Yahoo Tuning Groups Ultimate Backup tuning-math "harmonicity" ranking

Hi All,

I recently posted the following reply to Danny Wier's post on the
Tuning List re: harmonicity rankings.

My question is: Is there a tidy formula that anyone knows of that
describes the process for finding the harmonicity ranking of a ratio
which I've outlined in the attached post.

I want to explain a bit more about the process as well. Imagine
driving out of your driveway onto a residential street, then driving
onto a thoroughfare, then a highway, then a freeway, then onto an
airplane..... you get the idea... Jumping from counting by ones to
counting by twos and threes and so on.

To get to a remote harmonic, you can start out in the driveway..
counting by ones... then, you can count by twos... and if it's
quickest, then count by threes or fours...and so on..

Then, once you arrive at the harmonic, you bring it home (between
1/1 and 2/1) by parachuting it down by a similar process.

Read on... I think that the results of this process are good even
if the process is a bit untidy at first glance.

Thanks in advance..

Robin

(original message follows_

Re: ranking ratios by harmonicity and all that

Hi Danny,

I've been pondering something around this issue for a while and
thought I'd toss it your way to see what you might think.

It's a relatively simple concept. Odd number harmonics are a
certain distance from 1/1 on a number line. In this system of
ranking, however, you can "jump" to distant numbers by not only 1's,
but by 2's. (and others) by following simple rules. (1.) You have
to jump upwards by doubling the current number, by adding one to it,
or by adding a factor of the number. So, from 2, you can jump to 4
or just add one to get to 3. From 3, you can go to 6 or 4. From 4,
you can go to 8, 5, or 6. (the jump from 4 to 6 is by a factor of
2). From 6, you can go to 12, 7, 8, or 9. The jump from 6 to 8 is
by a factor of 2. The jump to 9 is by a factor of 3.

The minimum number of steps it takes to get to a given harmonic is
it's harmonicity ranking. The higher the ranking, the lower the
harmonicity. That number of steps up plus the number of steps it
takes to get the harmonic back down to between 1/1 and 2/1 is the
ranking of the ratio.

For example: The 7 harmonic can be reached by jumping from 1 to 2
to 4 to (either 6 or 8) and then to 7, for a total of 4 steps. It
takes another 2 steps to arrive at 7/4 by halving 7 twice (3.5,
1.75). So the ranking of 7/4 is 6.

For subharmonics, it's basically the same except that they will
always have an added step to get up into the range between 1/1 and
2/1. Example: (1/5 subharmonic) Go from 1/1 down to ½, then to ¼,
then to 1/5, for a total of 3 steps. Multiply 1/5 times 2 cubed to
arrive at 8/5. The total number of steps is 6. This puts both 7/4
and 8/5 into the same ranking. So, a perfect fourth is not as
harmonic as a perfect fifth and so on.

Some common ratios and rankings based on this system: (not all in
order)

2/1 - 1
3/2 - 3 (2,3,3/2)
4/3 - 4 (1/2,1/3,2/3,4/3)
5/4 - 5 (2,4,5,5/2,5/4)
8/5 - 6 (1/2,1/4,1/5,2/5,4/5,8/5)
6/5 - 6 (1/2,1/4,1/5,2/5,4/5,6/5)
5/3 - 5 (1/2,1/3,2/3,4/3,5/3)
7/4 - 6 (2,4,6,7,7/2,7/4)
8/7 - 7 (1/2,1/4,1/6,1/7,2/7,4/7,8/7)
9/8 - 7 (2,4,8,9,9/2,9/4,9/8)
16/9 - 8 (1/2,1/4,1/8,1/9,2/9,4/9,8/9,16/9)
7/6 - 7 (1/2,1/4,1/6,2/6,4/6,6/6,7/6)
12/7 - 8 (1/2,1/4,1/6,1/7,2/7,4/7,8/7,12/7)
10/9 - 8 (1/2,1/3,1/6,1/9,2/9,4/9,5/9,10/9)
9/5 - 7 (1/2,1/4,1/5,2/5,4/5,8/5,9/5)
7/5 - 7 (1/2,1/4,1/5,2/5,4/5,6/5,7/5)
10/7 - 8 (1/2,1/4,1/6,1/7,2/7,4/7,5/7,10/7)

This goes on and on, of course. A simple scale based on all the
harmonic and subharmonic pairs up through level 6 would look like:

1/1, 6/5, 5/4,4/3,3/2,8/5,5/3,7/4,2/1.

I'm interested in hearing what you might have to say about this
idea especially if you listen to the pairs and can tell me if you
agree, or not, that those on the same `level' are equally consonant
or dissonant (as the case may be).

Regards,

Robin Perry

🔗Carl Lumma <ekin@lumma.org>

The problem with Euler's function, and apparently with yours,
is that they're based on factoring. While this comes into play
when harmonies are generated from scales, it does not have a
signficant role in the perception of isolated dyads.

In your ranking below, I disagree with where you put 5/3.
I put it above 5/4.

A concordance ranking method that puts 5/3 there is n*d. It's
the measure favored by Galileo, and by years of consensus on
this list. "Good ol' numerator times denominator," we'd say.
One caveat: it only works until the product is about 100 or
something. Any trick based on ratios will have to falter around
here, as you can start approximating smaller ratios with more
complex ones.

