Yahoo Tuning Groups Ultimate Backup tuning Re: Iota and "centisteps" generally

Hello, there, Robert and Gene.

First, Gene, please let me emphasize that a unit like the iota (1/1700
octave) certainly isn't meant to replace many other useful units of
measure, including integer ratios and cents as well as your 3-5
schismas at 32805:32768 (~1.95 cents), which Manuel Op de Coul's
Scala, for example, uses as one choice.

For music using something like 612-EDO, or in various kinds of tunings
and styles where the 3-5 schisma has a strong relevance to what is
happening musically, your unit seems at least one obvious choice.

The iota, or 17-centistep as I'll explain below, is likely to be used
in a rather different musical setting, where semitones are not too far
from 1/17 octave, or major thirds from 6/17 octave, and so forth.

Thus I would see each standard as potentially "the best" where
applicable. Often the musical assumptions are so different that any
"comparison" seems to me a bit strange, although I would grant that
everything is connected to everything, and that unlikely connections
can sometimes be the most interesting.

Robert, your presentation on different bases and "centlike"
measurements suggests to me a term for a very useful unit you
demonstrated: maybe the "centistep" equal to 1/100n-octave in any
n-tET: thus 1700 centisteps to an octave in 17-EDO, or 3100 in 31-EDO,
etc.

The iota is simply a special case of this, and the advantages you note
and illustrate can apply not only to a 17-note system (equal or
unequal) where each step is some kind of "semitone" (or "thirdtone"),
but to a 31-note system with "fifthtone" steps, and so on. These units
might be of special interest where steps are slightly unequal, since
they show how intervals vary from the mean or "average" size.

Thus cents would be 12-centisteps, a special case of your more general
concept, just as iotas are 17-centisteps, and your 31-centisteps
follow the same pattern.

Incidentally, your heptadecimal or base-17 notation (if the first term
is correct, by analogy with "hexadecimal") shows a curious property of
17: as you explain, it goes

1 2 3 4 5 6 7 8 9 a b c d e f g

Here we have the same letters as the seven note names of traditional
European notation as described by people like Guido d'Arezzo
(c. 1030). This isn't the only note-naming system, of course, in the
early European tradition, since some systems continue h i k, etc., in
higher octaves (the distinct letter "j" not yet being standard in
medieval Latin, so that "h i k ..." would continue the series).

Anyway, centisteps are an interesting way of representing a range of
EDO steps, and in showing how far a given ratio or interval varies
from the average in a not-too-far-from-equal system of a given size.
They should be a very nice feature in FTS.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗genewardsmith <genewardsmith@juno.com>

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> For music using something like 612-EDO, or in various kinds of tunings
> and styles where the 3-5 schisma has a strong relevance to what is
> happening musically, your unit seems at least one obvious choice.

If I understand where you are coming from, which I may not, then my suggestion is not that it would be better for me, but that it would be better for you. I think at least from what you've said before that
Pythagorean tunings are of interest, so let's compare iotas and shismas in a 17-context in connection to pure fifths.

To the nearest iota, a fifth is 994 iotas. If we use this in connection with a Pythagorean scale, we get

0--288--576--706--994--1282--1570--1700

The correct values to two decimal places are

0--288.87--577.75--705.56--994.44--1283.31--1572.18--1700

On the other hand, doing the same thing with 612 gives us

0--104--208--254--358--462--566-612

with correct values to two decimal places of

0--103.99--207.99--254.00--358.00--461.99--565.99--612

Despite its smaller step size, the 612 is vastly more accurate. If we compare how well they represent 17-et intervals, the answer of course is equally well, so there seems no advantage to using iotas. Further, if we want to relate iotas to 12-et, we find that it has a step size of 141.67 iotas, which is hardly as convenient as saying 51 sk. I guess I am baffled by what possible advantage you (and Robert, it seems) see in iotas.

> The iota, or 17-centistep as I'll explain below, is likely to be used
> in a rather different musical setting, where semitones are not too far
> from 1/17 octave, or major thirds from 6/17 octave, and so forth.

Which shismas can do better than iotas! The iota major third is a not particularly convenient 527.28; shismas are more accurate, as well as simpler, in calling this 197.02, which is easier to take to be 197 than 527 is to be a major third when using iotas. Again, when asking how well each one deals with 1/17 of an octave or 6/17 of an octave, the answer is equally well. Iotas have it as 100 and 600, whereas shismas have it as 36 and 216; both are exact.

