9-5=4!

Hello! After calculating a bunch of difference tones for various
chords (the usual JI, 5-limit ones I find on the piano), I saw
something that interested me: the chord 4-5-9. (Another favorite of
mine :) Spelled c-e-b-d', it has a conjoined major and minor
seventh chord chord, just as the triad has a joined major and minor
third. Sort of a 'triad' in music built largely of 7th and 9th
chords.)

Any way, I can't find a second case where, given a 3 note tertial
chord, (with chord defined as closed position, no duplicates, outer
interval smaller that a 10th) where the bottom note (4) is the
difference tone of the the upper two (9-5). (Reminds me of the
previously discusedd 3-5-8-13, where the difference tones stay
within the bounderies of the chord. Oh, so concerning 3-5-8-13,
Paul, regardless of how many cases there are of these types of
differnce tones, do you see any psychoacousitcal effect from it? I
would imagine that the difference tones would be harder to hear,
nestled right inside the mother chord.)

FOr 4-5-9, I'm wondering if this difference tone arrangement could
be related to the fact that 9, in 5-limit harmony, is a sort of
ceiling, with 10, and all numbers above it, aside from 15, being
mere doubles (in 5-limit harmony)...(Just a shot in the dark...)

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

> Hello! After calculating a bunch of difference tones for various
> chords (the usual JI, 5-limit ones I find on the piano),

How do you find JI chords on the piano if it's tuned by a standard
piano tuner?

> I saw
> something that interested me: the chord 4-5-9. (Another favorite of
> mine :) Spelled c-e-b-d',

I'm confused. 4:5:9 could be c-e-d', but where does the b come in?

> it has a conjoined major and minor
> seventh chord chord, just as the triad has a joined major and minor
> third. Sort of a 'triad' in music built largely of 7th and 9th
> chords.)
>
> Any way, I can't find a second case where, given a 3 note tertial
> chord, (with chord defined as closed position, no duplicates, outer
> interval smaller that a 10th) where the bottom note (4) is the
> difference tone of the the upper two (9-5).

It's not even close if the piano is tuned in standard 12-equal.

But wouldn't 3:5:8 be another example? And 3:4:7?

> Oh, so concerning 3-5-8-13,
> Paul, regardless of how many cases there are of these types of
> differnce tones, do you see any psychoacousitcal effect from it?

Yes, but certainly none that could be demonstrated on a piano in
equal temperament. If you're talking JI, the chord is very coherent
and easy to tune by ear, and the combinational tones certainly help.
Fokker's book is full of such examples.

> I
> would imagine that the difference tones would be harder to hear,
> nestled right inside the mother chord.)

Maybe they're harder to hear, but then they're less disturbing to the
chord as a whole, a "good" thing.

> FOr 4-5-9, I'm wondering if this difference tone arrangement could
> be related to the fact that 9, in 5-limit harmony, is a sort of
> ceiling, with 10, and all numbers above it, aside from 15, being
> mere doubles (in 5-limit harmony)...(Just a shot in the dark...)

I don't know what you mean, but wouldn't 3:5:8, for example, have the
same difference tone property that supposedly makes 4:5:9 special
here in the first place?

> How do you find JI chords on the piano if it's tuned by a standard
> piano tuner?

You're Right.

> > I saw
> > something that interested me: the chord 4-5-9. (Another favorite
of
> > mine :) Spelled c-e-b-d',
>
> I'm confused. 4:5:9 could be c-e-d', but where does the b come in?

Excuse me. I meant major ninth chord.

> > it has a conjoined major and minor
> > seventh chord chord, just as the triad has a joined major and
minor
> > third. Sort of a 'triad' in music built largely of 7th and 9th
> > chords.)
> >
> > Any way, I can't find a second case where, given a 3 note
tertial
> > chord, (with chord defined as closed position, no duplicates,
outer
> > interval smaller that a 10th) where the bottom note (4) is the
> > difference tone of the the upper two (9-5).
> > It's not even close if the piano is tuned in standard 12-equal.

Right.

> But wouldn't 3:5:8 be another example? And 3:4:7?

I'm defining 'chord' as having an outer interval less than a 10th
(closed position), and in 5 limit harmony. (Sorry. Did I not say
5-limit harmony?) Looking at your list of 'coherent' chords from
your previous reply (and thank you for that reply), there are other
chords, like 5:6:11, which almost qualify, except that theyr'e not 5-
limit. (It's very arbitrary to stay with 5-limit, I know....You're
very patient...)

