simple tuning question

I have a question that is probably simple to answer.

Why is it that the frets on a "normal" guitar get closer together in
what appears to be a logarithmic patterning?

Does this have something to do with 12-equal, or does it have
something to do with the nature of logarithmic frequency??

I should know this, but I apparently don't...

Thanks!

J. Pehrson

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> I have a question that is probably simple to answer.
>
> Why is it that the frets on a "normal" guitar get closer together
in
> what appears to be a logarithmic patterning?
>
> Does this have something to do with 12-equal, or does it have
> something to do with the nature of logarithmic frequency??
>
> I should know this, but I apparently don't...
>
> Thanks!
>
> J. Pehrson

any equal temperament would show this pattern (just look at my 22-
equal and 31-equal guitars). let's talk N-equal.

string length is inversely proportional to frequency.

the frequencies get higher and higher, by a factor of 2^(1/N) each
time . . . the string length gets shorter and shorter by a factor of
2^(1/N) each time -- that is, a constant *proportion* of the string
length gets chopped off each time you go up a step -- so the frets
get closer and closer, since each time you're chopping off the same
proportion off a *smaller* string length . . .

get it?

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_40590.html#40601

wrote:
> --- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> > I have a question that is probably simple to answer.
> >
> > Why is it that the frets on a "normal" guitar get closer together
> in
> > what appears to be a logarithmic patterning?
> >
> > Does this have something to do with 12-equal, or does it have
> > something to do with the nature of logarithmic frequency??
> >
> > I should know this, but I apparently don't...
> >
> > Thanks!
> >
> > J. Pehrson
>
> any equal temperament would show this pattern (just look at my 22-
> equal and 31-equal guitars). let's talk N-equal.
>
> string length is inversely proportional to frequency.
>
> the frequencies get higher and higher, by a factor of 2^(1/N) each
> time . . . the string length gets shorter and shorter by a factor
of
> 2^(1/N) each time -- that is, a constant *proportion* of the string
> length gets chopped off each time you go up a step -- so the frets
> get closer and closer, since each time you're chopping off the same
> proportion off a *smaller* string length . . .
>
> get it?

***Why, of course. I actually *knew* this... I knew it had something
to do with the logarithmic frequency situation, but momentarily
forgot about the corresponding inverse association with string length.

That's quite something to think that one sees logarithms every time
one looks at a standard guitar. I wonder how many people actually
think about that. It's rather "in your face..."

By the way, *why* does frequency work like that?? Why is there a
greater change in frequency to make the same pitch difference as one
goes up the scale.

I should probably know *this* too...

THANKS!!!!!!!!!

Joseph

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:

> By the way, *why* does frequency work like that?? Why is there a
> greater change in frequency to make the same pitch difference as
one
> goes up the scale.

our ears are designed this way. if they weren't, a single voice with
its harmonics would sound more "crowded" if it were lower down than
higher up . . . well it sort of does in a way, but at least the pitch
differences between the harmonics remain the same whether you're
talking about a low voice or a high voice . . .

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_40590.html#40601

wrote:
> --- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> > I have a question that is probably simple to answer.
> >
> > Why is it that the frets on a "normal" guitar get closer together
> in
> > what appears to be a logarithmic patterning?
> >
> > Does this have something to do with 12-equal, or does it have
> > something to do with the nature of logarithmic frequency??
> >
> > I should know this, but I apparently don't...
> >
> > Thanks!
> >
> > J. Pehrson
>
> any equal temperament would show this pattern (just look at my 22-
> equal and 31-equal guitars). let's talk N-equal.
>
> string length is inversely proportional to frequency.
>
> the frequencies get higher and higher, by a factor of 2^(1/N) each
> time . . . the string length gets shorter and shorter by a factor
of
> 2^(1/N) each time -- that is, a constant *proportion* of the string
> length gets chopped off each time you go up a step -- so the frets
> get closer and closer, since each time you're chopping off the same
> proportion off a *smaller* string length . . .
>
> get it?