-Carl

At 12:48 AM 9/3/2007, you wrote:
>Hi All,
>
>I recently posted the following reply to Danny Wier's post on the
>Tuning List re: harmonicity rankings.
>
>My question is: Is there a tidy formula that anyone knows of that
>describes the process for finding the harmonicity ranking of a ratio
>which I've outlined in the attached post.
>
>I want to explain a bit more about the process as well. Imagine
>driving out of your driveway onto a residential street, then driving
>onto a thoroughfare, then a highway, then a freeway, then onto an
>airplane..... you get the idea... Jumping from counting by ones to
>counting by twos and threes and so on.
>
>To get to a remote harmonic, you can start out in the driveway..
>counting by ones... then, you can count by twos... and if it's
>quickest, then count by threes or fours...and so on..
>
>Then, once you arrive at the harmonic, you bring it home (between
>1/1 and 2/1) by parachuting it down by a similar process.
>
>Read on... I think that the results of this process are good even
>if the process is a bit untidy at first glance.
>
>Thanks in advance..
>
>Robin
>
>(original message follows_
>
>
>Re: ranking ratios by harmonicity and all that
>
>
>Hi Danny,
>
>I've been pondering something around this issue for a while and
>thought I'd toss it your way to see what you might think.
>
>It's a relatively simple concept. Odd number harmonics are a
>certain distance from 1/1 on a number line. In this system of
>ranking, however, you can "jump" to distant numbers by not only 1's,
>but by 2's. (and others) by following simple rules. (1.) You have
>to jump upwards by doubling the current number, by adding one to it,
>or by adding a factor of the number. So, from 2, you can jump to 4
>or just add one to get to 3. From 3, you can go to 6 or 4. From 4,
>you can go to 8, 5, or 6. (the jump from 4 to 6 is by a factor of
>2). From 6, you can go to 12, 7, 8, or 9. The jump from 6 to 8 is
>by a factor of 2. The jump to 9 is by a factor of 3.
>
>The minimum number of steps it takes to get to a given harmonic is
>it's harmonicity ranking. The higher the ranking, the lower the
>harmonicity. That number of steps up plus the number of steps it
>takes to get the harmonic back down to between 1/1 and 2/1 is the
>ranking of the ratio.
>
>For example: The 7 harmonic can be reached by jumping from 1 to 2
>to 4 to (either 6 or 8) and then to 7, for a total of 4 steps. It
>takes another 2 steps to arrive at 7/4 by halving 7 twice (3.5,
>1.75). So the ranking of 7/4 is 6.
>
>For subharmonics, it's basically the same except that they will
>always have an added step to get up into the range between 1/1 and
>2/1. Example: (1/5 subharmonic) Go from 1/1 down to ½, then to ¼,
>then to 1/5, for a total of 3 steps. Multiply 1/5 times 2 cubed to
>arrive at 8/5. The total number of steps is 6. This puts both 7/4
>and 8/5 into the same ranking. So, a perfect fourth is not as
>harmonic as a perfect fifth and so on.
>
>Some common ratios and rankings based on this system: (not all in
>order)
>
>2/1 - 1
>3/2 - 3 (2,3,3/2)
>4/3 - 4 (1/2,1/3,2/3,4/3)
>5/4 - 5 (2,4,5,5/2,5/4)
>8/5 - 6 (1/2,1/4,1/5,2/5,4/5,8/5)
>6/5 - 6 (1/2,1/4,1/5,2/5,4/5,6/5)
>5/3 - 5 (1/2,1/3,2/3,4/3,5/3)
>7/4 - 6 (2,4,6,7,7/2,7/4)
>8/7 - 7 (1/2,1/4,1/6,1/7,2/7,4/7,8/7)
>9/8 - 7 (2,4,8,9,9/2,9/4,9/8)
>16/9 - 8 (1/2,1/4,1/8,1/9,2/9,4/9,8/9,16/9)
>7/6 - 7 (1/2,1/4,1/6,2/6,4/6,6/6,7/6)
>12/7 - 8 (1/2,1/4,1/6,1/7,2/7,4/7,8/7,12/7)
>10/9 - 8 (1/2,1/3,1/6,1/9,2/9,4/9,5/9,10/9)
>9/5 - 7 (1/2,1/4,1/5,2/5,4/5,8/5,9/5)
>7/5 - 7 (1/2,1/4,1/5,2/5,4/5,6/5,7/5)
>10/7 - 8 (1/2,1/4,1/6,1/7,2/7,4/7,5/7,10/7)
>
>This goes on and on, of course. A simple scale based on all the
>harmonic and subharmonic pairs up through level 6 would look like:
>
>1/1, 6/5, 5/4,4/3,3/2,8/5,5/3,7/4,2/1.
>
>I'm interested in hearing what you might have to say about this
>idea especially if you listen to the pairs and can tell me if you
>agree, or not, that those on the same `level' are equally consonant
>or dissonant (as the case may be).
>
>Regards,
>
>Robin Perry

🔗Robin Perry <jinto83@yahoo.com>

Hi Carl,

When you say you put 5/3 above 5/4, do you mean that 5/3 is more, or
less consonant?

