Three consecutive chords in a chain

[Paul Erlich:]*
"Many other theorists, including Schoenberg, have explained the diatonic
scale as arising from three consecutive triads in a chain of fifths. This
is another reversal of historical facts, as the chords were constructed
from the scale, and not the other way around. Von Hoerner constructed a
scale from three consecutive 7-limit tetrads in a chain of fifths, using
31-tone equal temperament. The structure of this scale is too bizarre for
it to function as a melodic entity."

How about taking the triads of: 4/3, 5/3, 1/1, 5/4, 3/2, 15/8, 9/8, and
recasting them as tetrads: 16/15, 4/3, 8/5, 1/1, 5/4, 3/2, 15/8, 75/64,
45/32, 225/128, and then 'ignore' the 225/224 (16/15, 4/3, 8/5, 1/1, 5/4,
3/2, 15/8, 7/6, 7/5, 7/4) and the 385/384:

1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
1/1, 12/11, 7/6, 5/4, 21/16, 7/5, 3/2, 18/11, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 6/5, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6, 2/1
1/1, 16/15, 9/8, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8, 2/1
1/1, 21/20, 9/8, 6/5, 21/16, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 25/14, 40/21, 2/1
1/1, 16/15, 7/6, 5/4, 4/3, 10/7, 14/9, 5/3, 16/9, 15/8, 2/1
1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 11/9, 4/3, 10/7, 32/21, 8/5, 12/7, 11/6, 2/1
1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 15/8, 2/1
1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1

I've used this with a variety n-tETs** to generate 10-note (decatonic***)
scales where:

16/15 4/3 8/5 2/1
1/1 5/4 3/2 15/8
[28/15] 7/6 7/5 7/4

is recast as:

[(log16-log15)*(12/log2)]*n/12, etc., etc.

Dan

*This quote is taken from Paul Erlich's "Tuning, Tonality, and
Twenty-Two-Tone Temperament" @:
<http://www-math.cudenver.edu/~jstarret/Question1.html>

**For example, in one of my pieces ("At a Day Job," during the bridge
before the first chorus, and during the 'tag'), a fuzz bass at LLsLsLLLsL
in 17-tET (0, 141, 282, 353, 494, 565, 706, 847, 988, 1059, 1200) carries
the main (melodic) line...

***If n=22 here, I believe this would be what Paul Erlich ["Tuning,
Tonality, and Twenty-Two-Tone Temperament"] refers to as an "Alternate
Pentachordal Major."

Dan Stearns wrote:

>[Paul Erlich:]*
>"Many other theorists, including Schoenberg, have explained the diatonic
>scale as arising from three consecutive triads in a chain of fifths. This
>is another reversal of historical facts, as the chords were constructed
>from the scale, and not the other way around. Von Hoerner constructed a
>scale from three consecutive 7-limit tetrads in a chain of fifths, using
>31-tone equal temperament. The structure of this scale is too bizarre for
>it to function as a melodic entity."
>
>
>How about taking the triads of: 4/3, 5/3, 1/1, 5/4, 3/2, 15/8, 9/8, and
>recasting them as tetrads: 16/15, 4/3, 8/5, 1/1, 5/4, 3/2, 15/8, 75/64,
>45/32, 225/128, and then 'ignore' the 225/224 (16/15, 4/3, 8/5, 1/1, 5/4,
>3/2, 15/8, 7/6, 7/5, 7/4) and the 385/384:
>
>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
>1/1, 12/11, 7/6, 5/4, 21/16, 7/5, 3/2, 18/11, 7/4, 15/8, 2/1
>1/1, 16/15, 8/7, 6/5, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6, 2/1
>1/1, 16/15, 9/8, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8, 2/1
>1/1, 21/20, 9/8, 6/5, 21/16, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 25/14, 40/21, 2/1
>1/1, 16/15, 7/6, 5/4, 4/3, 10/7, 14/9, 5/3, 16/9, 15/8, 2/1
>1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 7/4, 15/8, 2/1
>1/1, 16/15, 8/7, 11/9, 4/3, 10/7, 32/21, 8/5, 12/7, 11/6, 2/1
>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 15/8, 2/1
>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
>
>
>I've used this with a variety n-tETs** to generate 10-note (decatonic***)
>scales where:
>
>16/15 4/3 8/5 2/1
>1/1 5/4 3/2 15/8
>[28/15] 7/6 7/5 7/4
>
>is recast as:
>
>[(log16-log15)*(12/log2)]*n/12, etc., etc.
>
>
>Dan
>
>*This quote is taken from Paul Erlich's "Tuning, Tonality, and
>Twenty-Two-Tone Temperament" @:
><http://www-math.cudenver.edu/~jstarret/Question1.html>
>
>**For example, in one of my pieces ("At a Day Job," during the bridge
>before the first chorus, and during the 'tag'), a fuzz bass at LLsLsLLLsL
>in 17-tET (0, 141, 282, 353, 494, 565, 706, 847, 988, 1059, 1200) carries
>the main (melodic) line...
>
>***If n=22 here, I believe this would be what Paul Erlich ["Tuning,
>Tonality, and Twenty-Two-Tone Temperament"] refers to as an "Alternate
>Pentachordal Major."