***You know, this is really just so fascinating, I can't stand it.
Seriously. I'm looking at the guitar now, and seeing the fretboard
in an entirely different light, and I'm not even smoking anything...

I mean I always knew this, but it's hitting me more. The math is
right there to see! But, unfortunately, I'm a little mystified by
the 2^(1/N). How does that work? Sorry for the moronicity.

I'm still not getting why there is a larger number of cycles per
second difference between, let's say C and C# in a higher octave than
there is in a lower octave. Does it have something to do with the
number of cycles per second you're starting with (probably not...)

And why does it all work out when you *multiply?* ratios rather than
adding and subtracting. And why is it *logs...*??

This is pretty basic stuff, and some of it I *knew*, but I'd better
be refreshed on this fast, since this is a little embarassing...

J. Pehrson

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:

> ***You know, this is really just so fascinating, I can't stand it.
> Seriously. I'm looking at the guitar now, and seeing the fretboard
> in an entirely different light, and I'm not even smoking anything...
>
> I mean I always knew this, but it's hitting me more. The math is
> right there to see! But, unfortunately, I'm a little mystified by
> the 2^(1/N). How does that work? Sorry for the moronicity.

that's the frequency ratio between adjacent notes in N-equal. do we
need to review this now?
>
> I'm still not getting why there is a larger number of cycles per
> second difference between, let's say C and C# in a higher octave
than
> there is in a lower octave. Does it have something to do with the
> number of cycles per second you're starting with (probably not...)

i guess it does . . . i'm not sure what you're asking. maybe we
should sleep on it and try again tomorrow?

> And why does it all work out when you *multiply?* ratios rather
than
> adding and subtracting. And why is it *logs...*??
>
> This is pretty basic stuff, and some of it I *knew*, but I'd better
> be refreshed on this fast, since this is a little embarassing...

isn't there a FAQ on all this? if there really isn't, we can take
another stab tomorrow.

hi Joe,

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, November 05, 2002 6:41 PM
> Subject: [tuning] Re: simple tuning question
>
>
> ***You know, this is really just so fascinating, I can't stand it.
> Seriously. I'm looking at the guitar now, and seeing the fretboard
> in an entirely different light, and I'm not even smoking anything...
>
> I mean I always knew this, but it's hitting me more. The math is
> right there to see! But, unfortunately, I'm a little mystified by
> the 2^(1/N). How does that work? Sorry for the moronicity.

2^(1/N) is simply the way an EDO (equal-temperament which is based
on dividing the "8ve") is calculated.

for examples:
. 2^(1/12) is the 12edo semitone;
. 2^(1/72) is the 72edo morion (one step of 72edo);
. 2^(1/24) is the quarter-tone, etc.

N is the variable paul chose to designate the cardinality
of all EDOs.

> I'm still not getting why there is a larger number of cycles per
> second difference between, let's say C and C# in a higher octave than
> there is in a lower octave. Does it have something to do with the
> number of cycles per second you're starting with (probably not...)

the fact that our hearing is multiplicative rather than additive
explains why the cycles-per-second differences are larger among
high pitches than they are among low pitches.

example: let's use A-440 as our starting point, since it's
a handy tuning standard. the "8ve" will be defined as the
2:1 ratio. therefore, the "A" an "8ve" above 440 Hz will
be 880 Hz, a difference of 440. but the "A" an "8ve" *below*
440 Hz will be 220 Hz, a difference of only 220. and the
"A" an "8ve" below that will be 110 Hz, a difference of
only 110.

but as you can see, 110:220:440:880 = 1:2:4:8, or as i would
prefer to express it, 2^0 : 2^1 : 2^2 : 2^3. expressing the
proportions as exponents of 2 clearly shows their relationship
as "8ves", because it changes the *ratio* into a *logarithm*.

> And why does it all work out when you *multiply?* ratios rather
> than adding and subtracting. And why is it *logs...*??

a logarithm is simply a way of expressing values which
increase multiplicatively (? -- is that a word?) on a scale
which shows them as equal measurements. in other words, a
constant *ratio* or proportion is used as a constant *measurement*.