As far as N*D goes: That would make a 7/5 more consonant than an
8/5. Do you agree that a 7/5 sounds more consonant than an 8/5?

Just trying to clarify.. There might be more personal preference
involved in this than I imagined.

Thanks,

Robin

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> The problem with Euler's function, and apparently with yours,
> is that they're based on factoring. While this comes into play
> when harmonies are generated from scales, it does not have a
> signficant role in the perception of isolated dyads.
>
> In your ranking below, I disagree with where you put 5/3.
> I put it above 5/4.
>
> A concordance ranking method that puts 5/3 there is n*d. It's
> the measure favored by Galileo, and by years of consensus on
> this list. "Good ol' numerator times denominator," we'd say.
> One caveat: it only works until the product is about 100 or
> something. Any trick based on ratios will have to falter around
> here, as you can start approximating smaller ratios with more
> complex ones.
>
> -Carl
>
> At 12:48 AM 9/3/2007, you wrote:
> >Hi All,
> >
> >I recently posted the following reply to Danny Wier's post on the
> >Tuning List re: harmonicity rankings.
> >
> >My question is: Is there a tidy formula that anyone knows of
that
> >describes the process for finding the harmonicity ranking of a
ratio
> >which I've outlined in the attached post.
> >
> >I want to explain a bit more about the process as well. Imagine
> >driving out of your driveway onto a residential street, then
driving
> >onto a thoroughfare, then a highway, then a freeway, then onto an
> >airplane..... you get the idea... Jumping from counting by ones
to
> >counting by twos and threes and so on.
> >
> >To get to a remote harmonic, you can start out in the driveway..
> >counting by ones... then, you can count by twos... and if it's
> >quickest, then count by threes or fours...and so on..
> >
> >Then, once you arrive at the harmonic, you bring it home (between
> >1/1 and 2/1) by parachuting it down by a similar process.
> >
> >Read on... I think that the results of this process are good
even
> >if the process is a bit untidy at first glance.
> >
> >Thanks in advance..
> >
> >Robin
> >
> >(original message follows_
> >
> >
> >Re: ranking ratios by harmonicity and all that
> >
> >
> >Hi Danny,
> >
> >I've been pondering something around this issue for a while and
> >thought I'd toss it your way to see what you might think.
> >
> >It's a relatively simple concept. Odd number harmonics are a
> >certain distance from 1/1 on a number line. In this system of
> >ranking, however, you can "jump" to distant numbers by not only
1's,
> >but by 2's. (and others) by following simple rules. (1.) You have
> >to jump upwards by doubling the current number, by adding one to
it,
> >or by adding a factor of the number. So, from 2, you can jump to 4
> >or just add one to get to 3. From 3, you can go to 6 or 4. From 4,
> >you can go to 8, 5, or 6. (the jump from 4 to 6 is by a factor of
> >2). From 6, you can go to 12, 7, 8, or 9. The jump from 6 to 8 is
> >by a factor of 2. The jump to 9 is by a factor of 3.
> >
> >The minimum number of steps it takes to get to a given harmonic is
> >it's harmonicity ranking. The higher the ranking, the lower the
> >harmonicity. That number of steps up plus the number of steps it
> >takes to get the harmonic back down to between 1/1 and 2/1 is the
> >ranking of the ratio.
> >
> >For example: The 7 harmonic can be reached by jumping from 1 to 2
> >to 4 to (either 6 or 8) and then to 7, for a total of 4 steps. It
> >takes another 2 steps to arrive at 7/4 by halving 7 twice (3.5,
> >1.75). So the ranking of 7/4 is 6.
> >
> >For subharmonics, it's basically the same except that they will
> >always have an added step to get up into the range between 1/1 and
> >2/1. Example: (1/5 subharmonic) Go from 1/1 down to ½, then to ¼,
> >then to 1/5, for a total of 3 steps. Multiply 1/5 times 2 cubed to
> >arrive at 8/5. The total number of steps is 6. This puts both 7/4
> >and 8/5 into the same ranking. So, a perfect fourth is not as
> >harmonic as a perfect fifth and so on.
> >
> >Some common ratios and rankings based on this system: (not all in
> >order)
> >
> >2/1 - 1
> >3/2 - 3 (2,3,3/2)
> >4/3 - 4 (1/2,1/3,2/3,4/3)
> >5/4 - 5 (2,4,5,5/2,5/4)
> >8/5 - 6 (1/2,1/4,1/5,2/5,4/5,8/5)
> >6/5 - 6 (1/2,1/4,1/5,2/5,4/5,6/5)
> >5/3 - 5 (1/2,1/3,2/3,4/3,5/3)
> >7/4 - 6 (2,4,6,7,7/2,7/4)
> >8/7 - 7 (1/2,1/4,1/6,1/7,2/7,4/7,8/7)
> >9/8 - 7 (2,4,8,9,9/2,9/4,9/8)
> >16/9 - 8 (1/2,1/4,1/8,1/9,2/9,4/9,8/9,16/9)
> >7/6 - 7 (1/2,1/4,1/6,2/6,4/6,6/6,7/6)
> >12/7 - 8 (1/2,1/4,1/6,1/7,2/7,4/7,8/7,12/7)
> >10/9 - 8 (1/2,1/3,1/6,1/9,2/9,4/9,5/9,10/9)
> >9/5 - 7 (1/2,1/4,1/5,2/5,4/5,8/5,9/5)
> >7/5 - 7 (1/2,1/4,1/5,2/5,4/5,6/5,7/5)
> >10/7 - 8 (1/2,1/4,1/6,1/7,2/7,4/7,5/7,10/7)
> >
> >This goes on and on, of course. A simple scale based on all the
> >harmonic and subharmonic pairs up through level 6 would look like:
> >
> >1/1, 6/5, 5/4,4/3,3/2,8/5,5/3,7/4,2/1.
> >
> >I'm interested in hearing what you might have to say about this
> >idea especially if you listen to the pairs and can tell me if you
> >agree, or not, that those on the same `level' are equally
consonant
> >or dissonant (as the case may be).
> >
> >Regards,
> >
> >Robin Perry
>

