Yahoo Tuning Groups Ultimate Backup tuning what appear to be fractions separated by colons

> [Paul Erlich, TD 489.]
> for the third time I suggest the possibility of
> 1/24:1/19:1/16 for your locked major triad with high third.

> [Jerry Eskelin, TD 490.11]
> For now the second time in a post to you (and the third
> if you count another to someone else) I am requesting a
> clarification of what appear to be fractions separated by
> colons. Please unlock the code. For example, how does
> the expression 4/5 relate to the expression 4:5? Also,
> what does 1/x represent? I really would like to know what
> you are saying here.

I'll try to clarify here, Jerry.

The numbers that 'appear to be fractions separated by colons'
are simply the *utonal* proportions for the tones in Paul's
chord.

The term 'utonal' was coined by Partch, and is short for
'under-tonal'. The definition in my online Tuning Dictionary
http://www.ixpres.com/interval/dict/utonal.htm
is the one Partch gives in his book:

> [Partch 1974, p 75]
> Otonality:
> one of those tonalities expressed by the under numbers
> [denominators] of ratios having a Numerary Nexus
> [in the numerator] -- in current musical theory,
> "minor" tonality.

(It wasn't until I looked at this (and Partch's definition
for the counterpart, otonality) this morning that I
realized that, without my editorial addition of '[in
the numerator]', Partch's definition is so inadequate
as to be misleading.)

And of course you also need the definition for
'numerary nexus':

> [Partch 1974, p 75]
> Numerary Nexus:
> the number common to all identities in the ratios of
> one tonality -- the common anchor; the characteristic
> of a series of ratios that determines them as a tonality.
> In the 8/5 Otonality the Numerary Nexus is 5, as seen in
> the sequence of the six Odentities:
>
> 11/10 [= 11/5 in Partch's theory]
> 9/5
> 7/5
> 6/5
> 1/1 (= 5/5)
> 8/5
>
> In the 5/4 Utonality the Numerary Nexus is also 5, as seen
> in the sequence of the six Udentities:
>
> 20/11 [= 5/11]
> 10/9 [= 5/9]
> 10/7 [= 5/7]
> 5/3
> 1/1 (= 5/5)
> 5/4

I'll work thru an obvious example here to help you understand.

You're already clear on how 'otonality' works: simple
integers between colons shows the proportions in a chord
as they are related otonally; for example, the typical
just 'major' chord is 4:5:6, meaning that the middle note
is a 4:5 above the 'root' (represented by the 4), and the
top note is a 4:6 [= 2:3] above the 'root' and a 5:6 above
the middle note.

And you're probably already familiar with the way a
typical just 'minor' chord is expressed with this kind
of proportional notation: 10:12:15, meaning that the
middle note is a 10:12 [= 5:6] above the 'root' (represented
here by the 10), and the top note is a 10:15 [= 2:3] above
the 'root' and a 12:15 [= 4:5] above the middle note.

These proportions are otonal because they all assume
a denominator for the ratios which is the same (or an
'octave' equivalent, i.e., a doubled or quadrupled number).

But this particular minor chord can be expressed more
simply (that is, with smaller numbers) by assuming a
*numerator* which is the same for all the ratios, and
looking at the proportions in the denominators. Here's
how it's done:

First, we know that the 10:12:15 proportion is an otonal
one. So we can simply put '1' as our denominator for all
three ratios, and that will still give us the otonal
proportions, because their Numerary Nexus (in the
denominator) is 1.

Now, to convert to utonal proportions, we have to find
the LCM (least common multiple - the lowest number which
can be divided into all of the intended factors) of
our three proportions. The LCM of 10, 12, and 15 is 60.

So if we multiply each of the numerators by the required
number to obtain the LCM, which in this case is 6, 5, and 4
respectively, and multiply the denominators by the same
numbers, we obtain a set of ratios which now have a
Numerary Nexus in the *numerator* and thus spell out a
utonal proportion in the denominators:

10 12 15
-- -- --
1 1 1

60 60 60
-- -- --
6 5 4

Just as we 'throw out' the Numerary Nexus in our description
of otonal proportions, we may simply reduce the numerator
here to 1, and we get 1/6:1/5:1/4 as our utonal description
of the 10:12:15 'minor' chord.

So now let's reverse that process to determine the otonal
proportions in Paul's 1/24:1/19:1/16 chord.

The LCM of the denominators is 912, so we make this our
numerator:

912 912 912
--- --- ---
24 19 16

Regardless of what number actually *is* the numerator,
as long they're all the same, the denominators will
spell out the utonal proportion.

Now we simply divide the ratios as fractions (which is
what they look like here anyway), and we obtain the
otonal proportion 38:48:57. That's a description of
Paul's chord that you should easily recognize. We see
that it's made up of a middle note which is 38:48 [= 19:24]
above the 'root' (represented by 38), and a top note
which is 38:57 [= 2:3] above the 'root' and 48:57 [= 16:19]
above the middle note.

What Paul presented to you is simply the utonal counterpart
(which is actually a 'major' chord) to the typical nondecimal
'minor' chord, which is expressed otonally as 16:24:19.

You should be able to see the analogy between these two
19-limit chords and the two 5-limit chords with which I
began my explanation, the otonal 'major' chord of 4:5:6 and
utonal 'minor' chord of 1/4:1/5:1/6. But the relationships
are reversed, because this 19-limit *odentity* (that is, otonal
identity, ratio 19/16) sounds like a 'minor 3rd', while this
19-limit *udentity* (utonal identity, ratio 24/19) sounds like
a 'major 3rd'.

The main point is that *any chord may be written as
either an otonal or a utonal proportion*. Usually
one of them gives smaller numbers and that is the
one normally used. In some cases, for example a
typical 5-limit 'minor 7th' tetrad, either description
gives the same numbers, so either one may be used.
Also, there may be other reasons why the version with
higher numbers would be chosen.

> [Jerry]
> Can someone whose math training ended at college algebra
> have a fighting chance of understanding? I hope this is
> simply a matter of form rather than of depth.

There's certainly hope for you.

My math training ended with difficult struggle to make
a barely-passing grade of 'D' in 11th-grade Algebra 2,
and I've been able to make it this far in tuning theory.

Regarding your questions about 27/16 and 45/32:

Daniel Wolf [TD 489.7] and Paul Erlich [TD 489.17] have
both given responses that should help clarify, but if
you're still mystified, it might help for you to understand
prime-factor notation of ratios. Try the original paper
I wrote outlining my theory:
http://www.ixpres.com/interval/monzo/article/article.htm

If we're assuming 'octave'-equivalence, which is *almost*
always the case, prime-factoring eliminates all the
'unnecessary' powers of 2, which often are the cause for
such large numbers.

45/32, eliminating all powers of 2, reduces to 9*5
= 3^2 * 5^1.

27/16, eliminating all powers of 2, reduces to 3^3.

(The caret sign [^] means 'raised to the power of'.)

Prime-factoring the ratios is how we lattice theorists
determine where to put the ratios on our lattice diagrams.
And the lattice diagrams make it very easy to see the
numerical relationships that may not be immediately
apparent from an examination of the numbers themselves.

Hope that helps.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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what appear to be fractions separated by colons

🔗Joe Monzo <monz@xxxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>