> This is pretty basic stuff, and some of it I *knew*, but I'd better
> be refreshed on this fast, since this is a little embarassing...

Joe, this is simply the way our hearing works.

what we perceive as an equal difference in pitch is always
multiplicative/divisive rather than additive/subtractive.

adding and subtracting ratios simply *isn't* perceived as
equal by our brains.

it's kind of like the fact that everything we see is tranmitted
by the lenses of our eyes upside-side, and the brain has to do
a "correction" to see it all right-side-up.

biologists can explain how it works, but *why* is it like that?
simply because God or Mother Nature (or whatever) made us that way.
it's kind of like asking "why does 2+2=4?". or in other words,
it's metaphysics and not tuning.

-monz
"all roads lead to n^0"

--- In tuning@y..., "monz" <monz@a...> wrote:

> biologists can explain how it works, but *why* is it like that?
> simply because God or Mother Nature (or whatever) made us that way.
> it's kind of like asking "why does 2+2=4?". or in other words,
> it's metaphysics and not tuning.

if biologists can explain it, it's certainly not metaphysics :)

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, November 06, 2002 12:07 AM
> Subject: [tuning] Re: simple tuning question
>
>
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> > biologists can explain how it works, but *why* is it like that?
> > simply because God or Mother Nature (or whatever) made us that way.
> > it's kind of like asking "why does 2+2=4?". or in other words,
> > it's metaphysics and not tuning.
>
> if biologists can explain it, it's certainly not metaphysics :)

scientists can explain *how*.
it's up to the philosophers to consider *why*.

-monz

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_40590.html#40663

wrote:
> --- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
>
> > ***You know, this is really just so fascinating, I can't stand
it.
> > Seriously. I'm looking at the guitar now, and seeing the
fretboard
> > in an entirely different light, and I'm not even smoking
anything...
> >
> > I mean I always knew this, but it's hitting me more. The math is
> > right there to see! But, unfortunately, I'm a little mystified
by
> > the 2^(1/N). How does that work? Sorry for the moronicity.
>
> that's the frequency ratio between adjacent notes in N-equal. do we
> need to review this now?
> >

***Oh sure. Now I remember this. This uses the octave, of course,
as the basic unit.

> > I'm still not getting why there is a larger number of cycles per
> > second difference between, let's say C and C# in a higher octave
> than
> > there is in a lower octave. Does it have something to do with the
> > number of cycles per second you're starting with (probably not...)
>
> i guess it does . . . i'm not sure what you're asking. maybe we
> should sleep on it and try again tomorrow?
>
> > And why does it all work out when you *multiply?* ratios rather
> than
> > adding and subtracting. And why is it *logs...*??
> >
> > This is pretty basic stuff, and some of it I *knew*, but I'd
better
> > be refreshed on this fast, since this is a little embarassing...
>
> isn't there a FAQ on all this? if there really isn't, we can take
> another stab tomorrow.

***There really should be a FAQ on this. I looked through the Backus
book and also the Doty _Just Intonation Primer_ and many of these
concepts are rather *assumed* but not spelled out.

Seriously, though, somebody *must* be able to explain these things in
words and a few math expressions so it sticks in the brain...

I guess the multiplication works because multiplication would take
into account the varying *proportions* created by the pitch-log
situation.

And, I guess the log situation is, fundamentally, caused by the
nature of hearing in the ear-- particularly the way it hears octave
*equivalence...*

Now why it is that the ear hears "logs" is another question...

If a tree falls in the forest, does one hear any logs?? :)

JP

--- In tuning@y..., "monz" <monz@a...> wrote:

/tuning/topicId_40590.html#40664

>> 2^(1/N) is simply the way an EDO (equal-temperament which is based
> on dividing the "8ve") is calculated.
>
> for examples:
> . 2^(1/12) is the 12edo semitone;
> . 2^(1/72) is the 72edo morion (one step of 72edo);
> . 2^(1/24) is the quarter-tone, etc.
>
> N is the variable paul chose to designate the cardinality
> of all EDOs.
>