🔗Carl Lumma <ekin@lumma.org>

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning-math@yahoogroups.com, "Robin Perry" <jinto83@...> wrote:
>
Dears Carl & Robin,
>
let's calculate some N*D examples:
> When you say you put 5/3
5*3 = 15
> above 5/4,
5*4 = 20
> do you mean that 5/3 is more, or
> less consonant?

Some people conclude from that for triads respectively:
3:4:5 = G:C:E with 3*4*5=60
would sound even twice cosonant as:
4:5:6 = C:E:G with 4*5*6=120 = 60*2

because that corresponds to:
3:4:5 = G:C:E is lower located in the harmonic series of partials:
1:2:3:4:5:6.... than the usually preferred triad:
4:5:6 = C:E:G
that is shifted about one position to the less consonant
position higher away from basic root one 1:1, the unison.
>
not to mention the stretched triad over 2 octaves:
1:3:5 = 15 = 5/3 !
alike concrete in absolute frequencies for instance:
100Hz : 300Hz : 500Hz ==>> Gradus=15
versus
400Hz : 500Hz : 600Hz ==>> Gradus=120

that computations
lead to even more questionable divergences in ranking triads,
than of 2-component intervals.

> As far as N*D goes: That would make a 7/5 more consonant than an
> 8/5. Do you agree that a 7/5 sounds more consonant than an 8/5?
That strange way of counting ratios yields for:
5*1 = 5 and
7*1 = 7
That do appear even early
before the above mentioned products of
5*3=15 or 5*4=20.

> > >2/1 - 1
How about putting the duodecime 3/1 consequently already here,
at second position inbetween octave: 2/1 and 5th: 3/2
due to the even simpler product of:
N*D = 3 , that term turns out to be concrete:
3*1 = 3 ?
even listed before the more complicated ratios of:
> > >3/2 - 3 (2,3,3/2)
> > >4/3 - 4 (1/2,"instead so late afterwards here: 1/3?",2/3,4/3)
> > >5/4 - 5 (2,4,5,5/2,5/4)
......

My objections against such random ranking:
There arouses from such highanded product
calculations an strong suspicion:

All that numerical attempts in guessing
about any alleged 'gradus-suavitis' claims
depends barely on arbitrarily personal taste,
howsoever individual preferred?

Conclusion:
"De gustibus non est disputandum"

kind regards
A.S.

🔗Robin Perry <jinto83@yahoo.com>

Well.. no, not really. I have been opening up Scala and listening
to a lot of ratios of late. I hear 5/4 and 5/3 more or less equally
consonant. And, I hear 8/5 as being more than 7/5 and I don't think
that's due to acculturation. I certainly like to use both 7/5 and
10/7 in my own work, so am pretty accustomed to the sounds.

Robin

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 12:22 AM 9/5/2007, you wrote:
> >Hi Carl,
> >
> >When you say you put 5/3 above 5/4, do you mean that 5/3 is more,
or
> >less consonant?
>
> More. Don't you think so?
>
> >As far as N*D goes: That would make a 7/5 more consonant than an
> >8/5. Do you agree that a 7/5 sounds more consonant than an 8/5?
>
> That's the classical counterexample / trouble spot with n*d.
> It's a hard judgement to make, and they're really pretty close
> on the measure in the first place (and near the threshold of
> where it stops working). The situation certainly isn't helped
> by acculturation to 8/5 in common-practice music. To answer
> your question, in some ways I think 7/5 is more consonant, or
> at least less tonally ambiguous.
>
> -Carl
>

🔗Carl Lumma <ekin@lumma.org>

Hi Andreas,

>Some people conclude from that for triads respectively:
>3:4:5 = G:C:E with 3*4*5=60
>would sound even twice cosonant as:
>4:5:6 = C:E:G with 4*5*6=120 = 60*2

I did leave out the notion of taking the log of
the result, which according to Paul Erlich is the
right way to scale the scores. I left this out since
the original thread on the tuning list was just about
getting a ranking.

