Why are there 7 discrete notes? A possibly stupid question about sound...
Michael Levin
me if this is a stupid question. Why are there 7 discrete notes? The pitch
we hear is a function of frequency, which is a continuous, not discrete
quantity. So, at first I thought it was kind of like light and colors - we
make up arbitrary names for sections of the spectrum, but in fact there are
an infinity of colors (or however many the human eye can distinguish within
the EM spectrum). So, maybe each note is just an arbitrary part of the
frequency spectrum which people have agreed to divide into 7 sections. But
one thing bothers me: why does it "wrap around"? That is, on a piano
keyboard, after you've gone A,B,C,D,E,F,G, the next one is A again, one
octave higher, but in some sense, A again. How does this work? And, why does
it wrap around like that? If there is a real sense in which the sounds after
"G" are "A" again, does this mean that there is something to the "7 notes"
beyond just convention? Is there any discreteness in the sound spectrum
which is real (a real feature of the physics as opposed to arbitrary
convention)? I am interested in special numbers which come up in various
areas of math and science. Is the "7" here real, in the sense that the value
of Pi is "real" and not simply arbitrary human convention? Probably not, but
I would appreciate an explanation of where I've gotten confused. How do the
7 notes and the octaves relate to the continuous spectrum of air wave
frequency? Thanks in advance for any info.
--
Mike Levin
mlev...@comcast.net
Robert J. Kolker
Michael Levin wrote:
> I'm a biologist, with no background in music and limited physics, so excuse
> me if this is a stupid question. Why are there 7 discrete notes?
Hello? What about the 12 tone scale? The human ear is capable of
discerning more than seven notes in an octave. And what is sacred about
octaves?
Bob Kolker
John T Lowry
the 60's I (and a few others) started playing the guitar. There's
nothing special about our scale; I've heard the ear can fairly readily
distinguish intervals only about one-fourth the size of our current
semitones. Now an octave has real meaning in that going up an octave the
frequency precisely doubles.
An interesting fact I can up with (likely not original) is that the
circle of fifths (which means -- illiterate musicians! -- up FOUR whole
notes, 7 semitones) works because there are 12 (now) equally spaced
semitones and 7 (and 5, going down) are relatively prime to twelve. The
others (2, 3, 4, 6, 8, 9, 10) divide 12 or have common factors with 12.
HTH
John
--
John T Lowry, PhD
Flight Physics
5217 Old Spicewood Springs Rd, #312
Austin, Texas 78731
(512) 231-9391
jlow...@earthlink.net
"Michael Levin" <mlev...@comcast.net> wrote in message
news:BCCBD9DE.17384%mlev...@comcast.net...
Sam Wormley
Dirk Van de moortel
The simple explanation goes back to the Pythagoreans...
Start from C, take a quint (frequency of C times 3/2) and go to G.
This C and G sound great when played together, or one after the
other.
Now take another quint from G and go to D, then to A, then to E,
then to B, then to F, and finally back to C - You now have 4 times
the frquency of your original C.
C x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 = 17.09 C
That is slightly over 16 C, i.o.w. you now span 4 octaves.
Now collapse them all together in one octave by taking half of
the frequencies of all the ones that didn't end up in the first octave.
That is your scale.
Not quite - It would be nice if (3/2)^7 were 16.
Now, do we have some other way of producing a power of 2
by taking a power of 3/2?
Yes.... try (3/2)^12 = 129.75 =~ 128.
Does 12 ring a bell?
http://www.musemath.com/flash/contents.swf
http://www.musemath.com/flash/math.swf
hth
Dirk Vdm
Michael Levin
BItpc.1514$H_3...@newsread1.news.pas.earthlink.net, "John T Lowry"
<jlow...@earthlink.net> wrote:
> Well, I used to be a biologist (geneticist) too. But somewhere back in
> the 60's I (and a few others) started playing the guitar. There's
> nothing special about our scale; I've heard the ear can fairly readily
> distinguish intervals only about one-fourth the size of our current
> semitones. Now an octave has real meaning in that going up an octave the
> frequency precisely doubles.
Ah! That answers my question exactly. The number of notes in an octave is
indeed arbitrary, and the octaves wrap around because the ear detects
doublings of frequency as similar notes. Makes sense. Thanks!!
--
Mike Levin
mlev...@comcast.net
Don Pearce
wrote:
The notes in the scale aren't really arbitrary. They are based on
harmonic relationships that make many of the overtones coincident. In
some cultures, this results in a very limited scale, with just five
notes. Play C, D, E, G, A on a piano to get the idea. Western music
has simply extended the scale to fill in all the blanks, so to speak.
d
Franz Heymann
excuse
> me if this is a stupid question. Why are there 7 discrete notes?