***Yes, I remember how this works now... But is the step of 72edo
really called a "morion?" Where does that term come from. Hopefully
it's not some kind of deprication... :)

>
> the fact that our hearing is multiplicative rather than additive
> explains why the cycles-per-second differences are larger among
> high pitches than they are among low pitches.
>
> example: let's use A-440 as our starting point, since it's
> a handy tuning standard. the "8ve" will be defined as the
> 2:1 ratio. therefore, the "A" an "8ve" above 440 Hz will
> be 880 Hz, a difference of 440. but the "A" an "8ve" *below*
> 440 Hz will be 220 Hz, a difference of only 220. and the
> "A" an "8ve" below that will be 110 Hz, a difference of
> only 110.
>
> but as you can see, 110:220:440:880 = 1:2:4:8, or as i would
> prefer to express it, 2^0 : 2^1 : 2^2 : 2^3. expressing the
> proportions as exponents of 2 clearly shows their relationship
> as "8ves", because it changes the *ratio* into a *logarithm*.
>

***Got it! This is a really clear explanation. So, basically, it's
all based on the *ear.* I guess that's not surprising since it's
after all *sound* we're discussing... :)

That *very* clearly shows how the logarithms work, too!

>>
>
> Joe, this is simply the way our hearing works.
>
> what we perceive as an equal difference in pitch is always
> multiplicative/divisive rather than additive/subtractive.
>
> adding and subtracting ratios simply *isn't* perceived as
> equal by our brains.
>
> it's kind of like the fact that everything we see is tranmitted
> by the lenses of our eyes upside-side, and the brain has to do
> a "correction" to see it all right-side-up.
>

***You know, I remember learning this, but absolutely forgot about it!

> biologists can explain how it works, but *why* is it like that?
> simply because God or Mother Nature (or whatever) made us that way.
> it's kind of like asking "why does 2+2=4?". or in other words,
> it's metaphysics and not tuning.
>

***How did I ever get into metaphysics just by staring at this
guitar?? :)

Thanks for all the help and the clear explanation!

Joe

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_40590.html#40665

wrote:
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> > biologists can explain how it works, but *why* is it like that?
> > simply because God or Mother Nature (or whatever) made us that
way.
> > it's kind of like asking "why does 2+2=4?". or in other words,
> > it's metaphysics and not tuning.
>
> if biologists can explain it, it's certainly not metaphysics :)

***Well, maybe then it still *does* belong on the Tuning forum!

I find this part of it all quite fascinating, and I really think it
has a big impact upon how we construct scales, use them, and such
like...

JP

hi Joe,

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, November 06, 2002 7:39 AM
> Subject: [tuning] Re: simple tuning question
>
>

> ***Yes, I remember how this works now... But is the step
> of 72edo really called a "morion?" Where does that term
> come from. Hopefully it's not some kind of deprication... :)

i thought you were a "power user" of the Tuning Dictionary!

http://sonic-arts.org/dict/moria.htm

the term "morion" (plural "moria") comes from Cleonides's
description of Aristoxenus's intervals, in _Eisagoge_, c. 100 AD.

it refers to to (logarithmic) 1/30 of a "perfect 4th", which
in Aristoxenus's case appears to be equivalent to one degree
of 72edo.

> > the fact that our hearing is multiplicative rather than additive
> > explains why the cycles-per-second differences are larger among
> > high pitches than they are among low pitches.
> >
> > example: let's use A-440 as our starting point, since it's
> > a handy tuning standard. the "8ve" will be defined as the
> > 2:1 ratio. therefore, the "A" an "8ve" above 440 Hz will
> > be 880 Hz, a difference of 440. but the "A" an "8ve" *below*
> > 440 Hz will be 220 Hz, a difference of only 220. and the
> > "A" an "8ve" below that will be 110 Hz, a difference of
> > only 110.
> >
> > but as you can see, 110:220:440:880 = 1:2:4:8, or as i would
> > prefer to express it, 2^0 : 2^1 : 2^2 : 2^3. expressing the
> > proportions as exponents of 2 clearly shows their relationship
> > as "8ves", because it changes the *ratio* into a *logarithm*.
> >
>
> ***Got it! This is a really clear explanation. So, basically, it's
> all based on the *ear.* I guess that's not surprising since it's
> after all *sound* we're discussing... :)

once again, i tout the utility of prime-factor-vector notation!
(and this time, you concur!)

