How does audacity figure a transposition? I’ve tried and tested the transposition tools and found that it handles one tone alright. But a series of tones will become distorted at about the fourth or fifth in the scale. The transposer seems to use addition, but that can not be a true transposition because A400 transposed by an addition of 4 up to A444 becomes A884 at the octave, when it should be A888. I haven’t completely checked this out, but it sounds wrong to me.

Does audacity allow for a retuning up to 444hz so I can use it with my 444hz melodica?

I know that much. If you are a musician who has studied the physics of sound, you might understand the question better. The transposing tools work by addition. Sure, I can change the pitch of one note. But I want to change the pitch of a series of tones - a complete melody. If I transpose a complete series of one octave to another octave, it does not seem to come out correctly. And I want to change the standard that the transposing tools work on from A440 to A444 - so that it affects all transposition. Audacity is set for A440. I want to tune it to my melodica. I can also record the melodica and use the pitch control to bring it down to standard 440 - but as I said - audacity can not transpose a series of tones correctly simply by the function of addition. A harmonic series is a combination of addition and multiplication.

One of Audacity’s two pitch-shift tools [#1, #2] can make that 1% change.
However I’m not sure it will be worth the effort (or the inevitable digital-artifacts) :
the resolution of human-hearing is about 4Hz …

Only noticeable via the beats effect.
If the pitch of a recording was 1% out, listeners would not be able to tell without a reference-tone to give the beats-effect.

Without a reference, most people would not know if an instrument was tuned within half an octave of “standard” tuning.
On the other hand, if I’m playing an instrument along with a backing track, then a 1 cent difference in tuning sounds clearly “out of tune” to me. So really, it depends what we’re talking about

Yes, you are correct. It can work either by multiplication or by addition. But the harmonic series is a combination of both, as page 5 of the pdf says:
“The frequencies of octaves form a geometric sequence.
The frequencies of harmonics form an arithmetic sequence.”

This is an old problem that was partially solved by the introduction of equal temperment. My question includes an inquiry into how the guts of audacity works when presented with these calculations. Note that I stated the question clearly in the OP. An addition of 4 cents to a series based upon 440 results in the octave being transposed to 484, where the proper function is a multiple (444 x 2 = 888) and not an addition. I am talking about a series of tones and not a single tone. 444 can easily be multiplied up to 888 as a single tone. I am speaking of transposing a series based upon 440 as a tonic up to 444 as the tonic.

Give me a little time to look at what everyone has said here. I am also preparing a sound file that will demonstrate the problem.

Let me save you some time:
Generate a “Square, no alias” tone at 440 Hz (A4).
View with Plot Spectrum and notice the harmonic series:
A4, E6, C#7, G7, B7, D#8, F8 …
Note that these are “Natural harmonics” (exact multiples):
440, 1320, 2200, 3080, 3960,…

Use the Change Pitch effect to transpose up by 1 semitone.
You now have:
A#4, F6, D7, G#7, C8…
Each harmonic has been transposed by 1 semitone, which is equivalent to multiplying each frequency by 2^(1/12).

Pick any frequency you like, and with Equal Temperament tuning, multiplying the frequency by 2^(1/12) is a pitch shift of 1 semitone.
Does that help?
(a handy table: http://www.phy.mtu.edu/~suits/notefreqs.html)

Change Pitch works by addition when you raise the number of Hz. It works by multiplication by raising it in percentages. Change pitch cannot use both simultaneously, unless the addition is a duplocation of the multiplication. But that does not occur in the harmonic series - the additions are separate from the multiplications because only octaves are in geometrical rato, while only harmonics are in arithmetical ratio.

Steve’s system used both, but Steve didn’t do the figuring. Audacity figured it and a quick glance at the numbers should raise a red flag. What’s the problem that I posed in the OP, and why won’t Audacity’s transposition work? The numbers are too good to be true. And if it’s too good to be true…

What kind of harmonic series did I mean in the title and OP? What kind of harmonic series is represented in these numbers?

I don’t think you’re really interested audacity novice, but for the benefit of other users, that’s wrong.

Transforming sound in a manner that changes frequencies by a specified amount (addition) is generally called “Frequency shifting”. This is distinctly different from “Pitch shifting”, which transforms the frequency by a specified ratio (multiplication). One semitone is defined as a ratio of 1:1.059463094, or to be exact, a ratio of 1:2^(1/12) https://en.wikipedia.org/wiki/Twelfth_root_of_two

@audacity novice, if you genuinely need help using Audacity, then we are happy to assist, but please note that for the benefit of all our users, trolling is not tolerated on this forum. Topic locked.