[rsaavedra]

After some other threads on the audible range, % not played back by certain speakers, % of audible range not heard because of hearing loss, and the issue of computing % of the audible range logarithmically instead of linearly, I thought it would be informative to elaborate a bit more on %'s of the audible range using the piano as an example.

What % of the audible range does the piano play?

First a bit on info on the audible range. Usually it is said to consist of all frequencies from 20 Hz to 20000 Hz. To fully include 10 octaves, the range is usually enlarged to reach up to 20480 Hz. The octaves and names usually asigned to them are the following:

(Octave) Frequencies (Range name)
(0) 20 - 40 Hz (Low Bass)
(1) 40 - 80 Hz (Low Bass)
(2) 80 - 160 Hz (Upper Bass)
(3) 160 - 320 Hz (Upper Bass)
(4) 320 - 640 Hz (Midrange)
(5) 640 - 1280 Hz (Midrange)
(6) 1280 - 2560 Hz (Midrange)
(7) 2560 - 5120 Hz (Upper Midrange)
(8) 5120 - 10240 Hz (Treble)
(9) 10240 - 20480 Hz (Treble)

As you can see, each octave goes from a specific frequency to double that frequency, and starting from 20 Hz, the last octave will reach up to 20480Hz. Notice the octaves numbered from 0 to 9 (not 1 to 10).

The whole range starts at 20 Hz because frequencies below 20 Hz are infrasound, e.g. below the audible range. Frequencies above 20KHz are ultrasound, e.g. beyond the audible range (though some young humans with healthy hearing reportedly can hear up to 22kHz)


What frequencies are played by a piano? Well, here are some useful links on that:

http://hyperphysics.phy-astr.gsu.edu...ic/pianof.html
http://www.physlink.com/Education/AskExperts/ae165.cfm

Using excel to compute the geometric series appropriately, I created a table of all the frequencies associated to each key in a grand piano:

Key/Octave/Note/Frequency in Hz
--------------------------
1/ 0/A0/27.50
2/ 0/A#0/29.14
3/ 0/B0/30.87
--------------------------
4/ 1/C1/32.70
5/ 1/C#1/34.65
6/ 1/D1/36.71
7/ 1/D#1/38.89
8/ 1/E1/41.20
9/ 1/F1/43.65
10/ 1/F#1/46.25
11/ 1/G1/49.00
12/ 1/G#1/51.91
13/ 1/A1/55.00
14/ 1/A#1/58.27
15/ 1/B1/61.74
--------------------------
16/ 2/C2/65.41
17/ 2/C#2/69.30
18/ 2/D2/73.42
19/ 2/D#2/77.78
20/ 2/E2/82.41
21/ 2/F2/87.31
22/ 2/F#2/92.50
23/ 2/G2/98.00
24/ 2/G#2/103.83
25/ 2/A2/110.00
26/ 2/A#2/116.54
27/ 2/B2/123.47
--------------------------
28/ 3/C3/130.81
29/ 3/C#3/138.59
30/ 3/D3/146.83
31/ 3/D#3/155.56
32/ 3/E3/164.81
33/ 3/F3/174.61
34/ 3/F#3/185.00
35/ 3/G3/196.00
36/ 3/G#3/207.65
37/ 3/A3/220.00
38/ 3/A#3/233.08
39/ 3/B3/246.94
--------------------------
40/ 4/C4/261.63 <---- C4 == The famous "Middle C"
41/ 4/C#4/277.18
42/ 4/D4/293.66
43/ 4/D#4/311.13
44/ 4/E4/329.63
45/ 4/F4/349.23
46/ 4/F#4/369.99
47/ 4/G4/392.00
48/ 4/G#4/415.30
49/ 4/A4/440.00 <-- A4 == the famous tuning tone of 440 Hz
50/ 4/A#4/466.16
51/ 4/B4/493.88
--------------------------
52/ 5/C5/523.25
53/ 5/C#5/554.37
54/ 5/D5/587.33
55/ 5/D#5/622.25
56/ 5/E5/659.26
57/ 5/F5/698.46
58/ 5/F#5/739.99
59/ 5/G5/783.99
60/ 5/G#5/830.61
61/ 5/A5/880.00
62/ 5/A#5/932.33
63/ 5/B5/987.77
--------------------------
64/ 6/C6/1046.50
65/ 6/C#6/1108.73
66/ 6/D6/1174.66
67/ 6/D#6/1244.51
68/ 6/E6/1318.51
69/ 6/F6/1396.91
70/ 6/F#6/1479.98
71/ 6/G6/1567.98
72/ 6/G#6/1661.22
73/ 6/A6/1760.00
74/ 6/A#6/1864.66
75/ 6/B6/1975.53
--------------------------
76/ 7/C7/2093.00
77/ 7/C#7/2217.46
78/ 7/D7/2349.32
79/ 7/D#7/2489.02
80/ 7/E7/2637.02
81/ 7/F7/2793.83
82/ 7/F#7/2959.96
83/ 7/G7/3135.96
84/ 7/G#7/3322.44
85/ 7/A7/3520.00
86/ 7/A#7/3729.31
87/ 7/B7/3951.07
--------------------------
88/ 8/C8/4186.01


(With respect to the middle C, see this).


