Re: Iotas and "centisteps" generally

Hello, there, Gene and everyone, and please let me explain how iotas
as "relative 17-EDO cents" -- a nice expression -- might be especially
useful in devising or analyzing _unequal_ 17-note tunings or
temperaments not too far from 17-EDO, just as cents can reflect some
of the parameters of a 12-note well-temperament or the like.

Specifically, let's consider the kinds of just or near-just thirds
likely to appear in a 12-note system or 17-note system.

Here I'll borrow a bit from Manuel Op de Coul and Dave Keenan, taking
full responsibility of all of my modifications or "creative
additions," giving some names and ratios for major and minor thirds
along with measurements in usual cents and relative 17-EDO cents or
iotas. These measurements may suggest which types of thirds are most
congenial to each type of system:

---------------------------------------------------------------
ratio name cents iotas
---------------------------------------------------------------
5/4 pental/classic major third 386.31 547.28
63/50 vicesimoprimal major third 400.11 566.82
24/19 smaller undevicesimal major third 404.44 572.96
81/64 ternary/Pythagorean major third 407.82 577.75
19/15 larger undevicesimal major third 409.24 579.76
33/26 tridecimal major third 412.75 584.72
14/11 undecimal major third 417.51 591.47
51/40 septendecimal major third 420.60 595.85
23/18 vicesimotertial major third 424.36 601.18
9/7 septimal major third 435.08 616.37
---------------------------------------------------------------
7/6 septimal minor third 266.87 378.07
27/23 vicesimotertial minor third 277.59 393.25
20/17 septendecimal minor third 281.36 398.59
33/28 undecimal minor third 284.45 402.97
13/11 tridecimal minor third 289.21 409.71
45/38 smaller undevicesimal minor third 292.71 414.67
32/27 ternary/Pythagorean minor third 294.13 416.69
19/16 larger undevicesimal minor third 297.51 421.48
25/21 vicesimoprimal minor third 301.84 427.61
6/5 pental/classic minor third 315.64 447.16
---------------------------------------------------------------

Here I've used "ternary" and "pental" as alternative descriptions for
Scala's "Pythagorean" and "classic," referring to intervals based
respectively on ratios of 3 and 5.

Let's assume that a "well-tempered or close to well-tempered" system
tends to have thirds staying within about 1/5 or 1/6 of a step of the
average sizes (i.e. the sizes in 12-EDO or 17-EDO).

Our table then suggests that in a 12-note system, with average sizes
for major and minor thirds of 400 cents and 300 cents respectively, we
might expect ratios ranging from around pental/classic (5/4, 6/5) to
ternary/Pythagorean (81/64, 32/27) or undecimal (14/11, 33/28).

In a 17-note system, with average sizes for major and minor thirds at
600 iotas and 400 iotas, we might likewise expect ratios from around
Pythagorean or undecimal to something close to septimal (9/7, 7/6).

To sum up, these relative cents could be especially useful for unequal
but "not-too-far-from-equal" systems in seeing how far certain ratios
of interest are from average sizes. With well-temperaments in either
12 notes or 17 notes, for example, some arrangements may be more
"balanced" (with intervals rather close to the average), and others
more "unbalanced" (with greater variations from the average).

Please feel most welcome to ask more questions, and I suspect that if
you're puzzled, others might be also, so this is an opportunity for
communication I much value.

Most appreciatively,

Margo Schulter
mschulter@value.net