So it isn't twice as consonant, but it's still more
consonant. And indeed, I think 3:4:5 may very well
be more consonant than 4:5:6. At least, 2:3:5 is more
consonant than either.

>not to mention the stretched triad over 2 octaves:
>1:3:5 = 15 = 5/3 !

When comparing n-ads of different n (if that is
meaningful), it would be advised to take the nth root of
the result (geometric mean). Since sqrt(15) > cbrt(15),
it is telling us that 1:3:5 is a more consonant triad,
as triads go, than 5/3 is a dyad as dyads go. Which
makes sense to me.

>My objections against such random ranking:
>There arouses from such highanded product
>calculations an strong suspicion:
>
>All that numerical attempts in guessing
>about any alleged 'gradus-suavitis' claims
>depends barely on arbitrarily personal taste,
>howsoever individual preferred?

Individuals will certainly disagree, but I think evidence
supports a belief that there is a common thread in the way
humans experience these sounds, and that n*d is a good
quick & dirty way to approximate it.

-Carl

🔗Robin Perry <jinto83@yahoo.com>

Hi Andreas,

Thank you. I appreciate the insight into this from the viewpoint of
extrapolating the concepts into triads. I'm interested in focusing
on diads for now, though. And, am still wondering if there is any
sort of formula that anyone might see in the process I described.

By the way, though.. I hear a 3,4,5,6 as even more consonant than a
3,4,5. I'm not really sure why... maybe because of the octave
present, it seems more complete.

Robin

--- In tuning-math@yahoogroups.com, "Andreas Sparschuh"
<a_sparschuh@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Robin Perry" <jinto83@> wrote:
> >
> Dears Carl & Robin,
> >
> let's calculate some N*D examples:
> > When you say you put 5/3
> 5*3 = 15
> > above 5/4,
> 5*4 = 20
> > do you mean that 5/3 is more, or
> > less consonant?
>
> Some people conclude from that for triads respectively:
> 3:4:5 = G:C:E with 3*4*5=60
> would sound even twice cosonant as:
> 4:5:6 = C:E:G with 4*5*6=120 = 60*2
>
> because that corresponds to:
> 3:4:5 = G:C:E is lower located in the harmonic series of partials:
> 1:2:3:4:5:6.... than the usually preferred triad:
> 4:5:6 = C:E:G
> that is shifted about one position to the less consonant
> position higher away from basic root one 1:1, the unison.
> >
> not to mention the stretched triad over 2 octaves:
> 1:3:5 = 15 = 5/3 !
> alike concrete in absolute frequencies for instance:
> 100Hz : 300Hz : 500Hz ==>> Gradus=15
> versus
> 400Hz : 500Hz : 600Hz ==>> Gradus=120
>
> that computations
> lead to even more questionable divergences in ranking triads,
> than of 2-component intervals.
>
> > As far as N*D goes: That would make a 7/5 more consonant than
an
> > 8/5. Do you agree that a 7/5 sounds more consonant than an 8/5?
> That strange way of counting ratios yields for:
> 5*1 = 5 and
> 7*1 = 7
> That do appear even early
> before the above mentioned products of
> 5*3=15 or 5*4=20.
>
> > > >2/1 - 1
> How about putting the duodecime 3/1 consequently already here,
> at second position inbetween octave: 2/1 and 5th: 3/2
> due to the even simpler product of:
> N*D = 3 , that term turns out to be concrete:
> 3*1 = 3 ?
> even listed before the more complicated ratios of:
> > > >3/2 - 3 (2,3,3/2)
> > > >4/3 - 4 (1/2,"instead so late afterwards here: 1/3?",2/3,4/3)
> > > >5/4 - 5 (2,4,5,5/2,5/4)
> ......
>
> My objections against such random ranking:
> There arouses from such highanded product
> calculations an strong suspicion:
>
> All that numerical attempts in guessing
> about any alleged 'gradus-suavitis' claims
> depends barely on arbitrarily personal taste,
> howsoever individual preferred?
>
> Conclusion:
> "De gustibus non est disputandum"
>
> kind regards
> A.S.
>

🔗Carl Lumma <ekin@lumma.org>

Here's how they sound on my system:

http://lumma.org/stuff/75.wav
http://lumma.org/stuff/85.wav

To me the sound of the first one *is* more
consonant in a sense. According to the
measure...

2/1 = log2(2) = 1
5/4 = log2(20) = 4.321928094887363
7/5 = log2(35) = 5.129283016944966
8/5 = log2(40) = 5.321928094887363

-Carl

At 12:27 PM 9/5/2007, you wrote:
>Well.. no, not really. I have been opening up Scala and listening
>to a lot of ratios of late. I hear 5/4 and 5/3 more or less equally
>consonant. And, I hear 8/5 as being more than 7/5 and I don't think
>that's due to acculturation. I certainly like to use both 7/5 and
>10/7 in my own work, so am pretty accustomed to the sounds.
>
>Robin
>
>
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>>
>> At 12:22 AM 9/5/2007, you wrote:
>> >Hi Carl,
>> >
>> >When you say you put 5/3 above 5/4, do you mean that 5/3 is
>> >more, or less consonant?
>>
>> More. Don't you think so?
>>
>> >As far as N*D goes: That would make a 7/5 more consonant than an
>> >8/5. Do you agree that a 7/5 sounds more consonant than an 8/5?
>>
>> That's the classical counterexample / trouble spot with n*d.
>> It's a hard judgement to make, and they're really pretty close
>> on the measure in the first place (and near the threshold of
>> where it stops working). The situation certainly isn't helped
>> by acculturation to 8/5 in common-practice music. To answer
>> your question, in some ways I think 7/5 is more consonant, or
>> at least less tonally ambiguous.
>>
>> -Carl

🔗Robin Perry <jinto83@yahoo.com>

Thanks, Carl,
Interesting voicing on those... what is that?

