# What other senses are logarithmic?

• Medical
Gold Member
So many of us know that our eyes detect brightness on a logarithmic scale. Something that is twice as bright to us is in fact 2.51x as bright. So I was curious as to other senses for humans are logarithmic?

The reason I was wondering is because I was wondering about how accurate you really need to be when you measure ingredients when cooking. "Hmm, if I accidentally put twice as much salt as needed, will my food taste twice as salty?" :P I also know that we hear things on a logarithmic scale but I don't remember how it goes...

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Simfish
Gold Member
The decibel scale is logarithmic. It needs to be because the loudest sound humans can hear is about 1 trillion times as powerful as the softest. But that is a measure of sound energy moving through an area per second (W/m^2). What we perceive is a function of that and frequency. We are most sensitive to frequencies from about 1 kHz to 5 kHz.

Pythagorean
Gold Member
we detect pitch logarithmically with respect to octaves (we hear $$f = f_0 2^n$$)

so the $$n^{th}$$ octave from $$f_0$$:

$$n = log_2 \frac{f}{f_0}$$

this of course, ignores the whole mess of the syntonic comma and equal-temperament tuning.

AlephZero
Homework Helper
we detect pitch logarithmically with respect to octaves (we hear $$f = f_0 2^n$$)
It is virtually impossible to confirm or deny that in humans by experiment now, since almost everybody on the planet has been regularly exposed to music based on western scales from birth onwards. Western music theory has been a playground for mathematical pedants deciding what things "ought to sound like" for thousands of yeas already.

But there are musical traditions which don't recognize the octave as anything special. For example gamelan, which is fairly easy to study historically because the instruments have stable pitches and can survive for hundreds of years without any regular maintenance which may have "updated" or "improved" them.

It is virtually impossible to confirm or deny that in humans by experiment now, since almost everybody on the planet has been regularly exposed to music based on western scales from birth onwards. Western music theory has been a playground for mathematical pedants deciding what things "ought to sound like" for thousands of yeas already.

But there are musical traditions which don't recognize the octave as anything special. For example gamelan, which is fairly easy to study historically because the instruments have stable pitches and can survive for hundreds of years without any regular maintenance which may have "updated" or "improved" them.
I could see this argument if we decided that frequencies that are an octave apart are in some sense the same because the have the same letter assigned. We could also say that octaves are two frequencies that, when played together, sets up a number of beats per second that numerically equals the lower frequency. Granted: we cannot hear beats that occur that rapidly, but we recognize the quality as distinct from other beat/frequency relationships. Is that property cultural or physical?

The link that Simfish provided, by the way, is about the Weber-Fechner Law. It led me to http://www.neuro.uu.se/fysiologi/gu/nbb/lectures/Stevens.html" [Broken] that cites the Stevens' formula about perception and what we perceive logarithmically.

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Pythagorean
Gold Member
It is virtually impossible to confirm or deny that in humans by experiment now, since almost everybody on the planet has been regularly exposed to music based on western scales from birth onwards. Western music theory has been a playground for mathematical pedants deciding what things "ought to sound like" for thousands of yeas already.

But there are musical traditions which don't recognize the octave as anything special. For example gamelan, which is fairly easy to study historically because the instruments have stable pitches and can survive for hundreds of years without any regular maintenance which may have "updated" or "improved" them.
western music doesn't consist of only octaves, yet non musicians in music appreciation class have the easiest time identifying the octave, describing it as "sounding the same, by higher/lower"

And of course, anytime you strike a note, the first harmonic after the fundamental is the octave. If you continue through the harmonic series like this, you recover the tonal centers of the western major and minor scales.

So I don't think it's as subjective as you make it out to be. I don't think it's a coincidence that the first set of integer ratios represent the majority of music theory (even beyond the west).

Modern music is based on a 12 tone scale which consists of rational fractions of the base frequency. All scales use rational fractions to try to approximate scales of equal temperament which are truly logarithmic. The equal tempered scale for the twelve tone scale is based on multiples of $$2^{x/12}$$ (where x is an integer $$0\leq x \leq 12)$$ of some base frequency such as A (fixed at 440 cps). .