> Thus I would see each standard as potentially "the best" where
> applicable. Often the musical assumptions are so different that any
> "comparison" seems to me a bit strange, although I would grant that
> everything is connected to everything, and that unlikely connections
> can sometimes be the most interesting.

So far as I can see the two systems are doing the same thing in the same way and are directly comparable. I don't see what advantage there is to iotas, and so far, you haven't suggested one. When measuring *relative* error, the idea of relative cents, which would be iotas in the particular case of 17-et, makes sense to me, but that does not seem to be what you are doing.

🔗M. Schulter <MSCHULTER@VALUE.NET>

Hello, there, Gene and everyone.

Please let me try better to explain, with some practical examples, why
I find the iota (1/1700 octave, or 1/100 of a 17-EDO step) a useful
unit specifically for a 17-note system with steps not too far from
17-EDO, that is, where every step can be used as some kind of
"diatonic semitone," with a size of at least 33:32 (~53.27 cents),
let's say, or maybe 1/22 octave (~54.55 cents).

While I can't speak for Robert Walker, or Manuel Op de Coul, I can
explain why I like the iota in this specific setting.

What I suspect is that we are approaching things on quite different
wavelengths, both musically and conceptually, and maybe the best thing
I can do is to provide examples of the kind of intervals I'm looking
for in a 17-note thirdtone system of the kind which iotas address, and
how their measure in iotas could be helpful in looking at such a
system.

A 17-note Pythagorean tuning, for which you'd prefer 612-EDO schismas as a
unit of measure and I would mostly use ratios or cents, is a different
type of system, because it represents an MOS where each whole-tone is
divided into limma-comma-limma (e.g. F-Gb-F#-G) with sizes of roughly
4/9-1/9-4/9 tone, something quite far from a division into three
more-or-less thirdtone-like steps.

For Pythagorean, I'd agree that one interesting type of unit of
measure could be based on some EDO with a very close approximation to
a pure 3:2, with 612 as one of these choices.

Anyway, I find iotas appealing because in a thirdtone type of 17-note
system, they can show how some ratios for various intervals compare
with the _average_ sizes. For example, major thirds will average at
600 iotas (~423.53 cents), minor thirds at 400 iotas (~282.35 cents),
and neutral thirds at 500 iotas (~352.94 cents).

In a typical 17-note system of this type for which iotas were
designed, a lot of usual just ratios (or close tempered
approximations) will tend to be not too far from these average
sizes. Some examples may help to illustrate this point, with interval
sizes given in iotas, cents, and also your 612-EDO units of
approximately 1.96 cents (with 1 cent equal to precisely 0.51 of these
units):

Major thirds (Average size 600 iotas or 423.53 cents)
------------
14:11 591.47 iotas 417.51 cents 212.93 612-EDO
23:18 601.18 iotas 424.36 cents 216.43 612-EDO
9:7 616.37 iotas 435.08 cents 221.89 612-EDO

Minor thirds (Average size 400 iotas or 282.35 cents)
------------
7:6 378.07 iotas 266.87 cents 136.10 612-EDO
20:17 398.59 iotas 281.36 cents 143.49 612-EDO
13:11 409.71 iotas 289.21 cents 147.50 612-EDO

Neutral or semi-neutral thirds (Average size 500 iotas or 352.94 cents)
------------------------------
17:14 476.18 iotas 336.13 cents 171.42 612-EDO
11:9 492.16 iotas 347.41 cents 177.18 612-EDO
27:22 502.28 iotas 354.55 cents 180.82 612-EDO
16:13 509.25 iotas 359.47 cents 183.33 612-EDO
21:17 518.25 iotas 365.83 cents 186.57 612-EDO

A point I might add is that units such as iotas -- or cents, as far as
I can tell -- are intended to measure intervals in terms of some equal
division of the octave, rather than to approximate just ratios as
evenly as possible. Of course, with lots of units per octave, the
maximum deviation of half a unit won't be too large.

The advantage of iotas is that they show in what I find an intuitive
way how close a given interval is to the average size: for example, a
major third at 23:18 is very close to the average, while 14:11 is a
bit smaller than average and 9:7 considerably larger.

That's the purpose of iotas: to show how just ratios or other interval
sizes compare with the average sizes for a 17-note thirdtone system,
these averages representing the sizes in 17-EDO.

Of course, we could also keep track of these things with a more
familiar unit such as cents, and the intonational artistry of a master
such as George Secor with his 17-tone well-temperament (17-WT) is
impressive in any unit of measure.

Most appreciatively,

Margo Schulter
mschulter@value.net