15 vs. 21

Considering two spellings of the chord e4-Bb4-f5 (great chord: tonic,
dominant, and subdominant combined!), the spellings being

7-10-15, or
10-14-21

They seem to have roughly equal dissonance levels (intervals 10/7,
3/2, 15/7; vs 7/5, 3/2, 21/10).

I wonder which spelling is more appropriate. In the latter, you have
the 'tautology' 7/5 x 3/2 =21/10, and if that is the correct spelling,
why? thanks! KElly

Hi Kelly,

On Thu, 09 Mar 2006, traktus5 wrote:
>
> Considering two spellings of the chord e4-Bb4-f5 (great chord: tonic,
> dominant, and subdominant combined!), the spellings being
>
> 7-10-15, or
> 10-14-21
>
> They seem to have roughly equal dissonance levels (intervals 10/7,
> 3/2, 15/7; vs 7/5, 3/2, 21/10).
>
> I wonder which spelling is more appropriate. ...

Traditional JI theory says that lower numbers in
the reduced fractions mean lower dissonance.

So, a rough guide would be the maximum of all the
numbers used in the ratios between chord tones.
For 7-10-15, that number is 15; for 10-14-21, that
number is 21. This calculation favours the first
form.

Another approach would be to average all the
numerators and denominators in the ratios between
chord tones. The first chord has three ratios, whose
six integers sum to 44, average 44/6 = 22/3 = 7 1/3.
The second chord has three ratios, whose six integers
sum to 48, average 48/6 = 8. Again, the first chord
has an edge.

A more psychoacoustically based approach would be
to calculate some measure of dissonance for each
inter-dyad interval in the chord (eg Helmholtz'
roughness or perhaps the inverse of HE), then sum
them. Here's a crude roughness measure for the
interval m/n, where m > n:
R = m /min {n, (m-n)}

(Please don't ask me to justify this!!! It just seems
right ...)

Chord 7-10-15, interval ratios 10/7, 3/2, 15/7, has R =
10/3 + 3/1 + 15/7 = (70+63+45)/21 = 178/21 = 8 10/21.

Chord 10-14-21, interval ratios 7/5, 3/2, 21/10, has R =
7/2 + 3/1 + 21/10 = (35+30+21)/10 = 86/10 = 8.6.
This is just a little larger than that for the first chord,
which is just under 8.5. Again, the first chord wins.

> ... In the latter, you have
> the 'tautology' 7/5 x 3/2 =21/10, ...

Irrelevant! In the former, you have the similar tautology:
10/7 x 3/2 = 30/14 = 15/7. These ARE tautolgies simply
because they MUST be true - the three ratios within a triad
ALWAYS obey such a relation.

> ... and if that is the correct spelling,
> why? thanks! KElly

What is "correct" in music depends entirely
on your objectives, I believe.

Regards,
Yahya

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hi Yayha - this is good information, as I'm studying roughness.
There's stuff here I've never seen textbooks at a college library.
One initial question regarding > > 7-10-15, or> > 10-14-21 is:

Isn't there a sort of 'lowness', or simplicity, about the second
spelling, specifically, that the product of the inner intervals
equals the outer interval, *already reduced*? (7/5 x 3/2 = 21/10,
as opposed to 10/7 x 3/2 = 30/14=15/7?) Is that ever a
consideration?

Another possible advantage of the second spelling is -- assuming for
the sake of argument that the 'seven-ness', from the tritone, is an
important part of this chord's character, regardless of how the
chord is spelled, then the second spelling, if you factor the
primary chord tones, is more 'loaded' with sevens (7-10-15 factors
to 7, 2x5, and 3x5, whereas 10-14-21 factors to 2x5, 2x7, and 3x7).

Finally, looking at the measures of dissonance and roughness you
calculated, they are actaully quite close, aren't they? The same
species, but with slightly different values, perhaps? Assuming
that, then I believe the second spelling would also be considered
more 'interesting', because it is basically just as low as the first
spelling in roughness and dissonance, but, at the same time, reaches
higher in the series (21 vs 15), with all the interesting
consequences that has (such as, according to Paul, multiple
allusions to similiarly spelled nearby chords) -- in addition to its
greater affinity with the number 7.

