It's highly likely I'm the only one who's been bothered by this, but my 3/14 post stretching a familiar 3/4 minuet into 3.14/4 time has stayed on my mind since the notation system I devised was a bit sneaky. I knew each 3/4 bar needed 0.14 extra beats, and I found a way within Lilypond to multiply note values in a way that achieved that (at least for playback purposes). But I cheated with the time signature. I just made my own time signature sign that showed a 3.14 over a 4 and then used Lilypond's "cadenza" function, which allows the user to place barlines wherever one wants. Essentially, this means the internal calculations act as if there is no time signature, so as long as the various parts line up rhythmically, the musical output will sound right, even though the barlines aren't generated by time signature calculations.
During idle moments this week, I tried to imagine a way to create a time signature which is precisely three beats plus 0.14 beats long. At some point, the old use of 22/7 as a fraction which approximates Pi came to mind, and that helped me to think about this a different way. After flirting a little with the idea of an irrational time signature like....well, 22/7, I realized that what I really need to do was add a single septuplet note to the three quarter notes in each 3/4 bar.
It's a curious and sometimes forgotten (by me, at least) fact that our names for note values are really just ways of dividing a whole note. So a quarter note means nothing more specific than that it is 1/4 the duration value of a whole note. I finally realized that dividing four quarter notes into septuplets means that it would take 28 septuplet notes to equal one whole note, so a septuplet value could also be called a 28th note.* One doesn't see a lot of 28th notes out in the wild, though curiously the name sounds a lot like a 128th note, which is a thing. (There are 32 128th notes in a quarter note.) But that's basically just a coincidence.
For my initial "Minuet in Pi" video, I used a special notehead with a pi symbol inside to indicate values which need to be stretched. It's now clearer to me this symbol could be defined as meaning the basic note value is stretched by 1/7. (This is similar to how a dotted note value extends the duration by 1/2.) This makes the notation pretty simple, but that's because I invented a sign to smooth things out.
Anyway, I finally realized that using Lilypond's wonderfully logical design, I could create a composite time signature of 3/4 + 1/28. Since each quarter note beat contains seven 28th notes, the signature could also be written as 22/28, but that doesn't show the additive process as clearly. Of course, one could notate more or less the same thing using 22/16 time, but that wouldn't allow the use of quarter notes to show simple beat values. Using 28 as denominator allows the use of quarter notes more or less as they'd be used in 3/4 time except on the third beat which requires stretching. (I don't think there's a truly elegant way to handle the ties required for that stretched beat.)
Is this all a little more than you want to think about? That would be fair, but I have found this a good challenge to help me think more clearly about how time signatures work. The two videos below are thus more precise than what I posted last week, though also a little less elegant. The first one is probably a little more proper, though the second one is maybe a little easier to read as it preserves more of a 3/4 look. [UPDATE (3/21): See also important footnote version in 22/7 below!]
Minuet in 3/4 + 1/28 Time, version 1
Minuet in 3/4 + 1/28 Time, version 2
* If you're like me and have flirted occasionally with irrational time signatures (which basically have denominators that aren't multiples of 2), you might be tempted to think a time signature with 7 on the bottom is the way to use septuplets as a primary division; but remember that a 7 would just mean a whole note is divided into 7, so that would provide a pulse which would be notated as quarter note septuplet. This would not fix my problem of needing a nice way to show standard quarter note beats which look like the way this minuet is usually notated. I needed a septuplet division of the quarter note itself, which is how we end up with 4 x 7 = 28.
In other words, although a 22/7 time signature would look cool, it would only work if the primary beats in each bar were indicated as whole notes like below. And, whaddya know? Having added this as a "day after" footnote, I now think this is my favorite way of notating this meter!
Minuet in 22/7 Time
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