There are other scales such as a 19 tone scale. In theory, music can be based on any logarithmic progression if you don't mind some built in dissonance. The 12 tone scale has the least dissonance of scales of similar size, but the 53 tone scale has even less. However it requires a rather large piano (or small range of frequencies).

http://thinkzone.wlonk.com/Music/12Tone.htm

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atyy
Is there really any music where the octave is not special? I would be very surprised if "unison" singing with men and women in all cultures is not in fact singing at an octave.

Nonetheless, I agree the theoretical octaves is not so special that real music does not deviate from it at all. Even the modern pianoforte has stretched octaves.

Anyway, the question doesn't have anything necessarily to do with octaves, since one can have logarithms to base 10 or base e. Octaves are the claim that base 2 is special.

Does anyone think this relationship for pure tones looks logarithmic?
http://en.wikipedia.org/wiki/Mel_scale

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atyy
And of course, anytime you strike a note, the first harmonic after the fundamental is the octave. If you continue through the harmonic series like this, you recover the tonal centers of the western major and minor scales.
Yes, it's well known that the various western scales can be derived by such logics. Interestingly, I once heard that gamelan scales obey similar logics, except with the assumption that the gamelan instruments are in fact inharmonic. Unfortunately, I don't know if that's true or not.

Edit: Googling gives Sethares's http://books.google.com/books?id=KChoKKhjOb0C&dq=gamelan+scale+inharmonic&source=gbs_navlinks_s , which states the above claim right at the top of p199.

Edit: Sethares's book has data!

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Pythagorean
Gold Member
Is there really any music where the octave is not special? I would be very surprised if "unison" singing with men and women in all cultures is not in fact singing at an octave.

Nonetheless, I agree the theoretical octaves is not so special that real music does not deviate from it at all. Even the modern pianoforte has stretched octaves.

Anyway, the question doesn't have anything necessarily to do with octaves, since one can have logarithms to base 10 or base e. Octaves are the claim that base 2 is special.

Does anyone think this relationship for pure tones looks logarithmic?
http://en.wikipedia.org/wiki/Mel_scale
I've never heard of the Mel scale before, so I'm still trying to wrap my head around it. It appears logarithmic to me (equations and alignment with western octaves every other mel octave).

it has an extra "octave" between each set of western "octaves", I wonder if it's the tritone... I wonder if this has anything to do with the syntonic comma:
http://en.wikipedia.org/wiki/Syntonic_comma

Yes, it's well known that the various western scales can be derived by such logics. Interestingly, I once heard that gamelan scales obey similar logics, except with the assumption that the gamelan instruments are in fact inharmonic. Unfortunately, I don't know if that's true or not.

Edit: Googling gives Sethares's http://books.google.com/books?id=KCh...gbs_navlinks_s [Broken] , which states the above claim right at the top of p199.

Edit: Sethares's book has data!
Couldn't get page 199 on the google site.

from what I've read so far, they have the same octaves as western music but different notes between:
http://en.wikipedia.org/wiki/Slendro
(the pluses and minuses mean they're slightly higher or lower pitch from the notated western note, but not quite the whole half-step to the next western note)

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Is there really any music where the octave is not special? I would be very surprised if "unison" singing with men and women in all cultures is not in fact singing at an octave.

Nonetheless, I agree the theoretical octaves is not so special that real music does not deviate from it at all. Even the modern pianoforte has stretched octaves.

Anyway, the question doesn't have anything necessarily to do with octaves, since one can have logarithms to base 10 or base e. Octaves are the claim that base 2 is special.

Does anyone think this relationship for pure tones looks logarithmic?
http://en.wikipedia.org/wiki/Mel_scale
Well, the mel scale still seems to adhere to the basic concept of base 2 log intervals, possibly approximated by other bases. From your link to the wiki article:

"As a result, four octaves on the hertz scale above 500 Hz are judged to comprise about two octaves on the mel scale. The name mel comes from the word melody to indicate that the scale is based on pitch comparisons."

In any case there are many subintervals in ordinary "consonant" music based on rational fractions and their inverses which when multiplied, yield 2.

harmonic (inverse): 2/1(1/1) , 3/2(4/3), 4/3(3/2), 5/3(6/5), 5/4(8/5). 6/5(5/3), 8/5(5/4).

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atyy
Well, the mel scale still seems to adhere to the basic concept of base 2 log intervals, possibly approximated by other bases. From your link to the wiki article:

"As a result, four octaves on the hertz scale above 500 Hz are judged to comprise about two octaves on the mel scale. The name mel comes from the word melody to indicate that the scale is based on pitch comparisons."

