perceptual gesast (question to Paul)

Hi Paul. Sorry to drag you into the rudiments, but I couldn't find
the answer after studying a Benade text. Concerning the 'acoustical
energy' of maj 7th chord (c-e-g-b), you wrote that the 'primary
difference tones' for the component intervals 5/4, 6/5, 5/4, 3/2,
3/2, 15/8, are 1/4, 1/4, 3/8, 1/2, 5/8, and 7/8. Can you explain
how you obtained the 3/8 and 5/8? thanks, Kelly

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> Hi Paul. Sorry to drag you into the rudiments, but I couldn't find
> the answer after studying a Benade text. Concerning
the 'acoustical
> energy' of maj 7th chord (c-e-g-b), you wrote that the 'primary
> difference tones' for the component intervals 5/4, 6/5, 5/4, 3/2,
> 3/2, 15/8, are 1/4, 1/4, 3/8, 1/2, 5/8, and 7/8. Can you explain
> how you obtained the 3/8 and 5/8? thanks, Kelly

Hi Kelly,

I should have said 'first order difference tones'. At loud volumes,
nonlinearities in the ear will create frequencies that are the
difference of the frequencies of pairs of objective tones. I can
explain this in more detail if you like. Also important are 'second
order difference tones', etc., but I'll ignore those here.

Let's write the major 7th chord as 1/1, 5/4, 3/2, 15/8.

1/1 and 5/4 will create a difference tone whose frequency is 5/4 -
1/1 = 1/4.

5/4 and 3/2 will create a difference tone whose frequency is 3/2 -
5/4 = 1/4.

3/2 and 15/8 will create a difference tone whose frequency is 15/8 -
3/2 = 3/8.

1/1 and 3/2 will create a difference tone whose frequency is 3/2 -
1/1 = 1/2.

5/4 and 15/8 will create a difference tone whose frequency is 15/8 -
5/4 = 5/8.

1/1 and 15/8 will create a difference tone whose frequency is 15/8 -
1/1 = 7/8.

Making sense?

on 6/2/04 8:09 AM, traktus5 <kj4321@hotmail.com> wrote:

> Hi Paul. Sorry to drag you into the rudiments, but I couldn't find
> the answer after studying a Benade text. Concerning the 'acoustical
> energy' of maj 7th chord (c-e-g-b), you wrote that the 'primary
> difference tones' for the component intervals 5/4, 6/5, 5/4, 3/2,
> 3/2, 15/8, are 1/4, 1/4, 3/8, 1/2, 5/8, and 7/8. Can you explain
> how you obtained the 3/8 and 5/8? thanks, Kelly

Paul seems to be off doing something else for the time being, but I had a
few spare moments and figured this out.

The chord could be spelled as:

C : E : G : B
1 : 5/4 : 3/2 : 15/8

Just as the component intervals are the *ratios* of all possible pairs taken
(in order) from this set, the difference tones are the corresponding
*differences* of all possible pairs. Thus:

pair ratio corresponding difference
C:E (5/4)/(1) = 5/4 (5/4)-(1) = 1/4
E:G (3/2)/(5/4) = 6/5 (3/2)-(5/4) = 1/4
G:B (15/8)/(3/2) = 5/4 (15/8)-(3/2) = 3/8
C:G (3/2)/(1) = 3/2 (3/2)-(1) = 1/2
E:B (15/8)/(5/4) = 3/2 (15/8)-(5/4) = 5/8
C:B (15/8)/(1) = 15/8 (15/8)-(1) = 7/8

The difference column matches the ratio column except for the "/" being
changed to a "-".

Does that make sense?

-Kurt

Well I guess Paul and I are on the same wavelength, or even the same wave!
Interesting to see the difference in styles.

-Kurt

It seems easier just to spell your tetrads as 8-10-12-15 and then figure out
the difference tones. being 2,2,3,4,5,7. The system i use to judge the con/dis
factor being adding only the unique numbers
2,3,4,5,7,8,10,12,15= 66

Kurt Bigler wrote:

