There could be a few reasons, but one of the key reasons is that the bass
sings so low. The radiating area of the singer is the opening of the mouth. Thus
the knee frequency of the singer's sound radiation is determined by the size of
the open mouth. Since the mouth of a bass, and of a tenor or of a soprano are
pretty comparable in wide open size , the knee frequencies for all of these will
be about the same. However the bass sings at least an octave lower than a tenor
and at least two or three octaves lower than a soprano. In all cases twice the
diameter of the mouth is about 8cm. wich would give a knee frequency of
340m/sec/.08 m= 4200Hz which is well above the note they sing ( a soprano can
sing up to about A5 or 880Hz, which is well below this knee frequency.)
If we imagine the bass singing three octaves below the soprano, this would
correspond to 3x6= 18dB less efficiency of sound radiated by the bass than the
soprano.
Another possible reason is that the ear is most sensitive in teh frequency
range of 1000-4000 Hz. The soprano's voice has a frequency that is closer to
this regime of maximum sensitivity, and thus would also seem louder because of
this. Both of these effects mean that a soprano in general is far easier to hear
than is a bass. A tenor lies between the two in terms of frequency and his
loudness would also lie between the two.
b) Why is an (unpowered) megaphone useful (consider the size of the
radiating area)?
While the megaphone simply spreads out the power of sound coming out of the mouth over a wider area. Thus, although the area is larger, the vibrtional velocity of the air there is less, the two just compensating. However, the megaphone by its larger diameter also has an increased efficiency of radiation. The knee frequency is brought down. If the outlet of the megaphone has a diameter of say 32 cm, rather than teh four cm of a wide open mouth, then the knee frequency will be about 8 times loweri (that is three octaves lower) . Thus the voice frequencies will be 8 times (3 octaves) nearer the knee frequency of the output to the megaphone than to the knee frequency of the mouth alone. This means that the sound output of the megaphone would be expected to be about 3(octaves)x6(db/octabe)= 18dB louder coming from the megaphone than from the open mouth.
Why do singers tend to sing with their mouths wide open? (It is not for the looks!)
Again it is all in the efficiency of radiation of the sound coming out of the mouth. The wider the mouth is open, the lower the knee frequency, and since the frequency of singer's singling is lower than the knee frequency of the widest mouth, this lowered knee frequency will mean a higher efficiency of radiation and a louder sound heard by the listener.
A Pythagorian whole tone has a frequency ration of 9/8. Two of them will
have a ration of 9/8x9/8= 81/64. A Just major third with a frequency ratio of
5/4 which is 80/64. Thus a Pythagorian third will be 81/80 times as high a
frequency as a just ajor third. This is about 1/5 of a semitone. a frequency
change which is audible to anyone.
Three major thirds (four semitones) could be said to be an octave (twelve
semitones). How mistuned would that octave
be if each of those major thirds were just major thirds?
Each major third increases the frequency by a factor of 5/4. Three of them would thus increase teh frequency by (5/4)x(5/4)x(5/4)= 125/64. A true octave increases the frequency by a factor of 2= 128/64. Thus a true octave is 128/125 higher in frequency than a three just major thirds. This is 1.024 which is about half of semitone. Ie, three just major thirds is almost half a semitone flatter than a true octave.
The full range of sounds which one wants from a speaker spans almost 10
octaves. For the lowest sounds one needs a large speaker, in order that the
efficiency can be high for those low tones, and that the main resonance
frequency of the speaker can be low. However, such a large area will have other
resonances which will distort the sounds if that speaker tries to play
frequencies around those frequencies. Thus, one wants to use the large speaker
at low frequencies where its efficiency and low main resonance helps, but stop
using it at higher frequencies. This arguement also is valid for the next
speaker which covers the next range of frequencies. Also, one wants one speaker
to cover teh center range of frequencies which contains the main frequencies
used by voices and most instruments ( eg 300Hz-2KHz) and one would rather not
have the frequencies handled by two different speakers. Thus one typically has a
bass woofer handling the lowest frequencies ( 30Hz- 300Hz) a midrange handling
the middle frequencies (300Hz- 3000Kz) and a small tweeter handling the highest
frequencies (3KHz-20KHz).
Of course these three speakers are expensive, and thus it is the more
expensive loudspeakers that will do this. Some of the cheaper good speakers will
use two (eg 50Hz-1KHz and 1KHz-10KHz). Usually this means that both the bass
speaker and the tweeter will have unwanted resonances in the cones, and will
have a rougher frequency response.
b)
What would happen to the sound straight ahead of a speaker if someone
miss-wired the midrange and the woofer to be exactly 180 out of phase with each
other? Give a sketch of what you would expect for the sound output of the
speaker as a function of frequency?
Since each speaker handles a separate range of frequencies, you would not expect this miswiring to do much over most of the frequency range. However, at the so called crossover frequencies, both speakers will carry the sound with approximately equal amplitude. In this case the two speakers will put out sound, but the sounds will be out of phase. Ie, as one speaker is pushing the air outward, the other is moving inward. These two sounds will cancel each other. Thus at these crossover frequencies one would expect very weak response. The spectral response would be the same at all frequencies except the crossover where you would get a huge notch in the frequency response.
The sound output goes as the area. Thus the 10cm speaker would have twice the output, just from the size. The larger speaker would have a knee frequency which is half that of the smaller. Thus at 440Hz, which is well below the knee frequency ( 340M/sec/(2x .1m)= 1700Hz for the larger speaker), it would be an octave closer to the larger speaker's knee frequency and thus would have a 6dB higher efficiency. The factor of two from the area correspondes to another 3dB. Thus the larger speaker would be 9dB louder, which is a factor of 8.
Again it is a matter of the knee frequency. When the earphone are on your ear, the sound motion of the speaker is directly transmitted to the ear. The efficiency is unity at all frequencies. When it is on the table however the the efficiency falls at 6dB per octave below the knee frequency, since headphone speakers tend to be only a few cm in diameter, teh knee frequecy is very high-about 5-10KHz. This means that the lower frequencies come out very much quieter. All you hear are the high frequencies.
In just tuning, the major third is a ratio of 5/4 while the perfect fifth is 3/2 Thus the ratio between the two will be (3/2)/(5/4) = 6/5= 1.2. However, the equal tempered minor third is (1 .059)(1.059)(1.059)=1.189. The just minor third is thus 1.009 higher than an equal tempered minor third ( this is about 1/6 of a semitone sharper).
The frequency ratio of a perfect fourth is 4/3. This means that each fourth harmonic of the lower note will have the same frequency as each third harmonic of the higher frequency. Ie, the fourth harmonic of the lower should be the same frequency as the third of the higher, and if the two a slightly mistuned these two harmonics will beat. Similarlty the eighth of the lower will be almost the same as the sixth of the higher, and will beat with it.