To my ear, 8/5 is more consonant.

Robin

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Here's how they sound on my system:
>
> http://lumma.org/stuff/75.wav
> http://lumma.org/stuff/85.wav
>
> To me the sound of the first one *is* more
> consonant in a sense. According to the
> measure...
>
> 2/1 = log2(2) = 1
> 5/4 = log2(20) = 4.321928094887363
> 7/5 = log2(35) = 5.129283016944966
> 8/5 = log2(40) = 5.321928094887363
>
> -Carl
>
> At 12:27 PM 9/5/2007, you wrote:
> >Well.. no, not really. I have been opening up Scala and
listening
> >to a lot of ratios of late. I hear 5/4 and 5/3 more or less
equally
> >consonant. And, I hear 8/5 as being more than 7/5 and I don't
think
> >that's due to acculturation. I certainly like to use both 7/5
and
> >10/7 in my own work, so am pretty accustomed to the sounds.
> >
> >Robin
> >
> >
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >>
> >> At 12:22 AM 9/5/2007, you wrote:
> >> >Hi Carl,
> >> >
> >> >When you say you put 5/3 above 5/4, do you mean that 5/3 is
> >> >more, or less consonant?
> >>
> >> More. Don't you think so?
> >>
> >> >As far as N*D goes: That would make a 7/5 more consonant than
an
> >> >8/5. Do you agree that a 7/5 sounds more consonant than an
8/5?
> >>
> >> That's the classical counterexample / trouble spot with n*d.
> >> It's a hard judgement to make, and they're really pretty close
> >> on the measure in the first place (and near the threshold of
> >> where it stops working). The situation certainly isn't helped
> >> by acculturation to 8/5 in common-practice music. To answer
> >> your question, in some ways I think 7/5 is more consonant, or
> >> at least less tonally ambiguous.
> >>
> >> -Carl
>

🔗Carl Lumma <ekin@lumma.org>

🔗Herman Miller <hmiller@IO.COM>

🔗Robin Perry <jinto83@yahoo.com>

Hi Carl,

I have been thinking about the 5/4 and 5/3 and think that a food
example is appropriate. Apricot jam and strawberry jam are equally
sweet. They just have different flavors. Following on that train
of thought: a 2/1 is pure glucose; a 3/2 is cane sugar; and a 4/3
is beet sugar.

I have begun thinking in terms of taste lately. My wife constantly
challenges me with new tastes (she's a consumate foodie) and I
challenge her tonal pallete with 'unusual' sounds.

I think that so much of this is a matter of taste.

Cheers,

Robin

🔗Carl Lumma <ekin@lumma.org>

🔗Robin Perry <jinto83@yahoo.com>

Hi,

No.. I had a feeling that that comparison might not work out..

Sweetness (most to least): 2/1,3/2,4/3

Thanks,

Robin

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 12:34 AM 9/7/2007, you wrote:
> >Hi Carl,
> >
> >I have been thinking about the 5/4 and 5/3 and think that a food
> >example is appropriate. Apricot jam and strawberry jam are
equally
> >sweet. They just have different flavors. Following on that
train
> >of thought: a 2/1 is pure glucose; a 3/2 is cane sugar; and a
4/3
> >is beet sugar.
>
> Actually glucose is substantially less sweet than cane sugar.
> And cane and beet sugar are both sucrose.
> Is that what you meant?
>
> Some advocate ranking ratios by denominator *when it's the top
> note that's changing* and by numerator *when the bottom note
> changes*. If you happen to know which note is changing, this
> may be better than n*d. Maybe Dave K. has something to add...
>
> -Carl
>

🔗Herman Miller <hmiller@IO.COM>

Robin Perry wrote:
> Hi Carl,
> > I have been thinking about the 5/4 and 5/3 and think that a food > example is appropriate. Apricot jam and strawberry jam are equally > sweet. They just have different flavors. Following on that train > of thought: a 2/1 is pure glucose; a 3/2 is cane sugar; and a 4/3 > is beet sugar.

I think a 2/1 is really pretty flavorless; at least not a sweet sort of flavor. Sweet would be a pretty good word to describe the 5-colored intervals, though.

> I have begun thinking in terms of taste lately. My wife constantly > challenges me with new tastes (she's a consumate foodie) and I > challenge her tonal pallete with 'unusual' sounds.
> > I think that so much of this is a matter of taste.