> on 6/2/04 8:09 AM, traktus5 <kj4321@hotmail.com> wrote:
>
> > Hi Paul. Sorry to drag you into the rudiments, but I couldn't find
> > the answer after studying a Benade text. Concerning the 'acoustical
> > energy' of maj 7th chord (c-e-g-b), you wrote that the 'primary
> > difference tones' for the component intervals 5/4, 6/5, 5/4, 3/2,
> > 3/2, 15/8, are 1/4, 1/4, 3/8, 1/2, 5/8, and 7/8. Can you explain
> > how you obtained the 3/8 and 5/8? thanks, Kelly
>
> Paul seems to be off doing something else for the time being, but I had a
> few spare moments and figured this out.
>
> The chord could be spelled as:
>
> C : E : G : B
> 1 : 5/4 : 3/2 : 15/8
>
> Just as the component intervals are the *ratios* of all possible pairs taken
> (in order) from this set, the difference tones are the corresponding
> *differences* of all possible pairs. Thus:
>
> pair ratio corresponding difference
> C:E (5/4)/(1) = 5/4 (5/4)-(1) = 1/4
> E:G (3/2)/(5/4) = 6/5 (3/2)-(5/4) = 1/4
> G:B (15/8)/(3/2) = 5/4 (15/8)-(3/2) = 3/8
> C:G (3/2)/(1) = 3/2 (3/2)-(1) = 1/2
> E:B (15/8)/(5/4) = 3/2 (15/8)-(5/4) = 5/8
> C:B (15/8)/(1) = 15/8 (15/8)-(1) = 7/8
>
> The difference column matches the ratio column except for the "/" being
> changed to a "-".
>
> Does that make sense?
>
> -Kurt
>
>
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-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Thanks Paul. Also, on some notation I'm unfamiliar with... when you
compared the chords 4:5:6:7 and 1/7, 1/6, 1/5, and 1/4 , what is the
actual chord spelling of the latter? (Are those subharmonics?)

thanks Kurt! Now I understand...Kelly

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 6/2/04 8:09 AM, traktus5 <kj4321@h...> wrote:
>
> > Hi Paul. Sorry to drag you into the rudiments, but I couldn't
find
> > the answer after studying a Benade text. Concerning
the 'acoustical
> > energy' of maj 7th chord (c-e-g-b), you wrote that the 'primary
> > difference tones' for the component intervals 5/4, 6/5, 5/4, 3/2,
> > 3/2, 15/8, are 1/4, 1/4, 3/8, 1/2, 5/8, and 7/8. Can you explain
> > how you obtained the 3/8 and 5/8? thanks, Kelly
>
> Paul seems to be off doing something else for the time being, but I
had a
> few spare moments and figured this out.
>
> The chord could be spelled as:
>
> C : E : G : B
> 1 : 5/4 : 3/2 : 15/8
>
> Just as the component intervals are the *ratios* of all possible
pairs taken
> (in order) from this set, the difference tones are the corresponding
> *differences* of all possible pairs. Thus:
>
> pair ratio corresponding difference
> C:E (5/4)/(1) = 5/4 (5/4)-(1) = 1/4
> E:G (3/2)/(5/4) = 6/5 (3/2)-(5/4) = 1/4
> G:B (15/8)/(3/2) = 5/4 (15/8)-(3/2) = 3/8
> C:G (3/2)/(1) = 3/2 (3/2)-(1) = 1/2
> E:B (15/8)/(5/4) = 3/2 (15/8)-(5/4) = 5/8
> C:B (15/8)/(1) = 15/8 (15/8)-(1) = 7/8
>
> The difference column matches the ratio column except for the "/"
being
> changed to a "-".
>
> Does that make sense?
>
> -Kurt

Hi Kurt/Paul ...I got the math...but wanted to confirm what the
actual difference tone notes are. (I'm comp/performer background).
(I'm assuming each interval is reckoned to it's own harmonic series,
irregardless of the lowest note of the chord?)

If we consider the octave registration of the major 7th chord to be
c4-e4-g4-b4, then, am I indicating (at far right) the correct actual
note which the difference ratios represent? And is this conected
with 'subharmonics', which I am having trouble finding a good
definition of on the web. thanks!
>
> pair ratio corresponding difference (tone)
> C:E (5/4)/(1) = 5/4 (5/4)-(1) = 1/4 (C2)
> E:G (3/2)/(5/4) = 6/5 (3/2)-(5/4) = 1/4 (C2)
> G:B (15/8)/(3/2) = 5/4 (15/8)-(3/2) = 3/8 (G2)
> C:G (3/2)/(1) = 3/2 (3/2)-(1) = 1/2 (C3)
> E:B (15/8)/(5/4) = 3/2 (15/8)-(5/4) = 5/8 (E3)
> C:B (15/8)/(1) = 15/8 (15/8)-(1) = 7/8 (C1)

Kelly

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> Thanks Paul. Also, on some notation I'm unfamiliar with... when
you
> compared the chords 4:5:6:7 and 1/7, 1/6, 1/5, and 1/4 , what is
the
> actual chord spelling of the latter?

I don't know what you mean by "actual chord spelling".

>(Are those subharmonics?)

Yes, the latter chord has been referred to as a "subharmonic"
or "utonal" chord.

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:

> Hi Kurt/Paul ...I got the math...but wanted to confirm what the
> actual difference tone notes are. (I'm comp/performer background).
> (I'm assuming each interval is reckoned to it's own harmonic
series,
> irregardless of the lowest note of the chord?)

I don't know what you mean by that question.

> If we consider the octave registration of the major 7th chord to be
> c4-e4-g4-b4, then, am I indicating (at far right) the correct
actual
> note which the difference ratios represent?

