How could you architecturally encourage singing in the shower? Bathroom resonance sounds great, but it’s not always in tune, and often there aren’t enough resonances to sing any but the simplest tunes in resonance.
So, perhaps you could provide carefully tuned resonator cavities that resonate pleasingly at in-tune frequencies; for example, concrete pipes of different lengths, sufficiently isolated from the room air as to have a reasonable Q, but sufficiently coupled to it that they can pick up a note in a reasonable period of time; something like Q = 10 or Q = 20 (half-power points a half-step apart) should be adequate.
Helmholtz resonators like the ocarina are another, more scalable possibility, but they only resonate at a single frequency. Not only does this deprive you of overtones, it also means you need a separate set of resonators for each octave.
The presence of such resonators would also enable you to play a tune on them without singing, just by energizing the resonators, for example by hitting them with your hands, by hitting them with Blue-Man-Group floppy paddles, by banging them with hammers, or by setting firecrackers off in them.
In some sense the simplest musical scale in common use is the Pythagorean major pentatonic scale, consisting of these intervals from the tonic in each octave (assuming the tonic is C):
(The Greek names are from Plato’s Republic and Timaeus.)
In some other tunings, 27:16 for the major sixth is replaced by 5:3, but the commonly-used equal temperament is only about six cents away from the Pythagorean 27:16, but 16 cents away from 27:16.
I’m not sure exactly how you should combine these Pythagorean intervals with ISO 16 A440 concert pitch, but one way to do it would be to use the equal-temperament A440 frequency for C as the tonic for each octave. So to get a C, you take 2-2/12 of 440 Hz (since C is two semitones above A) and get 392.00 Hz. Then you can go down by octaves from there: 196.00 Hz, 97.999 Hz, 48.999 Hz. That’s probably deep enough for singing, since the tubes will also resonate at harmonics; so your first octave is 48.999 Hz, 55.124 Hz, 65.333 Hz, 73.499 Hz, 82.687 Hz.
(440/2**(2/12)/2/2/2*numpy.array([1/1, 9/8, 4/3, 3/2, 27/16])).round(3)
I could be wrong here, but I think that for a pentatonic rather than diatonic or chromatic scale, the advantages of equal temperament do not really come into play, and for singing in the shower, the advantages of just intonation may be more important. But in that case it might be better to use 5:3 rather than 27:16 for the major sixth; the medieval tradition of using 27:16 for this interval led to theorists considering it unusably dissonant, and of course Pythagoras himself was tuning tetrachords, which is why there’s no Greek name for it above.
Nodes occur at fixed ends and anti-nodes at open ends. If fixed at only one end, only odd-numbered harmonics are available.
Having a node at one end of a tube and an antinode at the other end
means we only need a half-wavelength, which is nice, because the speed
of sound in sea-level, room-temperature air is about 343 m/s, though
it can increase by up to 0.6% with humidity, and vary considerably
with temperature (definitions.units says 331.46 m/s in dry air at
STP). So a whole wavelength at 48.89 Hz is 7 m! (And that’s why a
tuba has so many curls.)
A half-wavelength one-end-open tube at each of these frequencies would be 3.500 m, 3.111 m, 2.625 m, 2.333 m, and 2.074 m. But those tubes won’t resonate at an octave, since it isn’t an odd harmonic, so you need five more for the next octave, which is the one people most commonly talk in: 1.750 m, 1.556 m, 1.313 m, 1.167 m, 1.037 m. Alternatively, you could have another set of tubes of the same length, but open at both ends to raise the pitch by an octave — curved around to open again into the bathroom, of course.
Wikipedia says the speed of sound in dry air around room temperature is 331.3 m/s + 0.606 θ m/s, where θ is the temperature in °C, or more accurately 331.3 m/s √(1 + θ/273.15). This is larger than the variation with humidity, which will also be a consideration when singing in the shower. Long, narrow concrete pipes will tend to slow down variation in both temperature and humidity.
So it might be best to use an average temperature of, say, 20°, or
whatever is likely to be the average temperature of the house
containing the bathroom; using the 331.46 number from
definitions.units we get 343.38 m/s, with numbers ranging from
340.43 m/s at 15° to 349.19 m/s at 30°, about 44 cents (0.44
half-steps) of tuning variation.
331.46 * (1 + numpy.array([0, 15, 20, 25, 30])/273.15)**.5
Closing the mouths of the tubes a bit, so the aperture into the room is smaller than the body of the tube, would reduce the variability of temperature and humidity inside the tube by impeding air circulation. However, because of the wavelength-dependent participation of air just outside the mouth of the tube in the resonation process, this will also exacerbate the detuning of the higher harmonics. Perhaps a better solution is to flare the mouths of the tubes like brass instruments to counteract this detuning effect, then cover them with something fairly acoustically transparent like aluminum foil or waxed paper.
However, you aren’t going to notice the temperature-driven detuning of the pipes unless their Q is higher than about 40, so maybe you could just not worry about it.