another possible candidate for "wierisma" ;)

2,954,312,706,550,833,698,643/2,951,479,051,793,528,258,560
~ 1.661325 cents

This is a quinary schisma, based on 5/4 compared to a chain of 45 perfect
fifths. It's slightly smaller than the traditional (and much more practical)
32805/32768 schisma ~ 1.953721 cents.

I'm coming up with these strange intervals using the argument that if X number
of fifths is a good approximant of a certain prime interval such as 5/4, 7/4,
11/8 etc., then X + 53 or X - 53 fifths should work too, and possibly better.
I'm using 53-tet because it has such a great fifth, and it's easier to handle
than say, 665-tet. That's how I came up with the inferior septimal schisma based
on 39 fifths, and I'm currently calculating the other prime intervals up to
31/16. (For 53-tone, it's not really possible to produce a good periodicity-type
just scale using primes higher than 7; at that point one had better use a
symmetrical Farey approximation, and that would result in irregular step sizes
and problematic intervals such as the 17/16 augmented prime/chromatic step.)

~Danny~

p.s. I just found a good tredecimal schisma: 3489660928/3486784401, which is
13/8 compared to a chain of 20 perfect fifths, or approximately 1.427644 cents.

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Hello, there, Danny, and I'd agree that given your interest in
accurate 7-limit approximations within a 53-note Pythagorean "loop,"
as Joe Pehrson has neatly suggested this kind of circulating but not
precisely closed scheme, the named "wierisma" might neatly be applied
to the difference between 45 fifths up and a pure 5:4, or 44 fourths
down and a pure 6:5.

wierisma M3 -- from chain of 45 fifths up
2954312706550833698643:2361183241434822606848 = ~387.975 cents

wierisma m3 -- from chain of 44 pure fourths up
1180591620717411303424:984770902183611232881 = ~313.980 cents

wierisma -- difference between wierisma third and pure 5:4 or 6:5
2954312706550833698643:2951479051793528258560 = ~1.661 cents

As it happens, these thirds are closer to pure 5-based ratios than the
usual schisma thirds at 8192:6561 (~384.36 cents) and 19683:16384
(~317.60 cents) by a monzisma, the difference between the Pythagorean
"tricomma" at ~70.38 cents and 25:24 at ~70.67 cents (the latter
interval defining the difference between 5:4 and 6:5).

We could also define the impurity of "wierisma thirds" as equal to the
difference of the comma of Mercator (53 pure fifths vs 31 pure
octaves) at about 3.62 cents, and the 3-5 schisma at about 1.95 cents.

Having read your recent posts, I see that you are seeking 7-limit
approximations, including factors of 5 (2-3-5-7-9), as well as some
higher prime factors.

Thus given your evident goal of finding the most accurate 7-limit
approximatiosn within a 53-note loop based on pure fifths, applying
the name "wierisma" to the thirds you suggest, closer to pure than
usual schismic thirds, nicely fits this outlook.

In contrast, my interest in the tricomma approximations of what I might
call "7-flavor" intervals (2-3-7-9) such as 7:6, 9:7, and 7:4 focuses
on a different kind of setting where the idea is to combine regular
Pythagorean intervals (e.g. 81:64 and 32:27 thirds) and 7-flavor
intervals with close approximations of 17:14 and 21:17.

The tricomma tuning thus reflects a compromise and balance of sorts
between 7-flavor intervals and sonorities (e.g. 12:14:18:21) and
14:17:21.

As it happens, the tricomma tuning includes a few wierisma thirds,
although these are incidental to the main purposes of tuning. Using an
asterisk (*) to show the tricomma at around 70.38 cents, we have for
example the wierisma major third F4-G#*4 (schisma m3 plus tricomma);
and the wierisma minor third G#*4-C5 (schisma M3 less tricomma).

By the way, on your approximation of 13:8 as 20 pure fifths up, this
can easily occur in a regular 24-note Pythagorean tuning (two 12-note
chains a Pythagorean comma apart) in an idiom like this, with the
carat sign (^) showing a note raised by a Pythagorean comma of
531441:524288 (~23.46 cents):

C#^4 D4
F3 E3 D3

Here the near-13:8 interval F3-C#^4 is followed by the near-12:7 major
sixth E3-C#^4 (a comma wider than the regular Pythagorean sixth
E3-C#4), resolving to the octave D3-D4.

This is a neat coloristic touch, and one might wonder if
flexible-pitch ensembles in 14th-century Italy or France might have
arrived at something like this progression and intonation at certain
cadences where a vertical augmented fifth seems not unlikely, and the
major sixth before an octave might have been given a size of somewhere
around 12:7. Anyway, I like it very much for related 21st-century
styles.

To conclude, thank you for your latest messages -- and I agree that
recognizing "wierisma" thirds of 45 fifths up and 44 fifths down would
neatly fit your outlook.

Most appreciatively,

Margo Schulter
mschulter@value.net

Thanks for your input, Margo, though I'm not really trying to get an interval
named after me (yet) ;)

As an afterthought, I decided that it probably won't be necessary to use both 5-
and 7-based schismatic adjustments in my 53-tone scale, but it would work in
larger scales. 5:4 is much stronger than 7:4. But 13:8 is strong; it equals 37
steps in 53-tet, or a spiral of 20 fifths, like I stated earlier.

Now a mix of 5:4 and 7:4 would work in classical tuning, where the 5:4 ratio
equals a chain of four fifths.

~Danny~ (no more big intervals; my posts are already pretty boring)

>From: "M. Schulter" >Reply-To: tuning@yahoogroups.com>To:
tuning@yahoogroups.com>Subject: [tuning] Re: another possible candidate for
"wierisma" ;)>Date: Tue, 3 Sep 2002 15:15:14 -0700 (PDT)>>Hello, there, Danny,
and I'd agree that given your interest in>accurate 7-limit approximations within
a 53-note Pythagorean "loop,">as Joe Pehrson has neatly suggested this kind of
circulating but not>precisely closed scheme, the named "wierisma" might neatly
be applied>to the difference between 45 fifths up and a pure 5:4, or 44 fourths
>down and a pure 6:5.

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