The elusive second skhisma

Help me out; does anyone know the simplest 5-limit ratio, other than the
32805/32768 skhisma, smaller than 2 cents? Correct me if I'm wrong, but
doesn't there have to be one, since there's never a "best" rational
approximation to an irrational number (there's always a better one)?

Stay tuned,
Keenan P.

> [Keenan Pepper, TD 699.2]
>
> Help me out; does anyone know the simplest 5-limit ratio,
> other than the 32805/32768 skhisma, smaller than 2 cents?
> Correct me if I'm wrong, but doesn't there have to be one,
> since there's never a "best" rational approximation to an
> irrational number (there's always a better one)?

Keenan,

I reproduce here a lattice with the arbitrary exponent-limits of
3^(-15...15) * 5^(-6...6), showing the cent-values of intervals
which are smaller than 100 cents (it's from a Tuning List post
of last year). See my comments below.

power of 5
6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6

15 --- ---75--- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
14 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
13 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
12 --- --- --- --- --- ---23--- --- ---65--- --- ---
| | | | | | | | | | | | |
11 ---53--- --- ---94--- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
10 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
9 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
8 --- --- --- --- --- 2--- --- ---43--- --- ---84---
| | | | | | | | | | | | |
7 32--- --- ---73--- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
6 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
5 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
4 --- --- --- --- --- --- ---22--- --- ---63--- ---
| | | | | | | | | | | | |
3 --- ---51--- --- ---92--- --- --- --- --- --- ---
p | | | | | | | | | | | | |
o 2 --- --- --- --- --- --- --- --- --- --- --- ---
w | | | | | | | | | | | | |
e 1 --- --- --- --- --- --- --- --- --- --- --- ---
r | | | | | | | | | | | | |
0 --- --- --- --- --- --- 0--- --- ---41--- --- ---82
o | | | | | | | | | | | | |
f -1 ---30--- --- ---71--- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
3 -2 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
-3 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
-4 --- --- --- --- --- --- --- ---20--- --- ---61---
| | | | | | | | | | | | |
-5 8--- --- ---49--- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
-6 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
-7 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
-8 --- --- --- --- --- --- --- --- --- ---39--- ---
| | | | | | | | | | | | |
-9 --- ---28--- --- ---69--- --- --- --- --- --- ---
| | | | | | | | | | | | |
-10 98--- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
-11 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
-12 --- --- --- --- --- --- --- --- ---18--- --- ---59
| | | | | | | | | | | | |
-13 --- 6--- --- ---47--- --- ---88--- --- --- --- ---
| | | | | | | | | | | | |
-14 --- --- --- --- --- --- --- --- --- --- --- ---
| | | | | | | | | | | | |
-15 --- --- --- --- --- --- --- --- --- --- --- ---

To answer your first question,

> does anyone know the simplest 5-limit ratio, other than
> the 32805/32768 skhisma, smaller than 2 cents?

I'm assuming here that by 'simpler' you mean what we usually
mean: smaller-integer ratio terms. You can see clearly from
the lattice that there is no *simpler* ratio smaller than
2 cents; the skhisma [ = 3^8 * 5^1 ] is the smallest.

If you were to continue the lattice beyond my boundaries, with
the numbers in both the numerators and denominators becoming
progressively larger the farther you go from the central 1/1,
you would eventually find smaller intervals (i.e., 'smaller than
2 cents').

Concerning your second question,

> doesn't there have to be one [a simpler ratio?],
> since there's never a "best" rational approximation to an
> irrational number (there's always a better one)?

It's not clear to me what you meant by 'one', because

1) it seems to refer to 'a simpler ratio', which
2) doesn't make sense in the light of your next comment.

A better rational approximation will always be the result of
using larger numbers in the comparison. The 'largeness'
can be the result of either higher prime-factors or higher
exponents, or both. In this particular case, we're expressly
limiting our prime-factors to the 5-limit, so the more accurate
rational approximation will always be the result of higher
exponents.

My last comment is that I don't see what an 'irrational' number
has to do with anything. The skhisma, when it refers specifically
to 32805/32768 [ = 3^8 * 5^1 ] as it generally does, is
a precisely determined *rational* interval designating the
difference between the JI 'major 3rd' [ 5/4 = ~386 cents ]
and the Pythagorean 'diminished 4th' [ 8192/6561 = 2^12 * 3^-8
= ~384 cents ]. The '2 cents' is merely the *approximate*
cent-value of the 32805/32768 skhisma, which can be expressed
more accurately as ~1.9537208 cents.

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
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