Yahoo Tuning Groups Ultimate Backup mills-tuning-list Monzo-Erlich debate over the Hendrix chord

[5 April 1998]

Ever since I posted something about Jimi Hendrix bringing
the 19/16 ratio into the harmonic language of rock, Paul
Erlich and I have been involved in a debate over the
tuning, or at least the rational implications, of the
"dominant 7th, sharp 9th" or "Hendrix chord". Most of
the discussion took place off-list, so I've put it
together to post it here, along with some further
editorial additions/ explanations/ clarifications.
It's very long, but most of you should find it quite
interesting.

Paul's main point is that due to guitar and finger mechanics,
the only viable rational interpretation of the "Hendrix Chord"
is 1/1 : 5/4 : 7/4 : 7/6. However, our debate brought in
various thoughts about acoustics and mathematics, and we would
both like to see what other subscribers have to say about it.
I am particularly interested in what you guitarists have
to contribute, as I don't really play the instrument myself.

-Joe Monzo
============================================================
from TUNING DIGEST # 1354

From: Joseph L Monzo
To: tuning@eartha.mills.edu
Date: Sun, 15 Mar 1998 07:45:00 -0500
Subject: Re: TUNING digest 1353, Topic No. 4: non12 rock

> ... can anybody offhand list some people doing
> detwelvulated *pop music* ... i.e. "songs" in
> some kind of (fairly extended into folk- and
> psychedelic-, knowing the guy who's asking the
> question) "rock context"?

...I can provide an illustration of an incredibly
talented performer being able to coax any frequency he
wanted from an ordinary instrument. Listen very
carefully to Jimi Hendrix, particularly his "bent"
notes......

I also give Hendrix the credit for bringing the 19-identity
(the "sharp 9th" of the chord, i.e., the "minor 3rd" played
an octave above the "major 3rd" in a "dominant 7th chord")
definitely and securely into the harmonic language of rock
-- an excellent example is the main riff at the beginning
of "Foxey Lady" on "Are You Experienced" (vintage 1967).

-Monzo
==========================================================
[5 April 1998]

The cents values of the 4:10:14:19 chord would be
0, (octave +)386, (octave +)969, (2 octaves +)298.

-Monzo
==========================================================
from TUNING DIGEST # 1358
Topic No. 2

Date: Wed, 18 Mar 1998 15:57:57 -0500
From: "Paul H. Erlich"
To: "'tuning@eartha.mills.edu'"
Subject: Reply to Joe Monzo

As a rock guitarist who is both an avid Hendrix buff and a
constant investigator of tuning possibilities, I must
dispute this claim. If the bottom three notes of the
chord (well known to guitarists as "The Jimi Hendrix
Chord") are tuned 1/1:5/4:7/4, and a sharp 9th is added
above the chord, the most consonant (and most appropriate
on distorted guitar) tuning for the sharp 9th is 7/3
[1 octave + 267 cents]. Try it yourself if you don't
believe me. The 4/3 [498 cents, 3^-1 in JustMusic
notation] between the 7th and sharp 9th is very solid,
while a 19:14 [529 cents, 7^-1 * 19^1 in JustMusic] in
the same position is too complex a ratio and just sounds
out-of-tune.

The way the chord is played on guitar, the two lowest
notes are played with the two strongest fingers, which
a good player (like Jimi) will use to apply bending and/or
vibrato. This will bring the chord from 12-equal to
something approaching just intonation.

Not all chords are best analysed as portions of the harmonic
series!

-Erlich
==========================================================
[5 April 1998]

"Hendrix Chord" (I think)

E||----------|---------|--------|-------|-----4|A
B||----------|---------|--------|-------|-----3|E
G||----------|---------|-------2|-------|------|A#
D||----------|---------|--------|-------|------|
A||----------|---------|--------|-------|------|
E||----------|--------1|--------|-------|------|F#

Paul's argument is that because the E is 31 cents higher
than the 7th harmonic, in order for the E to be interpreted
as the 7th harmonic, since is can't be lowered, the other
notes in the chord must be raised. In order to make this
12-equal chord approach Paul's interpretation as
1/1 : 5/4 : 7/4 : 7/6, the first finger would have to
bend the F# up 31 cents, and the second finger would have
to bend the A# up 17 cents. This gives F#:A#:E the
proportions 2:5:7. The 12-equal A is 500 cents above
E, which makes it only 2 cents higher than 7/6.

-Monzo
==========================================================
From: Joseph L Monzo
To: PErlich@Acadian-Asset.com
Date: Thu, 19 Mar 1998 13:14:19 -0500
Subject: Hendrix and 7/3

Thanks for the correction. It points out the difficulty
of analyzing stuff when you don't play the instrument
under consideration. I can only be called a guitarist
by a very long stretch of the imagination, and I've always
*heard* this chord as having a 19/16. Can't say I ever
tested it on anything other than a keyboard.

One of the projects on which I will not give up (although
I've been told by nearly everyone that it's all but
impossible) is to create software that will analyze the
pitches being played on a CD. The data's there on the
disk, already in digital format, it's just a matter of
putting the .WAV file through the proper mathematical
formulas to break it down into individual instruments and
pitches. This way we would all finally have accurate
information on what tunings were actually played by
performers, with none of the I-could-be-wrong slips
of the ear.

Last year, all of my musical equipment (with the single
exception of my Rational Guitar, described below) and
almost all of my CDs and records were stolen. So I
don't have the Hendrix album to check it anymore, and
I'm limited these days to doing all my musical work on
a computer/soundcard combo. Makes it even more difficult
to check statements I've made on tuning.