In European music, there are 12 logarithmically equally spaced notes
to the octave, not 7.
In some Asian music, there are up to 24 notes to the octave.
A violin can play a complete continuum.
> The pitch
> we hear is a function of frequency, which is a continuous, not
discrete
> quantity. So, at first I thought it was kind of like light and
colors - we
> make up arbitrary names for sections of the spectrum, but in fact
there are
> an infinity of colors (or however many the human eye can distinguish
within
> the EM spectrum). So, maybe each note is just an arbitrary part of
the
> frequency spectrum which people have agreed to divide into 7
sections. But
> one thing bothers me: why does it "wrap around"? That is, on a piano
> keyboard, after you've gone A,B,C,D,E,F,G, the next one is A again,
one
> octave higher, but in some sense, A again. How does this work?
> And, why does
> it wrap around like that? If there is a real sense in which the
sounds after
> "G" are "A" again, does this mean that there is something to the "7
notes"
> beyond just convention?
THe "7 white notes" is a cultural convention. The relationship
between a note and its octave is what matters as far as your question
goes.
Take a pure sinewave note and one an octave higher. Play them
together. The second one now simply sounds the second harmonic of the
first. In the process, it loses its indentity as far as the listener
is concerned. It simply sounds as if it has modified the timbre of
the first note.
[snip]
Franz
Prai Jei
<BCCBD9DE.17384%mlev...@comcast.net> thusly:
May I refer the interested reader to the article
http://www.deniseswanson.com/stybr/stybr-mm.htm by contemporary American
composer David Stybr, which sets out the rationale behind the diatonic and
chromatic scales, simply but systematically.
--
Paul Townsend
I put it down there, and when I went back to it, there it was GONE!
Interchange the alphabetic elements to reply
Old Man
excuse
There's also sharp and flat, and "atonal" music has been around
for centuries. There's no accounting for harmonious taste.
[Old Man]
> ...
> Mike Levin
John O'Flaherty
wrote:
>I'm a biologist, with no background in music and limited physics, so excuse
There's an essay on the topic by Isaac Asimov. This is a paste from
www.asimovonline.com -
Music to My Ears
Subject: musical scale
First Published In: Oct-67, The Magazine of Fantasy and Science
Fiction
Collection(s):
* 1968 Science, Numbers, and I
It explains the matter very nicely.
--
john
Franz Heymann
"Don Pearce" <comp...@nonsense.com> wrote in message
news:g21da097bh3r85p9k...@4ax.com...
That is, of course, equivalent to playing on the black notes only,
starting at F#
Franz
Patrick Powers
> me if this is a stupid question. Why are there 7 discrete notes? The pitch
> we hear is a function of frequency, which is a continuous, not discrete
> quantity. So, at first I thought it was kind of like light and colors - we
> make up arbitrary names for sections of the spectrum, but in fact there are
> an infinity of colors (or however many the human eye can distinguish within
> the EM spectrum). So, maybe each note is just an arbitrary part of the
> frequency spectrum which people have agreed to divide into 7 sections. But
> one thing bothers me: why does it "wrap around"? That is, on a piano
> keyboard, after you've gone A,B,C,D,E,F,G, the next one is A again, one
> octave higher, but in some sense, A again. How does this work? And, why does
> it wrap around like that?
Each higher A is a vibration twice as fast as the next lower A. Its
called an octave.
>If there is a real sense in which the sounds after
> "G" are "A" again, does this mean that there is something to the "7 notes"
> beyond just convention? Is there any discreteness in the sound spectrum
> which is real (a real feature of the physics as opposed to arbitrary
> convention)? I am interested in special numbers which come up in various
> areas of math and science. Is the "7" here real, in the sense that the value
> of Pi is "real" and not simply arbitrary human convention? Probably not, but
> I would appreciate an explanation of where I've gotten confused. How do the
> 7 notes and the octaves relate to the continuous spectrum of air wave
> frequency? Thanks in advance for any info.
Most humans don't use the 7 note scale. Five might be the most
popular. Jazz uses (at least) 12 notes per octave. Ancient Thailand
used 33 or so.
Dirk Van de moortel
"Patrick Powers" <frisbie...@yahoo.com> wrote in message news:9511688f.04051...@posting.google.com...
But so terribly boooooooring.
Dirk Vdm
Dirk Van de moortel
"Angelo Campanella" <a.camp...@att.net> wrote in message news:j%Mpc.69963$Ut1.1...@bgtnsc05-news.ops.worldnet.att.net...