IMO, the single greatest roadblock to a mathematically-challenged
person trying to understand Partch's theory, is his dogged
retention of the use of ratios, when prime-factor-vectors make
it so much more transparent.

in fact, my difficulty (in the beginning) in understanding
the technical parts of _Genesis of a Music_ is *exactly*
what led me into tuning theory and into developing my
"JustMusic" concepts.

-monz
"all roads lead to n^0"

--- In tuning@y..., "monz" <monz@a...> wrote:
>
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> > To: <tuning@y...>
> > Sent: Wednesday, November 06, 2002 12:07 AM
> > Subject: [tuning] Re: simple tuning question
> >
> >
> > --- In tuning@y..., "monz" <monz@a...> wrote:
> >
> > > biologists can explain how it works, but *why* is it like that?
> > > simply because God or Mother Nature (or whatever) made us that
way.
> > > it's kind of like asking "why does 2+2=4?". or in other words,
> > > it's metaphysics and not tuning.
> >
> > if biologists can explain it, it's certainly not metaphysics :)
>
>
> scientists can explain *how*.
> it's up to the philosophers to consider *why*.

not if you believe in natural selection (for the issue of why we hear
that way).

2+2=4, on the other hand, was proved by russell and whitehead
starting from axioms that were basically nothing more than plausible
definitions of what we mean by natural numbers, addition, and
equality. philosophy yes, metaphysics, no.

i'm sure gene will correct some part of the last statement -- please
direct to metatuning or tuning-math.

--- In tuning@y..., "monz" <monz@a...> wrote:

/tuning/topicId_40590.html#40687

>>
> i thought you were a "power user" of the Tuning Dictionary!
>
> http://sonic-arts.org/dict/moria.htm
>

***Yes, generally. I knew I forgot to do something...

>
> in fact, my difficulty (in the beginning) in understanding
> the technical parts of _Genesis of a Music_ is *exactly*
> what led me into tuning theory and into developing my
> "JustMusic" concepts.
>

***Thanks for the update and explanation of factorization which, in
this case was very clear...

JP

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_40590.html#40688

>
> 2+2=4, on the other hand, was proved by russell and whitehead
> starting from axioms that were basically nothing more than
plausible
> definitions of what we mean by natural numbers, addition, and
> equality. philosophy yes, metaphysics, no.
>

***This is good. I like this. I *thought* somebody might actually
try to figure this kind of basic stuff out... Hopefully more on
metatuning...

JP

--- In tuning@y..., "monz" <monz@a...> wrote:

> once again, i tout the utility of prime-factor-vector notation!
> (and this time, you concur!)

Now if only someone could convince you that 2 is a prime number. :)

hi Joe,

here's a really great page about the decibel scale,
which is a logarithmic scale of loudness similar to
our musical pitch scales, which are logarithmic scales
of frequency.

http://physics.mtsu.edu/~wmr/log_1.htm

the author makes the same description of "8ves" that
i gave to you in my earlier post, and also notes that
our vision works logarithmically too.

(also note the lengthy reference to Fechner, a favorite
author of my man Mahler.)

-monz

--- In tuning@y..., "monz" <monz@a...> wrote:

/tuning/topicId_40590.html#40728

> hi Joe,
>
>
> here's a really great page about the decibel scale,
> which is a logarithmic scale of loudness similar to
> our musical pitch scales, which are logarithmic scales
> of frequency.
>
> http://physics.mtsu.edu/~wmr/log_1.htm
>
>
> the author makes the same description of "8ves" that
> i gave to you in my earlier post, and also notes that
> our vision works logarithmically too.
>
> (also note the lengthy reference to Fechner, a favorite
> author of my man Mahler.)
>
>
>
> -monz

****Thanks, Monz, for the interesting link...

Joe