So the piano keys reproduce 88 specific frequencies, ranging from 27.50 Hz, up to 4186.01 Hz. What % of the audible range is the range of all frequencies from 27.50 to 4186.01?

To compute this percentage we have to use logarithms. The reason for doing this is that we don't perceive sound linearly but logarithmically. Here's an attempt to bring explanation to that: we can easily differentiate 20 Hz from 30 Hz (a couple of frequencies that are just 10 Hz apart in linear terms), but we can't differentiate so easily between 10 kHz and 10.01 Khz (another couple of frequencies that are just 10 Hz apart in linear terms). Putting it loosely, what matters is not really the linear separation between frequencies, but their ratio.

If we did things incorrectly (e.g. linearly), we would think the range of a gran piano covers just 20.3% of the audible spectrum :
(4186.01 - 27.50) / (20480 - 20) = 20.33 %

But doing things properly (e.g. logarithmically), we can compute the real answer to this thread's title:
(ln(4186.01) - ln(27.50)) / (ln(20480) - ln(20)) = 72.5 %


[size=small]So a grand piano's range is 72.5% of the audible range.[/size]


PS. The easiest way to think about these %'s of the audible range is to ask the question in terms of # of octaves out of 10. The piano plays 7+ octaves. 7+ out of 10 would have straightforwardly provided a good estimate of slightly above 70%.

 

[Wodgy]

Interesting. You're only counting the notes themselves, though. When you add in a reasonable number of the harmonics that are produced at the same time as the notes themselves, the piano produces nearly the entire audible spectrum.

 

[rsaavedra]

Quote:

Originally Posted by Wodgy Interesting. You're only counting the notes themselves, though. When you add in a reasonable number of the harmonics that are produced at the same time as the notes themselves, the piano produces nearly the entire audible spectrum.

Good point, yes all the above just taking into account the fundamental frequencies associated to the keys, not resulting harmonics.

 

[daycart1]

Useful post; thanks!

It is interesting that when you hit key #1 on a grand piano, it doesn't sound THAT close to the bottom limit of human hearing.

 

[ooheadsoo]

That's because it wasn't a very good grand piano

 

[adhoc]

Quote:

Originally Posted by rsaavedra 49/ 4/A4/440.00 <-- A4 == the famous tuning tone of 440 Hz

afaik, some orchestra concertmasters now purposely use 442hz or 445hz to tune their instruments. apparently, the brighter sound is more fashionable/appealing now.

 

[CSMR]

Quote:

Originally Posted by adhoc afaik, some orchestra concertmasters now purposely use 442hz or 445hz to tune their instruments. apparently, the brighter sound is more fashionable/appealing now.

I don't think pitch has increased recently. Soviet orchestras apparently used the highest As. (Now Baroque pitch, that's in fashion!). America is generally on 440, Europe a bit higher, I think. It's a matter of preference.

 

[MaltaCross]

Hi rsaavedra ,
 
I have been looking at the article above, and I think that your reasoning is not quite accurate! Let me explain.
 
Your reasoning is based on the assumption that "we don't perceive sound linearly but logarithmically". But please note that we perceive sound logarithmically only in terms of LOUDNESS. The perception of frequency is something else, and is perceived almost linearly. (Besides, you did not explain why you used natural logarithms to the base of the natural exponent [e] [size=medium]— [/size]the decibel scale for instance is calculated with logarithms to base 10!)
 
Let me just offer an alternative calculation please. Elaine Nicpon Marieb and Katja Hoehn (2007), in their medical textbook Human anatomy & physiology, state in page 589 that:
 
Our ears are most sensitive to frequencies between 1500 and 4000 Hz and, in that range, we can distinguish frequencies differing by only 2-3 Hz. We perceive different sound frequencies as differences in pitch: the higher the frequency, the higher the pitch.
 

Let us therefore stick with the range in which we are most capable of distinguishing pitch, i.e. between 1500 and 4000 Hz. Let us also assume a generous 4 Hz difference between each frequency that we can uniquely distinguish.
 
This gives us (4000 - 1500) / 4   =  2500 / 4   =  625 different distinguishable frequencies.
 
However, looking at a table of musical frequencies (such as http://www.phy.mtu.edu/~suits/notefreqs.html), we find that between 1500 and 4000 Hz there are only 17 semitones! The first is   G6   at   1567.98 Hz, and the last is   B7   at   3951.07 Hz.
 
That leaves 625 - 17 = 608 distinguishable frequencies unaccounted for!  That is, the piano's range accounts for a mere 2.72% of the range that we can hear between 1500 and 4000 Hz !
 
Some commentators also made the point that the piano produces harmonics, that allegedly increase the piano's range. This is also incorrect! Harmonics are integer multiples of the fundamental frequencies, and as such each harmonic coincides with an already existant fundamental frequency. Thus harmonics do not actually increase the piano's range of frequencies at all.
 

 

[MaltaCross]

Further to the above, I just would like to add the human sensitivity to frequencies is described by the so-called Fletcher-Munson curves. These curves are not precisely linear, but they are definitely not exponential.