> To help quantify this thread, can someone make a short list of note
> frequencies in Hz A-G, do re mi....do, sharp and flat perturbations,
> etc.... An octave or so around middle C, or A would really help; state
> the pitch criterion ("piano"?, concert pitch, or what have you)...
Try this one:
http://www.phy.mtu.edu/~suits/Physicsofmusic.html
and specially
http://www.phy.mtu.edu/~suits/scales.html
Dirk Vdm
Richard Henry
"Angelo Campanella" <a.camp...@att.net> wrote in message
news:j%Mpc.69963$Ut1.1...@bgtnsc05-news.ops.worldnet.att.net...
> To help quantify this thread, can someone make a short list of note
> frequencies in Hz A-G, do re mi....do, sharp and flat perturbations,
> etc.... An octave or so around middle C, or A would really help; state
> the pitch criterion ("piano"?, concert pitch, or what have you)...
They are all over the web. For instance:
http://www.mjorch.com/hertz.html
Start with the A above middle C as 440 Hz. exactly. Compute from there.
Franz Heymann
"Angelo Campanella" <a.camp...@att.net> wrote in message
news:j%Mpc.69963$Ut1.1...@bgtnsc05-news.ops.worldnet.att.net...
> To help quantify this thread, can someone make a short list of note
> frequencies in Hz A-G, do re mi....do, sharp and flat perturbations,
> etc.... An octave or so around middle C, or A would really help;
state
> the pitch criterion ("piano"?, concert pitch, or what have you)...
A just above middle C is 440 Hz
Any other note n semitones above A then has a frequency 440 * 2^(n/12)
Franz
Herman Family
Michael
"Franz Heymann" <notfranz...@btopenworld.com> wrote in message
news:c88jjj$an$1...@sparta.btinternet.com...
John T Lowry
I recall reading, in Helmholtz's Sensations of Tone that, when he wrote
it (couple of centuries ago) scales of European bands differed in their
definition of say middle C by more than a note!
--
John T Lowry
5217 Old Spicewood Springs Rd, #312
Austin, Texas 78731
(512) 231-9391
jlow...@earthlink.net
"Angelo Campanella" <a.camp...@att.net> wrote in message
news:j%Mpc.69963$Ut1.1...@bgtnsc05-news.ops.worldnet.att.net...
> To help quantify this thread, can someone make a short list of note
> frequencies in Hz A-G, do re mi....do, sharp and flat perturbations,
> etc.... An octave or so around middle C, or A would really help; state
> the pitch criterion ("piano"?, concert pitch, or what have you)...
>
> Angelo Campanella
>
John T Lowry
Hertz! Sorry.
neurothing
But the answer is there aren't 7 discrete notes. The western tuning system
has 12 chromatic steps, other tuning systems range from 5 to several
hundred. However, there is substantial literature on why certain degress of
pitch discrimination are universal - it has to do with the feature of the
mammalian auditory system called the critical band. The critical band is a
filter function of the ear which describes how close in frequency two tones
are before they can no longer be resolved into discrete pitches. In order
to avoid a very lengthy message here, I would recommend you look at
Fletcher's experiment, in which listeners were asked to detect a pure tone
embedded in background noise - the tone was kept constant in frequency but
varied in amplitude, and the bandwidt of th enoise was varied. This allowed
the generation of a working definition of critical band as the bandwidth of
noise that alone contributes to the masking of a pure tone embedded within
that noise. In other words, the stretches of the basilar membrane in the
inner ear act as a series of bandpass filters - if two sounds are too close
in pitch or frequency, they will begin to sound "rough" (think of D and D#)
because they are imposing on each others filter edges. Thios also explains
why some intervals are universal in all tunings - they are separated by more
than a minimal fraction of the critical band, and often in complex sounds,
their harmonics or partials will overlap.
Hope this helps.
Seth Horowitz, PhD
CTO, NeuroPop
As a function of the width of the critical band
Gulliver
> I'm a biologist, with no background in music and limited physics, so excuse
An 'octive' on a piano consists of 7 (not eight - count 'em ABCDEFG(7)
A (the first note in the next 'octive' makes eight)) white keys and 5
black keys (12 notes total).
A note that is one 'octive' higher in pitch is exactly twice the
frequency of the same note one octive lower. That might be why the
notes might seem to sound similar in some ways when they may be an
octive higher or lower than a specific note.
I am not sure if notes are officially set at base 12 logarithmic
increments of frequency or not, or wheather they are set at specific
frequencies based upon some arbitrary criteria about 'sounding best'
by some official music organization.