>
> > Oh, so concerning 3-5-8-13,
> > Paul, regardless of how many cases there are of these types of
> > differnce tones, do you see any psychoacousitcal effect from it?
>
> Yes, but certainly none that could be demonstrated on a piano in
> equal temperament. If you're talking JI, the chord is very
coherent

"Coherent": does this mean, not disturbed by difference tones? Or
a specific type of sound? Could this be partly why 4:5:9 and 3:5:8
are so 'beautifully sonorous'? (I know that' subjective. Do you
personally like these two chords? Or this doesn't apply in your
higher-limit universe?) Or, could 4-5-9 and 3-5-8 share something
in the nature of their 'chord color' based on this shared difference
tone feature, this 'coherence'?

> and easy to tune by ear, and the combinational tones certainly
help.
> Fokker's book is full of such examples.
>
> > I
> > would imagine that the difference tones would be harder to hear,
> > nestled right inside the mother chord.)
>
> Maybe they're harder to hear, but then they're less disturbing to
the
> chord as a whole, a "good" thing.
>
> > FOr 4-5-9, I'm wondering if this difference tone arrangement
could
> > be related to the fact that 9, in 5-limit harmony, is a sort of
> > ceiling, with 10, and all numbers above it, aside from 15, being
> > mere doubles (in 5-limit harmony)...(Just a shot in the dark...)
>

>but wouldn't 3:5:8, for example, have the
> same difference tone property that supposedly makes 4:5:9 special
> here in the first place?

Except that it's not closed position. (I work with chords which are
fairly 'rooted' and tertial (some of the them!), so, in some cases,
I use the closed position critiria in my ruminations/analysis.)

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

> > > I saw
> > > something that interested me: the chord 4-5-9. (Another
favorite
> of
> > > mine :) Spelled c-e-b-d',
> >
> > I'm confused. 4:5:9 could be c-e-d', but where does the b come in?
>
>
> Excuse me. I meant major ninth chord.

So did you mean for a different set of numbers to go along with the
letters, or a different set of letters to go along with the numbers?
A C major ninth chord would be c-e-g-b-d', but it seems you meant a
subset of that.

> > > it has a conjoined major and minor
> > > seventh chord chord, just as the triad has a joined major and
> minor
> > > third. Sort of a 'triad' in music built largely of 7th and 9th
> > > chords.)
> > >
> > > Any way, I can't find a second case where, given a 3 note
> tertial
> > > chord, (with chord defined as closed position, no duplicates,
> outer
> > > interval smaller that a 10th) where the bottom note (4) is the
> > > difference tone of the the upper two (9-5).
> > > It's not even close if the piano is tuned in standard 12-equal.
>
> Right.

You should calculate, or listen, to see where the difference tones
actually fall in this tuning.

> > > Oh, so concerning 3-5-8-13,
> > > Paul, regardless of how many cases there are of these types of
> > > differnce tones, do you see any psychoacousitcal effect from it?
> >
> > Yes, but certainly none that could be demonstrated on a piano in
> > equal temperament. If you're talking JI, the chord is very
> > coherent
>
> "Coherent": does this mean, not disturbed by difference tones? Or
> a specific type of sound?

A little of both.

> Could this be partly why 4:5:9 and 3:5:8
> are so 'beautifully sonorous'?

Have you ever actually listened to these chords?

> (I know that' subjective. Do you
> personally like these two chords?

Sure, they're nice.

> Or this doesn't apply in your
> higher-limit universe?)

???

> Or, could 4-5-9 and 3-5-8 share something
> in the nature of their 'chord color' based on this shared
difference
> tone feature, this 'coherence'?

Maybe. Of course, if so, this would utterly fall apart on the piano
in 12-equal.

hi Paul.

> > Excuse me. I meant major ninth chord.
>
> So did you mean for a different set of numbers to go along with
the
> letters, or a different set of letters to go along with the
numbers?
> A C major ninth chord would be c-e-g-b-d', but it seems you meant
a
> subset of that.

Actually, I was thinking of c-e-b-d', but haven't calculated the
numbers for any quatrads yet.