The analogy could also be applied to dissonant intervals. Something like 35/24 could be more of a "bitter" flavor, while 36/25 is just a little "spicy". I'm not sure how far you could go with the taste analogy, but there's more than one kind of dissonance. There are ambiguous irrational intervals like phi or the square root of 2, which fall in between consonant intervals on the one hand, more complex integer ratios like 15/11 or 35/24 on the other hand, and also intervals like the meantone wolf fifth, which fall close to a more consonant interval but sound like mistuned versions of that interval.

🔗Carl Lumma <ekin@lumma.org>

🔗Robin Perry <jinto83@yahoo.com>

Hi Herman,

Good suggestions. There are what, six categories of taste? Sweet,
sour, bitter, salty, hot (spicy), and savory (umami)are the ones
that come to mind. I suppose that you could, to some extent,
categorize intervals that way. I see what you mean about the 2/1.
It's so close to 'home' that it is hard to categorize. Maybe it's
just bland.

I've started sketching on a 3-axis isometric graph the intervals in
their levels as I place them by the process outlined in the original
post. The numerator is X, the denominator is Y, and the level is Z.

I have then connected the dots of those intervals which have the
relationship of m*(x+1)/m*x; where m and x are positive integers.
That expanded form of a ratio defines the way that I see the steps
working. You can go from point A to point B by steps that satisfy
that form. You can't go from odd integer to odd integer in this
system because they don't conform.

Hopefully that is somwhat clearer than mud.

Cheers,

Robin

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...>
wrote:
>
> Robin Perry wrote:
> > Hi Carl,
> >
> > I have been thinking about the 5/4 and 5/3 and think that a food
> > example is appropriate. Apricot jam and strawberry jam are
equally
> > sweet. They just have different flavors. Following on that
train
> > of thought: a 2/1 is pure glucose; a 3/2 is cane sugar; and a
4/3
> > is beet sugar.
>
> I think a 2/1 is really pretty flavorless; at least not a sweet
sort of
> flavor. Sweet would be a pretty good word to describe the 5-
colored
> intervals, though.
>
> > I have begun thinking in terms of taste lately. My wife
constantly
> > challenges me with new tastes (she's a consumate foodie) and I
> > challenge her tonal pallete with 'unusual' sounds.
> >
> > I think that so much of this is a matter of taste.
>
> The analogy could also be applied to dissonant intervals.
Something like
> 35/24 could be more of a "bitter" flavor, while 36/25 is just a
little
> "spicy". I'm not sure how far you could go with the taste analogy,
but
> there's more than one kind of dissonance. There are ambiguous
irrational
> intervals like phi or the square root of 2, which fall in between
> consonant intervals on the one hand, more complex integer ratios
like
> 15/11 or 35/24 on the other hand, and also intervals like the
meantone
> wolf fifth, which fall close to a more consonant interval but
sound like
> mistuned versions of that interval.
>

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> > They just have different flavors. Following on that train
> > of thought: a 2/1 is pure glucose; a 3/2 is cane sugar; and a 4/3
> > is beet sugar.

> more complex integer ratios like
> 15/11 or 35/24 on the other hand,....
>
Simply by playing an
http://en.wikipedia.org/wiki/Natural_horn
you can taste the flavour of the:
http://en.wikipedia.org/wiki/Harmonic_series_%28music%29

Here my private keyboard emulation for blowing 2 swiss
http://en.wikipedia.org/wiki/Alphorn
's instead by lips,
touching with fingers on the keys corresponding

Horn_1: Fundamental pitch on C_0:
22Hz *(...13:14:15:16:[17]:18:[19]:20:21:22)
attend the intensional gaps @ lacking [17] & [19]

Horn_2. Fundamental pitch on C_1:
33Hz * (...8:9:10:11:12:13:14...)
preferably for beginners, with less trained lip muscles.

Both together represent the middle-C octave chromatically full in range.

For all those, who possess no own horns or
want to play in resonance together with hornists below 13-limit:

! alphorns.scl
!
Keyboard emulation 22Hz*(13...22) and 33Hz*(8...14)harmonic partials
!
12
!
! middle C = 264 Hz ; 132 66 C_0 = 33Hz = fundamental(horn_2)
!relative ratio versus the absolute frequencies on the keys
13/12 ! c# 286 (143)
9/8 ! D 297 '2nd'
7/6 ! d# 308 154 (77)
5/4 ! E 330 165 '3rd'
4/3 ! F 352 176 88 44 22Hz = fundamental(horn_1) (11) '4th'
11/8 ! f# 363 "Alphorn-fa" 11th partial on horn_2"
3/2 ! G 396 198 99 '5th'
13/8 ! g# 429
5/3 ! A 440 220 110 55 '6th' standard normal pitch: 440Hz
7/4 ! a# 231 septimal "blue-note"
11/6 ! B 242 "Alphorn-fa" fun on horn_1 (121=11*11=11^2) squared
2/1
!
!
!
The chosen keyboard-layout corresponds
partially to the harmonic series:

Chart:

Horn_1: Fundamental pitch on F_0=22Hz
B:d#:E:F:_:G:_:A:A#:B === 13:14:15:16:_:18:_:20:21:22
with lacking gaps at the [17th] and [19th] partials

Horn_2: Fundamental pitch on C_1: 33Hz
C:D:E:f#:G:g#:a# === 8:9:10:11:12:13:14

Attend the 12/11 exceptional wide "leading-tone" B-C at the end
in order to maximize the 'fun' that arises by
squareing 11^2*2=242Hz versus the real unison 1/1
just before jumping to the middle-C: 264Hz.

