Ennealimmal[45]

The standard 45-et val, h45 = [45, 71, 104, 126], has 81/80,
2401/2400 and 4375/4374 as a comma basis. The Fokker block for this is

[1, 49/48, 36/35, 21/20, 200/189, 27/25, 54/49, 10/9, 245/216, 8/7,
7/6, 25/21, 6/5, 49/40, 100/81, 63/50, 9/7, 35/27, 324/245, 4/3,
49/36, 243/175,7/5, 10/7, 350/243, 72/49, 3/2, 245/162, 54/35, 14/9,
100/63, 81/50, 80/49, 5/3, 42/25, 12/7, 7/4, 432/245, 9/5, 49/27,
50/27, 189/100, 40/21, 35/18, 96/49]

which in terms of the 612 et, is

[0, 18, 25, 43, 50, 68, 86, 93, 111, 118, 136, 154, 161, 179, 186,
204,222, 229, 247, 254, 272, 290, 297, 315, 322, 340, 358, 365, 383,
390, 408, 426, 433, 451, 458, 476, 494, 501, 519, 526, 544, 562, 569,
587, 594]

All of this is nicely regular, but adjusting by an 81/80 has some
interest but not as much as some smaller chromas.

However, 16/15 as a chroma also works in a sense; we have instead of
the standard val ennealimmal ^ 16/15 = [45, 73, 107, 128] as a val.
The corresponding Fokker block for [16/15, 2401/2400, 4375/4374] is

[1, 36/35, 200/189, 49/48, 21/20, 27/25, 10/9, 8/7, 54/49, 245/216,
7/6, 6/5, 216/175, 25/21, 60/49, 63/50, 35/27, 4/3, 9/7, 324/245,
49/36, 7/5, 36/25, 25/18, 10/7, 72/49, 245/162, 14/9, 3/2, 54/35,
100/63, 49/30, 42/25, 175/108, 5/3, 12/7, 432/245, 49/27, 7/4, 9/5,
50/27, 40/21, 96/49, 189/100, 35/18]

which in terms of the 612-et is

[0, 25, 50, 18, 43, 68, 93, 118, 86, 111, 136, 161, 186, 154, 179,
204,229, 254, 222, 247, 272, 297, 322, 290, 315, 340, 365, 390, 358,
383, 408, 433, 458, 426, 451, 476, 501, 526, 494, 519, 544, 569, 594,
562, 587]

Here things are not ordered by increasing size; in terms of Paul
theory this is not a valid val. However, if we reorder the above, we
see that the two scales, in 612-et, are identical. Ordering it as
above leads to 16/15 as a chroma in terms of circulating chords, and
we could, if we chose, approach the scale in this way.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> The standard 45-et val, h45 = [45, 71, 104, 126], has 81/80,
> 2401/2400 and 4375/4374 as a comma basis. The Fokker block for this
is
>
> [1, 49/48, 36/35, 21/20, 200/189, 27/25, 54/49, 10/9, 245/216, 8/7,
> 7/6, 25/21, 6/5, 49/40, 100/81, 63/50, 9/7, 35/27, 324/245, 4/3,
> 49/36, 243/175,7/5, 10/7, 350/243, 72/49, 3/2, 245/162, 54/35, 14/9,
> 100/63, 81/50, 80/49, 5/3, 42/25, 12/7, 7/4, 432/245, 9/5, 49/27,
> 50/27, 189/100, 40/21, 35/18, 96/49]
>
> which in terms of the 612 et, is
>
> [0, 18, 25, 43, 50, 68, 86, 93, 111, 118, 136, 154, 161, 179, 186,
> 204,222, 229, 247, 254, 272, 290, 297, 315, 322, 340, 358, 365, 383,
> 390, 408, 426, 433, 451, 458, 476, 494, 501, 519, 526, 544, 562,
569,
> 587, 594]
>
> All of this is nicely regular, but adjusting by an 81/80 has some
> interest but not as much as some smaller chromas.
>
> However, 16/15 as a chroma also works in a sense; we have instead of
> the standard val ennealimmal ^ 16/15 = [45, 73, 107, 128] as a val.
> The corresponding Fokker block for [16/15, 2401/2400, 4375/4374] is
>
> [1, 36/35, 200/189, 49/48, 21/20, 27/25, 10/9, 8/7, 54/49, 245/216,
> 7/6, 6/5, 216/175, 25/21, 60/49, 63/50, 35/27, 4/3, 9/7, 324/245,
> 49/36, 7/5, 36/25, 25/18, 10/7, 72/49, 245/162, 14/9, 3/2, 54/35,
> 100/63, 49/30, 42/25, 175/108, 5/3, 12/7, 432/245, 49/27, 7/4, 9/5,
> 50/27, 40/21, 96/49, 189/100, 35/18]
>
> which in terms of the 612-et is
>
> [0, 25, 50, 18, 43, 68, 93, 118, 86, 111, 136, 161, 186, 154, 179,
> 204,229, 254, 222, 247, 272, 297, 322, 290, 315, 340, 365, 390, 358,
> 383, 408, 433, 458, 426, 451, 476, 501, 526, 494, 519, 544, 569,
594,
> 562, 587]
>
> Here things are not ordered by increasing size; in terms of Paul
> theory this is not a valid val.

can you fill me in on what i need to know about paul theory to
understand this last assertion?