This scale seems to have evolved out of the middle ages, however the
'half-step' (base 24) and 'quarter-step' (base 48) is referred to from
classical times in ancient Greece and Rome, and other scales such as
the 'pentatonic' are known from other cultures.
There is nothing that might preclude, however, that a scale might not
exist or be used in other increments besides base 12. It can be done
on any nonexclusively tonically stepped instruments. (Like trombones,
string instruments without clefs(sp.?), synthesizers, etc.)
crynwulf
You forgot about the "H" note. (Yes, there was an H but it got replaced by
G#/Aflat on the tempered scale). The notes were based on ratios of small
integers from Greek philosophy. There were notes for going from C1 ... C2
and another set going from C2 ... C1. The tempered scale replsce these with
one set that doesn't quiet match either. For example, it replaces
B1(rising) and B1(lowering) with a sort of not too bad sounding median tone
called B. I've been told that American Indians find piano music terrible
because they hear the hammers and all the notes are a bit off. On the other
hand, their music is so different from ours, we can't even hear it.
> A note that is one 'octive' higher in pitch is exactly twice the
> frequency of the same note one octive lower. That might be why the
> notes might seem to sound similar in some ways when they may be an
> octive higher or lower than a specific note.
>
> I am not sure if notes are officially set at base 12 logarithmic
> increments of frequency or not, or wheather they are set at specific
> frequencies based upon some arbitrary criteria about 'sounding best'
> by some official music organization.
>
> This scale seems to have evolved out of the middle ages, however the
> 'half-step' (base 24) and 'quarter-step' (base 48) is referred to from
> classical times in ancient Greece and Rome, and other scales such as
> the 'pentatonic' are known from other cultures.
>
> There is nothing that might preclude, however, that a scale might not
> exist or be used in other increments besides base 12. It can be done
> on any nonexclusively tonically stepped instruments. (Like trombones,
> string instruments without clefs(sp.?), synthesizers, etc.)
--
Russ Lyttle
Not Powered by ActiveX
http://home.earthlink.net/~lyttlec/philosophy/logos.html
TimR
That assumption of an analogy to light, with a continuous spectrum, is
flawed. Real musical notes are never one frequency, they are a set of
frequencies related sufficiently that the brain assigns one pitch.
That A 440 you mentioned can be one frequency on a synthesizer, but
any real instrument plays the 440 and a series of higher notes,
probably at least five. (Also, though light is continuous across the
spectrum, color is determined by 3 receptors in the retina.)
It is true that when the ratios of frequency are in simple whole
numbers the ear assigns some qualties to them. That description of
just "tuning" posted above was almost correct and has been known since
the early greeks. The most perfect consonance is the octave, and
there is still no explanation for why. Even to nonmusicians A 220 and
A 440 both sound like A, and nobody knows why. So there is no answer
for your question about wrap around.
However your assumption of a major scale always fitting inside the
octave is only partially correct. All cultures seem to have the
octave but not all have 8 diatonic steps in it, nor 12 chromatic
steps. Indian musicians use 22 or soemthing like that. Interestingly
enough those with perfect pitch in other cultures have it only for the
notes that correspond to the Western chromatic scale. Perhaps there
is something physiological involved.
The error in the description of just and equal temperament is
understanding why. You can't make a scale with just intervals. When
you add up all the simple integer ratios you DON'T wrap around, you
have some left over. Temperament systems are just different ways to
apportion that error. Equal temperament puts it evenly across every
interval. Just assigns more to some than others. There are many
other ways to go about it, all have advantages and disadvantages.
Equal is in style now as you noted and key independence is one of the
advantages.
Robert J. Kolker
TimR wrote:
>
> It is true that when the ratios of frequency are in simple whole
> numbers the ear assigns some qualties to them. That description of
> just "tuning" posted above was almost correct and has been known since
> the early greeks. The most perfect consonance is the octave, and
> there is still no explanation for why. Even to nonmusicians A 220 and
> A 440 both sound like A, and nobody knows why. So there is no answer
> for your question about wrap around.
That is fascinating. You are saying the 2:1 ratio defining the octave
not merely a cultural artifact but is related to the way we process
sound in our brain. That is very deep . That is the kind of thing that
neuroscience should bust its gut (so to speak) to find an explanation
for. If we can find a physical/physiological explanation for things like
this we can reduce mental processes to brain processes in detail.
Bob Kolker
neurothing
explained neurologically - it has to do with overlap of processing of
harmonics by hair cells in the middle ear. Pitches made of complex tones
(rather than sine waves) share the most harmonics and hence the most
neurosensory overlap in any two note interval. This is also the basis of
the "octave errors" that occur in pitch identification.