Talk:Otonality and utonality

Definition Unclear - All Otonalities also Utonalities, and Vice-Versa?

"An Otonality is a collection of pitches which can be expressed in ratios that have equal denominators... Similarly, the ratios of an Utonality share the same numerator."

Can't any set of pitches meet either of these criteria, by multiplying their denominators/numerators respectively by each other? For example, the following pitches are given as an example of an Otonality, because they can be expressed with equal denominators:

1/1, 5/4, 3/2 -> 4/4, 5/4, 6/4

However, they can also be expressed with the same numerator, making them an Utonality:

1/1, 5/4, 3/2 -> 15/15, 15/12, 15/10

81.159.41.182 (talk) 17:57, 17 December 2009 (UTC)

The thing is, the numbers that Partch uses are odd-numbers up to and including 11. So these common numbers are limited to 1, 5, 3, 7, 9, 11 (arranged in thirds as in Partch's book). Also, the basis of the theory is small number ratios. 15/15, 15/12, 15/10 don't share a shared so-called Numerary Nexus that is one of those 11-limit numbers.

Otonality

Otonality on 1/1 (i.e., Odentities 1, 5, 3, 7, 9, 11 above the Numerary Nexus 1) 1/1 (1/1) - 5/4 - 3/2 - 7/4 - 9/8 - 11/8 (i.e., the Harmonic or Overtone Series generated upward from 1/1)

Otonality on 8/5 (i.e., Odentities 1, 5, 3, 7, 9, 11 above the Numerary Nexus 5) 8/5 - 5/5 (1/1) - 6/5 - 7/5 - 9/5 - 11/10 (i.e., the Harmonic or Overtone Series generated upward from 8/5)

Otonality on 4/3 (i.e., Odentities 1, 5, 3, 7, 9, 11 above the Numerary Nexus 3) 4/3 - 5/3 - 3/3 (1/1) - 7/6 - 3/2 - 11/6 (i.e., the Harmonic or Overtone Series generated upward from 4/3)

Otonality on 8/7 (i.e., Odentities 1, 5, 3, 7, 9, 11 above the Numerary Nexus 7) 8/7 - 10/7 - 12/7 - 7/7 (1/1) - 9/7 - 11/7 (i.e., the Harmonic or Overtone Series generated upward from 8/7)

Otonality on 16/9 (i.e., Odentities 1, 5, 3, 7, 9, 11 above the Numerary Nexus 9) 16/9 - 10/9 - 12/9 - 7/9 - 9/9 (1/1) - 11/9 (i.e., the Harmonic or Overtone Series generated upward from 16/9)

Otonality on 16/11 (i.e., Odentities 1, 5, 3, 7, 9, 11 above the Numerary Nexus 11) 16/11 - 20/11 - 12/11 - 14/11 - 18/11 - 11/11 (1/1) (i.e., the Harmonic or Overtone Series generated upward from 16/11)

Utonality

Utonality on 1/1 (i.e., Udentities 1, 5, 3, 7, 9, 11 below the Numerary Nexus 1) 1/1 - 8/5 - 4/3 - 8/7 - 16/9 - 16/11 (i.e., the Subharmonic or Undertone Series generated downward from 1/1)

Utonality on 5/4 (i.e., Udentities 1, 5, 3, 7, 9, 11 below the Numerary Nexus 5) 5/4 - 5/5 (1/1) - 5/3 - 10/7 - 10/9 - 20/11 (i.e., the Subharmonic or Undertone Series generated downward from 5/4)

Utonality on 3/2 (i.e., Udentities 1, 5, 3, 7, 9, 11 below the Numerary Nexus 3) 3/2 - 6/5 - 3/3 (1/1) - 12/7 - 12/9 - 12/11 (i.e., the Subharmonic or Undertone Series generated downward from 3/2)

Utonality on 7/4 (i.e., Udentities 1, 5, 3, 7, 9, 11 below the Numerary Nexus 7) 7/4 - 7/5 - 7/6 - 7/7 (1/1) - 14/9 - 14/11 (i.e., the Subharmonic or Undertone Series generated downward from 7/4)

Utonality on 9/8 (i.e., Udentities 1, 5, 3, 7, 9, 11 below the Numerary Nexus 9) 9/8 - 9/5 - 3/2 - 9/7 - 9/9 (1/1) - 18/11 (i.e., the Subharmonic or Undertone Series generated downward from 9/8)

Utonality on 11/8 (i.e., Udentities 1, 5, 3, 7, 9, 11 below the Numerary Nexus 11) 11/8 - 11/10 - 11/6 - 11/7 - 11/9 - 11/11 (1/1) (i.e., the Subharmonic or Undertone Series generated downward from 11/8)

Mbase1235 (talk) 04:46, 15 February 2012 (UTC)

I believe the problem the first guy had was with the quoted portion, which doesn't say anything about restricting to 1, 5, 3, 7, 9, 11. Rather than explaining it here, why don't you correct the article? I get what he's saying in that any just intonation chord of any complexity that you can imagine occurs naturally somewhere high up in the harmonic series. That's just basic math. So in a way that makes every shape look Otonal-ish since it can be expressed as a collection of numerators over 1. (And every possible just chord shape occurs somewhere deep in the undertone series too, at least mathematically, even though nature doesn't actually produce the undertone series in resonating bodies.) The problem is making the article describe Partch's definitions of Otonality and Utonality in such a way that clearly restricts his meaning. The current article text doesn't. Oh, also, your objection to the first guy's 15 in the numerator as not being in the 11 limit is another problem for the article. One meaning of limit is "involving primes no higher than" (which 15 fits, because it's 3*5), but if Partch's meaning is "using no odd numbers higher than", then 15 doesn't fit (because it's higher than 11), and somebody needs to explain that and cite it. 108.60.216.202 (talk) 06:30, 17 May 2015 (UTC)

Capitalization

Is there a reason the terms "Otonality" and "Utonality" should be capitalized in the article as they currently are? Why did Partch capitalize them? Hyacinth (talk) 09:28, 14 May 2010 (UTC)