Another "standard" unit: the iota

Hello, there, everyone, and here's a quick example of how to convert
between cents and yet another "standard" tuning unit: the iota, at
least unless or until someone comes up with a better name.

To illustrate the conversion process, and also provide a 21st-century
version of the "Hammond Organ" theme often brought up in "What is JI?"
threads, here's a rational tuning with values in cents provided by
Scala, and values in iotas added by "yours truly," as the Monz would
say, along with a mixed set of interval descriptions in part from
Scala, and in part added along with the iotas to fit with this unit of
measure. Note that "near-equal" means "close to some multiple of an
even 100 iotas":

|
New rational tuning of "Hammond organ type"
0: 1/1 0.000000 0.0000 unison, perfect prime
1: 25/24 70.67245 100.1193 near-equal diatonic semitone
2: 243/224 140.9491 199.6779
3: 26/23 212.2534 300.6922
4: 20/17 281.3584 398.5909
5: 38/31 352.4776 499.3430
6: 23/18 424.3645 601.1828
7: 141/106 493.9573 699.7725
8: 18/13 563.3825 798.1250
9: 13/9 636.6179 901.8750
10: 212/141 706.0432 1000.2275
11: 36/23 775.6360 1098.8172
12: 31/19 847.5230 1200.6570
13: 17/10 918.6421 1301.4091
14: 23/13 987.7471 1399.3078
15: 448/243 1059.051 1500.3221
16: 48/25 1129.328 1599.8807 near-equal major seventh
17: 2/1 1200.000 1700.0000 octave

A cent is approximately 1732:1731 (thank you, Manuel Op de Coul and
Scala), while an iota is approximately 2453:2452. To convert from one
unit to the other, here are some equivalents:

1 cent = 1.41666666... (precisely 1-5/12) iotas
1 iota = ~0.70588235294 (precisely 12/17) cents
100 cents = 141.666666... (precisely 141-2/3) iotas
100 iotas = ~70.588235294 (precisely 70-10/17) cents
1700 iotas = 1200 cents = 2:1 octave

As with any proposed term for a measure of intonation, I would find it
prudent to inquire if there is any established use of "iota" -- if so,
some other term would be better for this unit.

Curiously, while I didn't find any such established usage in Scala or
Monz's dictionary, I did find an item on the World Wide Web in which
someone suggested using the term "iota" for a semitone of 100 cents.
However, this might already be nicely covered by the Monzian "semitone,"
as in "100 iotas is around 0.7059 semitones."

Most appreciatively,

Margo Schulter
mschulter@value.net

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

> Hello, there, everyone, and here's a quick example of how to convert
> between cents and yet another "standard" tuning unit: the iota, at
> least unless or until someone comes up with a better name.

Instead of a better name, what about a better unit? I'd strongly recommend ditching the iota in favor of a shisma of size 2^(1/612),
which can be approximated in rational terms as 883/882, 1767/1765,
or 6184/6177. Now we have

1 cent = 0.51 sk
1 iota = 0.36 sk
1 fifth = 357.997 sk
1 fourth = 254.003 sk
1 major third = 197.02 sk
1 step of 34-et = 18 sk
1 step of 68-wt = 9 sk
1 step of 72-et = 8.5 sk

The advantages for someone interested in Pythagorean tuning, the
17/34/68 complex, and compatibility with 12 and 72 tunings should be clear.

Margo wrote:
>As with any proposed term for a measure of intonation, I would find it
>prudent to inquire if there is any established use of "iota" -- if so,
>some other term would be better for this unit.

Lindley and Turner-Smith use this letter as a symbol for "the
smallest interval representing a regular pitch-class relation
J, [...], is the least positive residue of the equivalence class J."
So for example the iota of 1/125 is 128/125, or the iota of
3^12 is the Pythagorean comma, etc.

But as this is a mapping and not a unit, I see no cause for
confusion and think the iota can be a useful interval measure.

Manuel