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There are some related answers (like "What is the formal definition of 'transpose'?", and "What is the idea of "transposing by perfect unison"?"), but they didn't answer my question, unfortunately.

That's a tricky question, let's give it some context first. Every note has three pieces of information related to its pitch, — octave, letter, and alteration (we're ignoring frequency):

  • A♮4
  • E♭2
  • G♯5, etc.

Similarly, every (conventional) interval has two pieces of information that define it completely, — type and kind (I'm not sure about the terms though):

  • minor third (a third, minor)
  • perfect unison (a unison, perfect)
  • doubly-augmented fifth (a fifth, doubly-augmented), etc.

If I were to be asked to transpose any note to any interval, that won't be too hard:

  • A♮4 transpose up by a minor third → C♮5
  • E♭2 transpose up by a perfect unison → E♭2
  • G♯5 transpose down by a doubly-augmented fifth → C♭5

I'm trying (and failing!) to pin-point the exact steps that I take to perform these transpositions, the way I'm manipulating these five pieces of information to produce the resulting note. Can somebody help me figure them out?

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  • The reason that there isn't a more well-known general method for this is that there are rather few different notes and interevals to begin with. Most professional musicians simply find it easier to know all possible combinations by heart than to construct them anew every time. Mar 27 at 19:40
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    @KilianFoth - How many people have their diminished union or augmented 2nd tables memorized, though? (Yes, I've used a diminished union transposition in Musescore.) Knowing an algorithm is useful.
    – Dekkadeci
    Mar 27 at 19:54
  • @KilianFoth Well, I did have to think a couple of seconds to transpose G♯5 down to C♭5, I didn't just took it from memory, there definitely was some algorithm-y activity going on :) Mar 27 at 20:49
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The ordinal designation of an interval indicates how the letter name changes: if it stays the same, the interval is a unison; if it changes by one, the interval is a second, and so on. Therefore, if you're transposing A flat up by some quality of sixth, you must go up the scale by five letters, arriving at F.

The quality of the interval tells you how many semitones it spans. This is basically a lookup. A major interval is always one semitone larger than the corresponding minor interval, augmented is always one more than major or perfect, and diminished is always one smaller than minor or perfect. Intervals one semitone larger or smaller than augmented or diminished are called doubly augmented, and this may be extended in theory to "triply" or more, but such intervals are rare in practice.

So, if that upward transposition of A flat by a sixth is to be a doubly diminished sixth, you just need to determine two things: how many semitones is a doubly diminished sixth, and which chromatic alteration of F gives that number of semitones? In this example, the answers are 6 and F-triple-flat.

To summarize:

  1. Determine the letter name of the target pitch from the kind of the transposition interval (unison, fifth, tenth, etc.).
  2. Determine the number of semitones separating the target pitch from the original pitch by considering both the kind and quality of the transposition interval (augmented unison, perfect fifth, minor tenth, etc).
  3. Determine which chromatic alteration of the target pitch yields the correct number of semitones.
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These are steps for performing a naive transposition by an interval, preserving notation mistakes in the original and all:

Step 1: Determine the interval you want to transpose by.

Step 2: Determine the direction you want to transpose by.

Step 3: Determine the note name (e.g. the "F" in "F♯") of the resulting note by adding the interval you want to the note you want. Feel free to ignore accidentals at this point. For example, at this point in transposing the note B♭4 down by an augmented second, we'd be looking at an A.

Step 4: Determine the accidental of the resulting note by combining the accidental on the original note with the quality (e.g. major, doubly diminished) and direction of the interval to transpose by. For example, when transposing the note B♭4 down an augmented second, now we calculate that the resulting note is A double flat instead of an A.

Step 5: Determine the octave number of the resulting note by combining the octave number of the original note with the size and direction of the interval. Note that interval sizes of an octave or more always change the octave number, while interval sizes smaller than that may not (or never do for unisons), and the octave number changes by 1 at the B-to-C transition. For example, when transposing the note B♭4 down an augmented ninth, now we determine that the resulting note is A double flat 3.


Alternately, you can change Steps 3 and 4 to these:

Step 3: Determine how many semitones you need to transpose by. Use the interval and direction to calculate this.

Step 4: Shift the original note up/down by that many semitones.

Step 5: Look at the quality-free size of the interval to transpose by (e.g. the "second" in "augmented second") and spell the resulting note with a note name that size away from the original note. For example, we'd spell the note an augmented second down from B♭4 "A double flat" and the note a minor third down from B♭4 "G".

The old Step 5 is now the new Step 6 (both are for determining octave number).