In any case there are many subintervals in ordinary "consonant" music based on rational fractions and their inverses which when multiplied, yield 2.

harmonic (inverse): 2/1(1/1) , 3/2(4/3), 4/3(3/2), 5/3(6/5), 5/4(8/5). 6/5(5/3), 8/5(5/4).
Yes, I of course believe (naively, as a musician) that the octave is special, as I indicated from my unison singing argument. But how can it be discerned from the approximately logarithmic Mel scale, since there's always the change of base formula?

atyy
Couldn't get page 199 on the google site.

from what I've read so far, they have the same octaves as western music but different notes between:
http://en.wikipedia.org/wiki/Slendro
(the pluses and minuses mean they're slightly higher or lower pitch from the notated western note, but not quite the whole half-step to the next western note)
Here's the quote:

"The gamelan "orchestras" of Central Java in Indonesia are one of the great musical traditions. The gamelan consists of a large family of inharmonic metallophones that are tuned to either the five-note slendro or the seven tone pelog scales. Neither scale lies close to 12-tet. The inharmonic spectra of certain instruments of the gamelan are related to the unusual intervals of the pelog and slendro scales in much the same way that the harmonic spectrum of instruments in the Western tradition is relaed to the Western diatonic scale." http://books.google.com/books?id=KChoKKhjOb0C&dq=sethares&source=gbs_navlinks_s

Of course that raises the question whether the scale came first or the instruments. The first instrument of any culture is almost certainly the voice, which is harmonic. OTOH, perhaps it shouldn't be such a surprise in music which doesn't stress harmony, to have drastically different scales, since melody is, even in Western music, in "opposition" to harmony, ie. what is melodically close is harmonically distant. I think on melodic considerations alone, I would ask whether any culture has melodic scale divisions much smaller than a semitone or much larger than a tone. However, because of arpeggiation, monody can have harmonic implications, so this is perhaps too naive.

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Pythagorean
Gold Member
Here's the quote:

"The gamelan "orchestras" of Central Java in Indonesia are one of the great musical traditions. The gamelan consists of a large family of inharmonic metallophones that are tuned to either the five-note slendro or the seven tone pelog scales. Neither scale lies close to 12-tet. The inharmonic spectra of certain instruments of the gamelan are related to the unusual intervals of the pelog and slendro scales in much the same way that the harmonic spectrum of instruments in the Western tradition is relaed to the Western diatonic scale." http://books.google.com/books?id=KChoKKhjOb0C&dq=sethares&source=gbs_navlinks_s

Of course that raises the question whether the scale came first or the instruments. The first instrument of any culture is almost certainly the voice, which is harmonic. OTOH, perhaps it shouldn't be such a surprise in music which doesn't stress harmony, to have drastically different scales, since melody is, even in Western music, in "opposition" to harmony, ie. what is melodically close is harmonically distant. I think on melodic considerations alone, I would ask whether any culture has melodic scale divisions much smaller than a semitone or much larger than a tone. However, because of arpeggiation, monody can have harmonic implications, so this is perhaps too naive.
Hrm, I never thought of arpeggios as melodies. They seem more like "broken chords". They don't seem to "tell a story" or represent a "phrase". They have more of a rhythmic fell to them.

Of course temporally, there's no difference between a melody and an arpeggio and this is obviously a very subjective judgment on my part. And of course, if you throw in minor/major third in your melody, you could argue that it's "half arpeggio".

It is virtually impossible to confirm or deny that in humans by experiment now, since almost everybody on the planet has been regularly exposed to music based on western scales from birth onwards. Western music theory has been a playground for mathematical pedants deciding what things "ought to sound like" for thousands of yeas already.

But there are musical traditions which don't recognize the octave as anything special. For example gamelan, which is fairly easy to study historically because the instruments have stable pitches and can survive for hundreds of years without any regular maintenance which may have "updated" or "improved" them.
Yes, but there is an interval between pure tonal notes given by the equation $$f = f_{0} (base)^{n}$$. Any musical piece can be transposed to a higher or a lower pitch with this formula and the melody will still sound the same. For western music the formula works out to $$f = f_{0} (2)^{n/12}$$. It should work for just about any music that is tonal or melodic.

Very interesting thread! Though I have nothing useful to provide...