(1 200 * ln(12 / 11)) / ln(2) = ~150.637059...Cents

~3 steps in 24-EDO.

Each new prime-limit provokes it's own special individual taste

Table:

1: basic fundamental on the ground
http://en.wikipedia.org/wiki/Sine_wave
2: banal trivial double
3: mild melting triple
5: sweet 3:4:5 triangle 3^2 + 4^2 = 5^2
7: blues
11: fun
13: sexy 5:12:13 sounding triangle 5^2 + 12^2 = 13^2
17/16: major semitone 8:15:17 contains triangle 8^2 + 15^2 = 17^2
19/18: minor semitone: less stable, due approx: (513:512)*(256/243) by
http://en.wikipedia.org/wiki/Eratosthenes
------------so-far-my-experience-on-horns--------------------------
23: mockery (trying the 23th partial on my horn caused an algospasm)
/bach_tunings/topicId_unknown.html#32
http://en.wikipedia.org/wiki/Momus

Further judgement results barely from passive listening experience:
29: almost indistinguishable from diagonal 20:21:29 20^2+21^2=29^2
31: little tiny: http://en.wikipedia.org/wiki/Mersenne_prime
37: smiles friendly fresh 12:35:37 triangle 12^2 + 35^2 = 37^2
41: Werckmeister, JSB & Co. 9:40:41 triangle 3^4 + 40^2 = 41^2
/bach_tunings/
&ct.....
.......
......
.....
....
...
..
.
Quest:
Who else in that group here has developed an
compareable catalouge item by item?

Anyhow:
Have a lot of fun in finding yours own personal mapping
when exploring to sound out the partials by blowing
or at least by checking out the above scala file

A.S.

🔗Dave Keenan <d.keenan@bigpond.net.au>

Here's the simplification you're after.

First a recap. The Perry harmonic complexity or Perry complexity (PC)
of a ratio n/d was defined by Robin Perry as the sum of the PC's of
the numerator and denominator. PC(n/d) = PC(n) + PC(d) where n and d
are whole numbers.

The PC's of whole numbers were then defined using an iterative
procedure up the whole numbers where PC(1) = 0 and PC(n+f) = PC(n) +
1, for all f which are factors of n (including the factors 1 and n),
provided that n+f had not already been assigned a (lower) PC.

It was clearly inconvenient to first have to find the PC's of all
whole numbers less than n in order to find PC(n). Robin asked if
anyone could offer a simplification.

Carl Lumma was on the right track when he suggested it would require
the prime factorisation of the ratio.

It turns out to be a simple a weighted sum of the absolute values of
the prime exponents where the weights are the PC's of the primes.

Of course that means you still have to iterate up the primes to get
the prime PC's. I note that PC(p) = PC(p-1) + 1 where p is prime.

However the first 5 weights (PC's of primes up to 11) are very easy to
remember since they are just the prime indexes pi(p).
PC(2) = 1
PC(3) = 2
PC(5) = 3
PC(7) = 4
PC(11)= 5

-- Dave Keenan

🔗Robin Perry <jinto83@yahoo.com>

Thank you very much, Dave. That is simple, elegant, brilliant. I'm
still looking at patterns in the process to refine the definition
for primes higher than 11.

Regards,

Robin

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...>
wrote:
>
> Here's the simplification you're after.
>
> First a recap. The Perry harmonic complexity or Perry complexity
(PC)
> of a ratio n/d was defined by Robin Perry as the sum of the PC's of
> the numerator and denominator. PC(n/d) = PC(n) + PC(d) where n and
d
> are whole numbers.
>
> The PC's of whole numbers were then defined using an iterative
> procedure up the whole numbers where PC(1) = 0 and PC(n+f) = PC(n)
+
> 1, for all f which are factors of n (including the factors 1 and
n),
> provided that n+f had not already been assigned a (lower) PC.
>
> It was clearly inconvenient to first have to find the PC's of all
> whole numbers less than n in order to find PC(n). Robin asked if
> anyone could offer a simplification.
>
> Carl Lumma was on the right track when he suggested it would
require
> the prime factorisation of the ratio.
>
> It turns out to be a simple a weighted sum of the absolute values
of
> the prime exponents where the weights are the PC's of the primes.
>
> Of course that means you still have to iterate up the primes to get
> the prime PC's. I note that PC(p) = PC(p-1) + 1 where p is prime.
>
> However the first 5 weights (PC's of primes up to 11) are very
easy to
> remember since they are just the prime indexes pi(p).
> PC(2) = 1
> PC(3) = 2
> PC(5) = 3
> PC(7) = 4
> PC(11)= 5
>
> -- Dave Keenan
>

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

🔗Robin Perry <jinto83@yahoo.com>

Hi,

Manuel has incorporated this attribute into the latest version of
Scala. I just wanted to let you know and to thank everyone who
contributed to this discussion.

Regards,

Robin