A variety of major seconds

From: Gerald Eskelin
>
>>It would be very interesting to know (as I mentioned earlier) to know if
>>experienced ears can consistently hear 8:9 as scale steps 1-2 and 9:10 as
>>scale steps 2-3.

John Link questioned:
>
> I don't see any reason why that would be the case, since the tuning of the
> tones of the scale steps depends upon the harmonization. You seem to have
> fallen prey to what you identified in your book "Lies My Music Teacher Told
> Me" as lie #6: A scale is a series of eight fixed pitches.
>
> Consider the following progression in the key of C major:
>
> Cmaj7 Dmin7 Emin7 Fmaj7
>
> S: B C D E
>
> A: G A B C
>
> T: E F G A
>
> B: C D E F
>
> Note that both the soparano(S) and the baritone(B) sing the sequence C, D,
> E. I expect that they would tune the steps differently. I would expect the
> soprano to sing C, D, E as a 9/8 followed by a 10/9, and the baritione to
> sing a 10/9 followed by a 9/8.

I appreciate that "dynamic tuning" goes on during harmonic changes (which is
of course the whole point of Lie #6), however the context of my comment
(quoted above) had to do with isolated major seconds. Since, as I reported,
hearing an isolated minor third seems to evoke an imagined "phantom root"
completing a "mistakenly perceived" major triad, I wondered if an isolated
9:10 major third would also evoke a "phantom root" which would logically
place the 9:10 major second where it "naturally" occurs in the partials
series over a fundamental.

In any case, your post appears to contain an interesting observation, albeit
misapplied to mine. Okay, I'll bite. Why would the soprano and baritone sing
those particular major seconds in that context?

Jerry

>From: "Gerald Eskelin" <stg3music@earthlink.net>

>Why would the soprano and baritone sing
>those particular major seconds in that context?

Consider the lattice of tones in which going up to the left means
multiplying by 5/4 and going up to the right means multiplying by 3/2:

B D

E G

A C

D F

Note that there are two different D's: the one in the lower left corner is
tuned 10/9 relative to C while the one in the upper right is tuned 9/8. The
former will be used in Dmin7 while the latter will be used in Emin7 (as
well as G7). I then works out, by multiplying the 3/2's and the 5/4's (and
then multipying or dividing by the appropriate power of 2 to bring the
results between 1 and 2), that the progression I gave in my previous post
would be tuned as follows:

Cmaj7 Dmin7 Emin7 Fmaj7

S: B 15/8 C 1/1 D 9/8 E 5/4

A: G 3/2 A 5/3 B 15/8 C 1/1

T: E 5/4 F 4/3 G 3/2 A 5/3

B: C 1/1 D 10/9 E 5/4 F 4/3

So the soprano sings C, D, E as 1/1, 9/8, 5/4, implying a major second of
9/8 followed by 10/9, while the baritone sings C, D, E as 1/1, 10/9, 5/4,
implying a major second of 10/9 followed by 9/8.

I hope that the lattice is helpful to you not only in this example but also
in understanding ratios involving higher numbers. I find it indispensible.

John Link

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