This actually brings up a question I've been meaning to ask,
regarding e-b-d', or 4:6:7. It's a chord I use alot, a gutted
dominant seventh chord! (like as part of c-e-b-d'), and realize it's
derived low in the series from 4:6:7, but was wondering if there is
any tendancy for the ear to also hear it as the utonal version of
5:6:9 (minor seventh chord e-g-d'.) I realize 4:6:7 is lower in
number, but recall Partch discussing the conundrum of assigning an
interval a low prime (eg, 7) vs. a higher composite number, like
9.

In addition to measuring ambiguity in ascribing a fundamental and
series for a given interval or chord, does harmonic entropy also
measure ambiguity in choosing between alternate numberings of the
same chord, like the two discussed above?

Another chord example of this problem, which I've thought a lot
about, is g-a-e. (It's in John Cage's organ piece 'Souvenir'.) 7-8-
12, or 1/6:9:10? (Or 9:10:15?)

(, in this regard, is g-a-e'; that is whether to consider it

> > Could this be partly why 4:5:9 and 3:5:8
> > are so 'beautifully sonorous'?
>
> Have you ever actually listened to these chords?
>

Not in just intonation. I'm thinking of buying the Sethares book.
Is it demonstrated in its cd? Can you direct me to a musical patch
on line that would demonstrate this, but which my very slow dial-up
connection, and worm-eaten PC, can handle? (For, example, I
couldn't download your "...swing" piece. (i can't recall the
title.))

How different do the JI versions actually sound from the ones I'm
using? (I realize that much of my chord-enjoyment experience is
very piano-centric, with all that that entails, such as long,
interesting decays, and - I read - inharmonicity.)

> > (I know that' subjective. Do you
> > personally like these two chords?
>
> Sure, they're nice.
>
> > Or this doesn't apply in your
> > higher-limit universe?)

> ???

I meant: 1) you and I probably hear them differently, as I haven't
heard them yet in JI or higher limit, yet. And, 2) are they even
considered as chord-types in the tonalites of alternate tunings?

thanks, Kelly

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
> hi Paul.
>
> > > Excuse me. I meant major ninth chord.
> >
> > So did you mean for a different set of numbers to go along with
> the
> > letters, or a different set of letters to go along with the
> numbers?
> > A C major ninth chord would be c-e-g-b-d', but it seems you meant
> a
> > subset of that.
>
> Actually, I was thinking of c-e-b-d', but haven't calculated the
> numbers for any quatrads yet.

But you mean 8:10:15:18, yes?

> This actually brings up a question I've been meaning to ask,
> regarding e-b-d', or 4:6:7. It's a chord I use alot, a gutted
> dominant seventh chord! (like as part of c-e-b-d'),

You mean like as part of e-g#-b-d'? c-e-b-d' isn't a dominant seventh
chord.

> and realize it's
> derived low in the series from 4:6:7, but was wondering if there is
> any tendancy for the ear to also hear it as the utonal version of
> 5:6:9 (minor seventh chord e-g-d'.) I realize 4:6:7 is lower in
> number, but recall Partch discussing the conundrum of assigning an
> interval a low prime (eg, 7) vs. a higher composite number, like
> 9.

In Partch's world, you're either playing one chord or the other. Both
(4:6:7 and 1/(9:6:5)) are available in his JI system. You should
listen to them both. They're both awesome chords, and sound quite
different from one another.

> In addition to measuring ambiguity in ascribing a fundamental and
> series for a given interval or chord, does harmonic entropy also
> measure ambiguity in choosing between alternate numberings of the
> same chord, like the two discussed above?

There is a chordal version of harmonic entropy, for which the answer
is: kind of, though it's biased toward otonal numberings.

> Another chord example of this problem, which I've thought a lot
> about, is g-a-e. (It's in John Cage's organ piece 'Souvenir'.) 7-
8-
> 12, or 1/6:9:10? (Or 9:10:15?)

Again, you should listen to both JI chords. Then you can listen and
compare and hear how much the original 12-ET chord is similar or
dissimilar to each of them.

> (, in this regard, is g-a-e'; that is whether to consider it
>
> > > Could this be partly why 4:5:9 and 3:5:8
> > > are so 'beautifully sonorous'?
> >
> > Have you ever actually listened to these chords?
> >
>
> Not in just intonation. I'm thinking of buying the Sethares
book.
> Is it demonstrated in its cd?

These chords? No, not really.

> Can you direct me to a musical patch
> on line that would demonstrate this, but which my very slow dial-up
> connection, and worm-eaten PC, can handle? (For, example, I
> couldn't download your "...swing" piece. (i can't recall the
> title.))