Five out of six. The last one, C1, is wrong, since that would be 1/8
instead of 7/8.

> And is this conected
> with 'subharmonics',

Not directly, no.

> which I am having trouble finding a good
> definition of on the web.

Subharmonics are tones whose *period* (or wavelength, or string
length, or air column length) is an integer multiple of the period
(or wavelength, or string length, or air column length) of a given
tone. Harmonics, by contrast, are tones whose *frequency* is an
integer multiple of the frequency of a given tone.

thanks!
> >
> > pair ratio corresponding difference
(tone)
> > C:E (5/4)/(1) = 5/4 (5/4)-(1) = 1/4 (C2)
> > E:G (3/2)/(5/4) = 6/5 (3/2)-(5/4) = 1/4 (C2)
> > G:B (15/8)/(3/2) = 5/4 (15/8)-(3/2) = 3/8 (G2)
> > C:G (3/2)/(1) = 3/2 (3/2)-(1) = 1/2 (C3)
> > E:B (15/8)/(5/4) = 3/2 (15/8)-(5/4) = 5/8 (E3)
> > C:B (15/8)/(1) = 15/8 (15/8)-(1) = 7/8 (C1)
>
> Kelly

hi Paul - (I'm referring to a tonalsoft excerpt which, according to
Joe's indendation, is from one of your postings, which reads "Our non-
laboratory experiments on the harmonic entropy list seem to
conclusively show that the dissonance of a chord can't be even close
to a function of the dissonances of the constituent intervals. For
example, everyone put the 4:5:6:7 chord near the top of their ranking
of 36 recorded tetrads from least to most dissonant, while everyone
put 1/7:1/6:1/5:1/4 much lower."

4:5:6:7 on the piano would be c-e-g-Bb.

How would I play 1/7, 1/6, 1/5, and 1/4 on the piano?

thanks, Kelly

on 6/11/04 4:35 PM, traktus5 <kj4321@hotmail.com> wrote:

> hi Paul - (I'm referring to a tonalsoft excerpt which, according to
> Joe's indendation, is from one of your postings, which reads "Our non-
> laboratory experiments on the harmonic entropy list seem to
> conclusively show that the dissonance of a chord can't be even close
> to a function of the dissonances of the constituent intervals. For
> example, everyone put the 4:5:6:7 chord near the top of their ranking
> of 36 recorded tetrads from least to most dissonant, while everyone
> put 1/7:1/6:1/5:1/4 much lower."
>
> 4:5:6:7 on the piano would be c-e-g-Bb.

If you've got a piano that will do that, you're lucky.

If your piano is tuned to 12et it probably won't do that so well.

> How would I play 1/7, 1/6, 1/5, and 1/4 on the piano?

However, in that case it will do 1/7:1/6:1/5:1/4 equally [not] well.

1/7:1/6:1/5:1/4 has exactly the same intervals as 4:5:6:7, but reversed in
order from top to bottom. The intervals are:

4:5 major 3rd
5:6 minor 3rd
6:7 septimal minor 3rd (?)

and in the 1/ chord these appear with this last interval at the bottom of
the chord.

On a 12et keyboard you get another interval closer to 5:6 instead of the
6:7, and so you can achieve that as C-Eb-Gb-Bb, which has the major 3rd at
the top instead of the bottom of the chord. But this untonal chord provides
no cue (that I know of) to allow you to hear the bottom interval as 6:7. On
the other hand, the 4:5:6:7 otonal chord has all the harmonics stacked up to
give a strong hint that the mistuned 7 is actually meant to be heard as
such. At least that is what a lot of us have gotten used to listening to
12et dominant 7th chords for all these years. I always thought they sounded
bad as a kid, but I grew used to it until recently.

-Kurt

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> hi Paul - (I'm referring to a tonalsoft excerpt which, according to
> Joe's indendation, is from one of your postings, which reads "Our
non-
> laboratory experiments on the harmonic entropy list seem to
> conclusively show that the dissonance of a chord can't be even
close
> to a function of the dissonances of the constituent intervals. For
> example, everyone put the 4:5:6:7 chord near the top of their
ranking
> of 36 recorded tetrads from least to most dissonant, while everyone
> put 1/7:1/6:1/5:1/4 much lower."

Yes.

> 4:5:6:7 on the piano would be c-e-g-Bb.

Well, that's as close as you can get to it on the piano, but of
course it's still very different, since pianos are normally in 12-
equal.

> How would I play 1/7, 1/6, 1/5, and 1/4 on the piano?

If we keep using C as the lowest note, the closest piano chord would
be C-Eb-Gb-Bb (1/1 would be the Bb two octaves higher), but again
there's quite a big difference since the piano is in 12-equal.
5:6:7:9 and 17:20:24:30 would also map to C-Eb-Gb-Bb on the piano but
really all three JI chords sound quite different.