I would _really_ love to understand what you're doing here, but I can't
follow you at all. Could you _please_ take us slowly, step by step, holding
our hand, through what you did?

[Paul Erlich:]
I would _really_ love to understand what you're doing here, but I can't
follow you at all.

I took the three consecutive triads of F, C, and G, constructed with a
major and minor third (@: 1/1*5/4*6/5) and separated by a fifth (the top
note):

4/3 5/3 2/1
1/1 5/4 3/2
3/2 15/8 9/8,

and made them into three consecutive tetrads; constructed + 5/4 + 6/5 +
5/4, and separated by the top note, a major seventh:

16/15, 4/3, 8/5, 2/1
1/1, 5/4, 3/2, 15/8
15/8, 75/64, 45/32, 225/128

Then I 'ignored' [...subtracted it from the 15/8, 75/64, 45/32, 225/128...]
the 225/224:

16/15, 4/3, 8/5, 2/1
1/1, 5/4, 3/2, 15/8
[28/15], 7/6, 7/5, 7/4

Then I used this scale:

1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1

as a generating (I.) scale. In the ('modal') example I gave:

1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8,
1/1, 12/11, 7/6, 5/4, 21/16, 7/5, 3/2, 18/11, 7/4, 15/8,
1/1, 16/15, 8/7, 6/5, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6,
1/1, 16/15, 9/8, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8,
1/1, 21/20, 9/8, 6/5, 21/16, 7/5, 3/2, 8/5, 7/4, 15/8,
1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 25/14, 40/21,
1/1, 16/15, 7/6, 5/4, 4/3, 10/7, 14/9, 5/3, 16/9, 15/8,
1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 7/4, 15/8,
1/1, 16/15, 8/7, 11/9, 4/3, 10/7, 32/21, 8/5, 12/7, 11/6,
1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 15/8,
1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8,

I once again 'ignored' the 225/224's, as well as the 385/384's...

What I primarily use this sort of a process for, is to recast various equal
divisions of the octave (n-tET's) into the 'generating scale' (here, that
'generating scale' would obviously be the three tetrad, 10-note scale,
outlined above). I do this by 'turning' the ratios into equally divided
scale steps (i.e., "n"-specific whole numbers):

[(log16-log15)*(12/log2)]*n/12
[(log4-log3)*(12/log2)]*n/12
[(log8-log5)*(12/log2)]*n/12
[(log1-log1)*(12/log2)]*n/12
[(log5-log4)*(12/log2)]*n/12
[(log3-log2)*(12/log2)]*n/12
[(log15-log8)*(12/log2)]*n/12
[(log7-log6)*(12/log2)]*n/12
[(log7-log5)*(12/log2)]*n/12
[(log7-log4)*(12/log2)]*n/12

Obviously the closer the n-tET is to the total number of scale degrees, the
more you run the chance of some pretty radical 'fits...' but at the actual
point of making music, I personally try to let my artistic intuitions and
intentions (And the endless repetitions of good old empirical trial and
error!) be the final arbiter of (these scales, pretty radical 'fits...",
etc., etc., etc.) musical 'worth.'

Dan

Earlier today I wrote in a reply to Paul Erlich I wrote...

>Then I used this scale:
>
>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
>
>as a generating (I.) scale. In the ('modal') example I gave:
>
>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8
>1/1, 12/11, 7/6, 5/4, 21/16, 7/5, 3/2, 18/11, 7/4, 15/8
>1/1, 16/15, 8/7, 6/5, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6
>1/1, 16/15, 9/8, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8
>1/1, 21/20, 9/8, 6/5, 21/16, 7/5, 3/2, 8/5, 7/4, 15/8
>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 25/14, 40/21
>1/1, 16/15, 7/6, 5/4, 4/3, 10/7, 14/9, 5/3, 16/9, 15/8
>1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 7/4, 15/8
>1/1, 16/15, 8/7, 11/9, 4/3, 10/7, 32/21, 8/5, 12/7, 11/6
>1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 15/8
>1/1, 16/15, 7/6, 5/4, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8
>
>I once again 'ignored' the 225/224's, as well as the 385/384's

The sixth degree of this example should have read:

1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 16/9, 40/21, 2/1

not:

1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 32/21, 5/3, 25/14, 40/21, 2/1

The 'modal' permutations of 3+4+3+3+2+3+3+4+3+4 (@: 0, 116, 271, 387, 503,
581, 697, 813, 968, 1084, 1200 in 31-tET) would be the first 10-note scales
[in an equal division of the octave] to give distinct
approximations/representations of all of these ratios.

Dan