The issue of additive vs. logarithmic hearing

   Consider that the harmonic series itself has the property of the HZ value between any two consecutive tones' being the same and the fact a perfect minor chord IE 1 1.2 1.5 has the property of the ratio of the HZ values being in proportion  (1.5-1.2)/(1.2-1) = .3/.2 = 3/2!

   I thought to myself "could the human ear really be working in an additive fashion and only use logarithms as a coping device to help it deal with the critical band?

  Observe the following scale (created in a matter of second: may well match an existing historical scale):
   1
   1.1111
   1.2222
   1.3333
   1.4444
        (1.5555: omitted to preserve critical band)
   1.6666
        (1.7777: omitted to preserve critical band)
   1.8888
   2.0000   (2/1 period)

    Notice how each tone = the last tone plus 1.1111...thus splitting the difference between consecutive tones into equal HZ ratios just like the harmonic series does.  Actually...it pretty much is a "stretched harmonic series".

.    And notice...how the above scale seems to sound bizarrely ordered and consonant despite not being created in any sort of variable^x type of fashion?

   A counter-example would be more interesting: why (if at all) do you think the ear looks for multiplicative and not additive symmetry (or do you think it looks both?

-Michael

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:

>   Observe the following scale (created in a matter of second: may
well match an existing historical scale):
>    1
>    1.1111
>    1.2222
>    1.3333
>    1.4444
>         (1.5555: omitted to preserve critical band)
>    1.6666
>         (1.7777: omitted to preserve critical band)
>    1.8888
>    2.0000   (2/1 period)
>
>     Notice how each tone = the last tone plus 1.1111...thus
splitting the difference between consecutive tones into equal HZ
ratios just like the harmonic series does.  Actually...it pretty much
is a "stretched harmonic series".

No, it is not stretched, if the decimal expansions are supposed to be
of the form 1.1111..., 1.2222... and so on, it is the same as
the harmonic series scale

9:10:11:12:13:15:17:18, also same as

1/1
10/9
11/9
4/3
13/9
5/3
17/9
2/1

Kalle Aho

Correction, the scale below is formally known as a harmonic series segment harmonics 9 to 18 with a few notes taken out...so it has been created before minus the fact I took out tones at strategic places to control beating amount.

--- On Fri, 5/15/09, Michael Sheiman <djtrancendance@...> wrote:

From: Michael Sheiman <djtrancendance@...>
Subject: [tuning] The issue of additive vs. logarithmic hearing
To: tuning@yahoogroups.com
Date: Friday, May 15, 2009, 9:54 AM

   Consider that the harmonic series itself has the property of the HZ value between any two consecutive tones' being the same and the fact a perfect minor chord IE 1 1.2 1.5 has the property of the ratio of the HZ values being in proportion  (1.5-1.2)/(1. 2-1) = .3/.2 = 3/2!

   I thought to myself "could the human ear really be working in an additive fashion and only use logarithms as a coping device to help it deal with the critical band?

  Observe the following scale (created in a matter of second: may well match an existing historical scale):
   1
   1.1111
   1.2222
   1.3333
   1.4444
        (1.5555: omitted to preserve critical band)
   1.6666
        (1.7777:
omitted to preserve critical band)
   1.8888
   2.0000   (2/1 period)

    Notice how each tone = the last tone plus 1.1111...thus splitting the difference between consecutive tones into equal HZ ratios just like the harmonic series does.  Actually...it pretty much is a "stretched harmonic series".

.    And notice...how the above scale seems to sound bizarrely ordered and consonant despite not being created in any sort of variable^x type of fashion?

   A counter-example would be more interesting: why (if at all) do you think the ear looks for multiplicative and not additive symmetry (or do you think it looks both?

-Michael

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote:
>
>    Consider that the harmonic series itself has the property of the HZ value between any two consecutive tones' being the same and the fact a perfect minor chord IE 1 1.2 1.5 has the property of the ratio of the HZ values being in proportion  (1.5-1.2)/(1.2-1) = .3/.2 = 3/2!
>
>    I thought to myself "could the human ear really be working in an additive fashion and only use logarithms as a coping device to help it deal with the critical band?
>
>   Observe the following scale (created in a matter of second: may well match an existing historical scale):
>    1
>    1.1111
>    1.2222
>    1.3333
>    1.4444
>         (1.5555: omitted to preserve critical band)
>    1.6666
>         (1.7777: omitted to preserve critical band)
>    1.8888
>    2.0000   (2/1 period)
>
>     Notice how each tone = the last tone plus 1.1111...thus splitting the difference between consecutive tones into equal HZ ratios just like the harmonic series does.  Actually...it pretty much is a "stretched harmonic series".
>
> .    And notice...how the above scale seems to sound bizarrely ordered and consonant despite not being created in any sort of variable^x type of fashion?
>
>    A counter-example would be more interesting: why (if at all) do you think the ear looks for multiplicative and not additive symmetry (or do you think it looks both?
>
> -Michael

In my opinion a good idea is to get some huge sheets of paper and tape them down so that they cover an entire tabletop or a good size floor space, then write everything out by hand- chart out the harmonic partials, see where they line up, imagine where they're coming from and going to, and so on.

Like this...
600
500
400
300
200
100

would be the first six harmonic partials of a tone at 100 Hz,(you're looking at it like a picture so the higher ones are higher) and

750
625
500
375
250
125

the 1st six of a tone at 125 Hz (aka 1 and 1.25, or 1/1 and 5/4)

then you see, hey the fourth partial of the second tone is the same as the fifth of the first. Ta-da, a ratio gives you quantity and position of the first coincident harmonic partials as well, and tells you all the rest to infinity if you're up to reckoning them. Irrationals? No problem, you can write them out to whatever, say 4 decimal places, and compare them with the harmonic series, tells you all kinds of things too, like where the beats are going to be most audible.

Continuing with this example it is clear that without other information, for all you know these two lists of number could actually just be decribing higher harmonics of a tone at 25 Hz, or 5Hz, or 1Hz (and right there you have the clue to the scale you just posted and what you were saying about it).

Does this seem too elementary? I don't think so at all. It's more like the truly advanced approach. And a monochord- just rip the frets out of a cheap acoustic guitar like I did and you've got a fine monochord. The back edge of a heavy butter-knife makes a good and surprisingly precise slide.

Well that's my opinion.

     BTW, to correct myself my example is simply a higher part harmonic series with certain tones removed.
  It's obviously nothing advanced...and a side point would be that you can add irrational numbers to each other IE 1.05946 and keep adding the decimal part it to itself to get 1.11892,  1.17838, etc. ....and it will still sound good.  1.1111 was a lousy number to pick for my example as it simply equals 10/9...the 10th harmonic of the harmonic series.  That really would be a stretched harmonic series, gets close to the point I am aiming for about the possibility of ear viewing tones as additive.  Or, to make it more obvious, try something like the scale 1, 1.2, 1.5, 1.7, 2.0 where there exists 3/2 proportions between the differences between consecutive tones instead of 1/1 proportions.

>"And a monochord- just rip the frets out of a cheap acoustic guitar like I did and you've got a fine monochord."
???
Huh?  Here I have no clue what you are talking about....unless it's just intentional sarcasm.

-Michael