for the answer
Twelve pitch classes but seventeen usable key names means the arithmetic has more than one correct-sounding answer, and picking between them is editorial. Pitch class 6 is F# major with six sharps or Gb major with six flats. Pitch class 1 is Db major with five flats or C# major with seven sharps - the same sound, but one signature has two fewer accidentals to read. The default rule, and the one this calculator applies on Automatic, is to take the spelling with the smaller signature. Six-against-six ties resolve toward F# major and toward Eb minor, the spellings engravers use most.
Two things override that rule in practice. The first is the instrument. Brass and woodwind players read flat keys more fluently because the concert flat keys that suit those instruments transpose into flat or natural written keys; string players and guitarists prefer sharps for the opposite reason. The second is the surrounding music. If the transposed section sits inside a movement in Db, spell it in flats even where sharps would be shorter, because a signature change mid-movement costs the reader more than two extra accidentals.
Watch for the enharmonic trap when the shift lands you at the far edge of the circle of fifths. Transposing a piece in B major up a semitone gives pitch class 0, which is plain C major - a relief. Transposing it up a semitone the other way, to Bb, and then again, lands you at pitch class 1, where the choice between Db and C# genuinely matters. The calculator warns you whenever the chosen spelling carries six or more accidentals and names the alternative.
If you are working from frequencies rather than note names - checking a recording's actual key, say - the note to frequency calculator will identify the pitch, and the cents and interval calculator will tell you how far a source has drifted from the reference before you decide what key it is in at all.
Transposition of the common instruments
| Instrument | Key | T (semitones) | Interval | Concert C is written |
|---|---|---|---|---|
| Flute, oboe, piano, guitar, voice | C | 0 | Unison | C |
| Trumpet, clarinet, tenor sax, soprano sax | Bb | 2 | Major 2nd | D |
| Clarinet in A | A | 3 | Minor 3rd | Eb |
| Alto flute | G | 5 | Perfect 4th | F |
| French horn, English horn | F | 7 | Perfect 5th | G |
| Alto sax, baritone sax | Eb | 9 | Major 6th | A |
| Trumpet in D | D | -2 | Major 2nd down | Bb |
Tenor sax and baritone sax also sound an octave (tenor) or an octave (baritone, on top of the sixth) lower than the pitch class shown, which affects range and octave placement but not the key signature. Double bass, guitar and piccolo transpose by whole octaves only, so their key signatures match concert pitch.
Mistakes that produce a wrong part
- Adding when you should subtract. Writing a part means adding T; reading a part back to concert means subtracting it. Route everything through concert pitch and the direction takes care of itself.
- Transposing the notes but not the key signature. A part shifted two semitones with the original signature left in place forces an accidental on nearly every note and will be sight-read wrong.
- Forgetting the octave. Pitch-class arithmetic is blind to register. A line that sits at the top of a tenor sax's range transposes into a written key an alto can read and a tessitura it cannot sustain.
- Spelling from the chromatic scale instead of the key. In Gb major the fourth degree is Cb, not B. Mechanical sharp-or-flat lookup gives a symbol that sounds right and reads as an accident.
- Assuming a capo can transpose downwards. A capo only shortens the string, so it only raises pitch. To play a song two semitones lower you change shapes or retune - and if you retune, recheck the load with the string tension calculator before you leave it there.
- Treating baritone sax as just an Eb instrument. Its pitch class transposition is a major sixth like the alto, but it also sounds a further octave down, so a doubled line will land in a different register than the arithmetic alone suggests.
Where transposition sits among the alternatives
Transposition is only one of the pitch operations you might want. Modulation moves the key within a piece and is a compositional act, not a mechanical one. Retuning - dropping the whole instrument a semitone rather than rewriting the part - gets you the same sounding pitch with the original fingerings, at the cost of slacker strings and a different timbre. Pitch shifting in software moves recorded audio and, unlike notation, cannot be undone without artefacts; it also changes formants unless the algorithm corrects for them.
For fretted instruments a capo is the fastest transposition tool there is, but it is directional. It raises pitch by one semitone per fret and cannot lower it. It also shortens the vibrating length, which raises tension slightly at the fretted pitch and changes the instrument's timbre - brighter and tighter the further up you go. If you are choosing between a capo and a genuine retune, the fret spacing calculator shows exactly how the scale length shortens at each fret.
Historic instruments complicate the picture further. Natural horns and trumpets were built in a fixed key and changed key with crooks, which is why nineteenth-century horn parts appear in a dozen different transpositions inside one symphony; the player was expected to read them all. Modern horn players read those parts on an F horn by transposing at sight, which is exactly the arithmetic on this page done in real time. The harmonic series calculator shows which notes those valveless instruments could produce at all, and why their parts look the way they do.
Finally, remember that everything here assumes twelve-tone equal temperament, where an enharmonic pair is literally the same pitch. In historic meantone or in just intonation, Db and C# are different notes, and transposition is not the pitch-neutral operation described above - it changes the intervals inside the music. That is precisely why equal temperament won.
Key terms
- Concert pitch
- The pitch that actually sounds, regardless of what any player is reading. A score in concert pitch shows every instrument at sounding pitch, which is how a conductor and an arranger prefer to see it.
- Transposing instrument
- An instrument whose written notes differ from its sounding notes by a fixed interval, so that one set of fingerings serves the whole family. Bb, Eb, F and A instruments are the common cases.
- Pitch class
- A note name with the octave discarded, numbered 0-11 from C. All Cs share pitch class 0. Transposition is addition in this system, taken modulo 12.
- Enharmonic equivalent
- Two spellings of the same sounding pitch, such as F# and Gb. Identical in equal temperament; distinct in meantone and in just intonation.