What do you think? thanks, Kelly

> > They seem to have roughly equal dissonance levels (intervals
10/7,
> > 3/2, 15/7; vs 7/5, 3/2, 21/10).
> >
> > I wonder which spelling is more appropriate. ...
>
> Traditional JI theory says that lower numbers in
> the reduced fractions mean lower dissonance.
>
> So, a rough guide would be the maximum of all the
> numbers used in the ratios between chord tones.
> For 7-10-15, that number is 15; for 10-14-21, that
> number is 21. This calculation favours the first
> form.
>
> Another approach would be to average all the
> numerators and denominators in the ratios between
> chord tones. The first chord has three ratios, whose
> six integers sum to 44, average 44/6 = 22/3 = 7 1/3.
> The second chord has three ratios, whose six integers
> sum to 48, average 48/6 = 8. Again, the first chord
> has an edge.
>
> A more psychoacoustically based approach would be
> to calculate some measure of dissonance for each
> inter-dyad interval in the chord (eg Helmholtz'
> roughness or perhaps the inverse of HE), then sum
> them. Here's a crude roughness measure for the
> interval m/n, where m > n:
> R = m /min {n, (m-n)}
>
> (Please don't ask me to justify this!!! It just seems
> right ...)
>
> Chord 7-10-15, interval ratios 10/7, 3/2, 15/7, has R =
> 10/3 + 3/1 + 15/7 = (70+63+45)/21 = 178/21 = 8 10/21.
>
> Chord 10-14-21, interval ratios 7/5, 3/2, 21/10, has R =
> 7/2 + 3/1 + 21/10 = (35+30+21)/10 = 86/10 = 8.6.
> This is just a little larger than that for the first chord,
> which is just under 8.5. Again, the first chord wins.
>
>
> > ... In the latter, you have
> > the 'tautology' 7/5 x 3/2 =21/10, ...
>
> Irrelevant! In the former, you have the similar tautology:
> 10/7 x 3/2 = 30/14 = 15/7. These ARE tautolgies simply
> because they MUST be true - the three ratios within a triad
> ALWAYS obey such a relation.
>
>
> > ... and if that is the correct spelling,
> > why? thanks! KElly
>
> What is "correct" in music depends entirely
> on your objectives, I believe.
>
> Regards,
> Yahya
>
>
> --
> No virus found in this outgoing message.
> Checked by AVG Free Edition.
> Version: 7.1.375 / Virus Database: 268.2.1/277 - Release Date:
8/3/06
>

Hi Kelly,

You wrote:
>
> hi Yayha - this is good information, as I'm studying roughness.
> There's stuff here I've never seen textbooks at a college library.
> One initial question regarding > > 7-10-15, or> > 10-14-21 is:
>
> Isn't there a sort of 'lowness', or simplicity, about the second
> spelling, specifically, that the product of the inner intervals
> equals the outer interval, *already reduced*? (7/5 x 3/2 = 21/10,
> as opposed to 10/7 x 3/2 = 30/14=15/7?) Is that ever a
> consideration?

The intervals of the second chord, *before reduction*,
are 14/10, 21/14 and 21/10. Wouldn't that also be a
consideration? No, honestly, I don't think it makes a
whit of difference that you need to express these
intervals using compound rather than prime numbers,
since that need only arises from the need to show them
all as integers in the wider context of a chord.

Here's another way of looking at this situation.
Suppose we take the complexity of any exact ratio as
simply the largest integer in it (*). Then:

The *reduced* intervals of the second chord are (in
order of complexity) 3/2, 7/5 and 21/10.

The *reduced* intervals of the first chord are (in
order of complexity) 3/2, 10/7 and 15/7.

Ignoring the 3/2, the complexities of all these align as:
... 7/5 ... 10/7 ... 15/7 ... 21/10 ...
The first chord has the two mid-most values, and the
second chord has the two outermost values, of complexity.

Those complexity numbers would (3,) 7, 10, 15, 21. Their
sums would be 31 for the second chord and 28 for the
first. Again, the second seems more complex. Or is that
"interesting"? ;-)

(*) This is a very over-simplified measure of complexity
or roughness. I recall having some conversation with Paul
Erlich on this and other measures (under what name I
don't recall) on the tuning list in, I think, around October
or Novemeber last year.

> Another possible advantage of the second spelling is -- assuming for
> the sake of argument that the 'seven-ness', from the tritone, is an
> important part of this chord's character, regardless of how the
> chord is spelled, then the second spelling, if you factor the
> primary chord tones, is more 'loaded' with sevens (7-10-15 factors
> to 7, 2x5, and 3x5, whereas 10-14-21 factors to 2x5, 2x7, and 3x7).

That 3x7 certainly acts together with the 2x7 to reinforce
the hearing of a perceived root of 1x7 for the dyad 14-21;
which also coincides with their difference tone. Comparably,
the first chord is more "loaded" with "fiveness". So the main
difference between the two chords may well be that the first
"feels like fives (major thirds)" whilst the second "fells like
sevens (harmonic sevenths)". But to be sure of this, I'd want
to tune up both chords *exactly* (not as 12-EDO approxima-
tions!), and then play them a few times.

> Finally, looking at the measures of dissonance and roughness you
> calculated, they are actaully quite close, aren't they? The same
> species, but with slightly different values, perhaps? Assuming
> that, then I believe the second spelling would also be considered
> more 'interesting', because it is basically just as low as the first
> spelling in roughness and dissonance, but, at the same time, reaches
> higher in the series (21 vs 15), with all the interesting
> consequences that has (such as, according to Paul, multiple
> allusions to similiarly spelled nearby chords) -- in addition to its
> greater affinity with the number 7.

More complex, certainly. More interesting? Depends on what
you're interested in ... ;-)

> What do you think? thanks, Kelly

Regards,
Yahya

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