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  • Step 3 - aug.2nd? Or dim 2nd?
    – Tim
    Mar 27 at 14:34
  • @Tim - The first Step 3 intentionally uses an A for an augmented 2nd. The double flat is added in the first Step 4.
    – Dekkadeci
    Mar 27 at 15:03
  • (Step 4, the first) I've tried the first approach actually, before submitting the answer! :) There's a flaw though. Imagine transposing E up a major third. We should get G#. Original alteration = 0 (♮), direction = + (up), interval accidental = 0 (not augmented, not diminished). The resulting alteration is 0 + 0 = 0, instead of 1. Mar 27 at 18:51
  • (Step 5, the second) I've tried that algorithm as well! Wow, we really think alike 🙌 I assume you mean picking an enharmonic equivalent? The problem here is not as obvious, and it is the ambiguity of the resulting note's octave. I can re-spell A♭♭4 as both G♮4 and G♭♭♭♭♭♭♭♭♭♭♭♭5 (twelve flats). Yes, G♮4 is obviously a better fit, nobody sanely uses twelve accidentals, but I don't impose alteration limits in other algorithms, and the fact that both G's are a valid answer makes the algorithm probabilistic, rather than deterministic. Mar 27 at 19:10
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    @DimaParzhitsky - Regarding Step 5-2, the note's octave paired with number of accidentals is unambiguous when taking the interval name into account, and Step 6-2 should make this certain. I'm only using a G-12 flats-5 as an answer when I'm told to transpose C6 down a 12-fold diminished 5th (or something like that).
    – Dekkadeci
    Mar 27 at 19:48
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To supplement to other answers, it should be made explicit why there isn't a strictly algorithmic way to calculate transpositions in the manner described.

There are two problems:

  1. The note alterations don't adhere to a strict pattern, and
  2. The interval naming conventions also don't adhere to a strict pattern.

The problem with note alterations

Some notes have two names: one involving a ♯ and one involving a ♭ — G♯/A♭, for example. But other notes only have one of these. F can be E♯, but there's not an equivalent (single) ♭ designation. We could call it [G♭]♭, but that would presume that ♭ in G♭ is not an alteration of G, but rather part of the letter designation.

If letter names are treated in a "pure" way, then we can't distinguish between flat and double-flat notes except by looking them up. That is, note alterations are context dependent.

The problem with intervals

This is a variation on the same problem with note-naming. To see the problem, consider augmented intervals. We think of "augmented" as meaning "one half step bigger". However, this implicitly takes Major and Perfect intervals as normative, which creates a problem as diminished intervals. We tend to think of those as "one half step smaller", which works for Minor and Perfect intervals, but it's two half-steps smaller than a Major interval.

Put another way, there is only one normative version of each perfect interval: unison/octave, fourth, fifth. But all other intervals come in two versions, major and minor.

The upshot is that there are two different meanings for augmented and diminished kinds of intervals, which, like note names, are context dependent.

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Transposing a note is an academic action and to all intents and purposes, it doesn't really matter what you call the resulting note. C transposed up by 3 semitones gives E♭ - or D♯!

However, when a piece is transposed, it will generally be in a key, and transposed to another key. There's the rub. Start with the original, and each note will have a relationship with each other note - within that key. So when a piece is transposed, it must be that in its new setting, those note intervals must remain the same.

A quick example. Key Am. Two consecutive notes, A and C. Transpose into key C minor. Both go up by three semitones - BUT that interval needs to be a minor third. So, the A changes up m3 to C, and the C changes up to E♭. Call that E♭ D♯, and although you've raised it by 3 semitones, you have NOT raised it by m3.

I just hope I caught the flavour of your question...

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    Well, it is fairly easy to transpose a note by a couple of semitones: a) define a note as a number of semitones from some reference note, b) do the math. The point of my question is to figure out how the "name" of the note changes. Given your example, I'm interested in the "[A4] changes up m3 to [C5]" and "C changes up to E♭" parts. Mar 27 at 18:28
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    Also, the name of the note doesn't matter when played (as a sound, I mean), but it does matter when shown on sheet music. Mar 27 at 19:15
  • @DimaParzhitsky indeed, transposing C up by three semitones may yield E flat or D sharp, but in the context of your question these are two different upward transpositions: one by a minor third and one by an augmented second. Tim: it seems the whole point of the question is to respect spelling differences in the face of enharmonic equivalence. The first paragraph of this answer seems to miss that point.
    – phoog
    Mar 27 at 19:29
  • @phoog - my point is that transposing a random note is not the same as transposing from one key to another. The former has two answers, the latter only one.
    – Tim
    Mar 28 at 7:58

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