Ouch. I can create JI chords as .wav files, which I've done before,
but I don't know if that'll be too much bandwidth for you.
Perhaps .mp3 files?

> How different do the JI versions actually sound from the ones I'm
> using?

Very different, at least for someone like me who's worked with
different tuning systems.

> (I realize that much of my chord-enjoyment experience is
> very piano-centric, with all that that entails, such as long,
> interesting decays, and - I read - inharmonicity.)

You can still tune a piano into something reasonably close to JI, at
least for the purposes of these comparisons.

> > > (I know that' subjective. Do you
> > > personally like these two chords?
> >
> > Sure, they're nice.
> >
> > > Or this doesn't apply in your
> > > higher-limit universe?)
>
> > ???
>
> I meant: 1) you and I probably hear them differently, as I haven't
> heard them yet in JI or higher limit, yet.

What do you mean by "higher limit", exactly?

> And, 2) are they even
> considered as chord-types in the tonalites of alternate tunings?

Sure they are!

> > Actually, I was thinking of c-e-b-d', but haven't calculated the
> > numbers for any quatrads yet.
>
> But you mean 8:10:15:18, yes?

Yes.

> > regarding e-b-d', or 4:6:7. It's a chord I use alot, a gutted
> > dominant seventh chord! (like as part of c-e-b-d'),

>You mean like as part of e-g#-b-d'? c-e-b-d' isn't a dominant
seventh > chord.

I mean, e-b-d', alone, or even when it's part of c-e-b-d', reminds
me of e-g#-b-d without the 3rd (gutted, and liking its absence!)

> You can still tune a piano into something reasonably close to JI,
at > least for the purposes of these comparisons.

(I've inquired about just such a tuner.) I'm writing a piano
piece. Can certain types of alternate tunings be indicated based on
certain chord types? (Or maybee it's best to consider alternate
tunings at the inception, and not in the middle of a project ..)..

>>Paul asked: 'what do you mean by 'upper limit'?

Above 5-limit, and loosely refering to writings I've skimmed, but
not studied, on '19-limit', 31-limit", etc.

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
> > > Actually, I was thinking of c-e-b-d', but haven't calculated
the
> > > numbers for any quatrads yet.
> >
> > But you mean 8:10:15:18, yes?
>
> Yes.
>
> > > regarding e-b-d', or 4:6:7. It's a chord I use alot, a gutted
> > > dominant seventh chord! (like as part of c-e-b-d'),
>
> >You mean like as part of e-g#-b-d'? c-e-b-d' isn't a dominant
> seventh > chord.
>
> I mean, e-b-d', alone, or even when it's part of c-e-b-d', reminds
> me of e-g#-b-d without the 3rd (gutted, and liking its absence!)

I'm suprised you hear a g# rather than a g missing in c-e-b-d'.

> > You can still tune a piano into something reasonably close to JI,
> at > least for the purposes of these comparisons.
>
> (I've inquired about just such a tuner.) I'm writing a piano
> piece. Can certain types of alternate tunings be indicated based
on
> certain chord types? (Or maybee it's best to consider alternate
> tunings at the inception, and not in the middle of a project ..)..

If you're interested in harmonies up to a prime-limit of 7, probably
the best approach is to draw lattices of the tunings you're
interested in, so you can see all the JI intervals at a glance, and
move things around as you see fit . . .

> >>Paul asked: 'what do you mean by 'upper limit'?
>
> Above 5-limit, and loosely refering to writings I've skimmed, but
> not studied, on '19-limit', 31-limit", etc.

Personally, I've done very little in theory or practice beyond 11-
limit.

> I'm suprised you hear a g# rather than a g missing in c-e-b-d'.

I hear the absence of g# ...(as well as the 'turned around' minor
seventh chord, with 6/5 on top).... it's hard to describe, but it's
sort of my modas operandi, to avoid dominants, so it effects the way I
hear chords or parts of chords which resemble it. (And avoiding
dominants can lead, interestingly, to progressions where the 'tendency
tones' occur between chords other than I and V.)

Regarding not hearing the g...I tend to omit the fifths from chords,
an undue influence, perhaps, of part-writing. (I'm experimenting with
getting away from the contrapuntal influence on chords. I was really
impressed with John Brion's guitar work in several sections of I Love
Huckabees: how strummed guitar chords can be so refreshingly removed
from nit-picky linear activity.) I would imagine that guitarists fill
out chords differently than pianists.