The first specifically microtonal instrument I had
was my "Rational Guitar", an acoustic guitar on which
I filed off all the frets and placed new ones,
individually for each string. It's a purely experimental
instrument, meant to supply different identities for
one chord, up to a 27-Limit. It's virtually useless
as a *musical* instrument, in that it will only play one
chord (other chords can be coaxed out of the higher
identities, for example, 3 - 15 - 9 make a 1 - 5 - 3
triad on the 3/2), but it does give me rock-solid
identities up to 27 for that "tonic" chord. On this
instrument, 4:10:14:19 sounds gorgeously in tune.

You mentioned distortion in the guitar's sound as a
factor, and this *is* an acoustic instrument, so maybe
that's why it sounds so good to me. Unfortunately,
this guitar has no notes built upward from 4/3, so
I can't check how 7/6 would sound in place of 19/16.

I'll be the first to admit that a particular
high-harmonic note may fall within a lower PRIME-limit
-- manipulation of primes is what my theory is all about,
and it's my belief that most ears take a while to adjust
to higher primes -- a good argument for the static pace
of La Monte Young's extremely high-prime sound
installations. Without proper tools with which to
analyze and check, it's tough.

-Monzo
============================================================
From: "Paul H. Erlich"
To: "'monz@juno.com'"
Date: Thu, 19 Mar 1998 14:58:46 -0500
Subject: RE: Hendrix and 7/3

Clearly the 12-equal sharp 9 [300 cents], which is
what Hendrix plays, is a very close approximation
of the 19/8 [298 cents]. However, what I'm trying
to say is that the relevant acoustical ratio is the
4/3 between the sharp 9 and the 7th. Since the
7th (7/4) is so high in 12-equal, the 7/3 is very
high as well -- in the neighborhood of 19/8. That
doesn't make 19/8 the relevant ratio for describing
the chord, however.

When you tune a 1/1:5/4:7/4 on a keyboard, where do
your ears tell you the sharp 9 should be?

Again, you're not _wrong_ -- I just think you're
using ratios too high to be acoustically relevant
in this context.

Actually, distortion will make that sound even better,
as it creates lots of difference tones, which will all
be integers. Distortion favors exact harmonic-series
based chords, while on an acoustic guitar many other
chords, such as subharmonic (utonal) chords, are
important. However, even with distortion, 19 is
usually a bit too high up the harmonic series to
be important.

I suspect that, _especially on an acoustic instrument_,
you will like 7/6 in place of 19/16.

Just to clarify -- even if Jimi did bend/vibrato the
root up to form a 7/3 with the sharp 9, the bass
would still be playing the root in 12-equal, so the
12-equal sharp 9 is still there, and that is only
3 cents off the 19th harmonic.

That still doesn't make the 19/8 the most relevant
ratio for describing the pitch of that note, however.

Surely you knew/heard that the seventh that Jimi
plays, relative to the bass, is 31 cents sharp of
the 7th harmonic, and yet you allowed that error
to be ignored in your analysis. Similarly, I would
argue, the 7/3 can sustain a fairly large error
in tuning.

If you tune 1/1:7/4:7/3 on an acoustic guitar,
and you listen closely for the seventh harmonic
of the lowest note, you will hear it quite clearly,
since it is also the 4th harmonic of the middle note
and the 3rd harmonic of the highest note. This
is called a subharmonic (or utonal) chord and is
the principle behind minor triads. Distortion
tends to weaken these types of harmonies and
strengthen ones with a harmonic (or otonal) genesis.

..I don't think our ears are particularly sensitive
to primes; when it comes to intervals, each new ODD
number presents something new for our ears to get used
to, whether it is prime or not. Johnny Reinhard gave
me a copy of your paper several years ago, and that
is one of the major problems I have with it.

-Erlich
============================================
[5 April 1998]

As Paul alluded to, it's interesting to note that
the 19th harmonic is 31 cents above 7/6 -- almost
exactly the same distance or "error" as that of the
12-equal 7th above 7/4.

-Monzo
===========================================
From: Joseph L Monzo
To: PErlich@acadian-asset.com
Date: Thu, 19 Mar 1998 17:55:38 -0500
Subject: Re: Hendrix and 7/3, accurate notation

By the way, it's important to realize that utonal
chords can also be analyzed as harmonic-series
chords -- the proportional numbers become different
(and much larger), but the intervallic relationships
are still the same. For example:

Build a chord from the bottom up with the ratios
1/1 - 7/6 - 7/5 - 7/4. The standard Partchian
analysis calls this a Utonality built downward from
7/4, with the udentities (downward) 1 - 5 - 3 - 7.
However, this chord can also be called an Otonality
built upward from 16/15, with the odentities (upward)
15 - 35 - 21 - 105 (with the "root" missing).
It's exactly the same chord.

> However, even with distortion, 19 is usually a bit
> too high up the harmonic series to be important.

I have to disagree with this -- I've analyzed many
pieces where others called a note 7/6 (or some
other utonal derivation) and I hear it clearly as
19/16. There are numerous arguments among "tuning
people" about what the "blue notes" are.

> ...I don't think our ears are particularly sensitive
> to primes...

Each new odd number provides a new *identity* in a
chord, but it's my theory that the emotional and
acoustical effect and affect of that interval is
determined by its prime-number constituents. The
point of what I wrote above, about being able to
analyze chords in either of the two ways, is that
what LOOKS like a note that's high in the harmonic
series may actually be a much lower utonal harmonic.
Of course, I realize that 19 is quite a high prime
-- to me, this bears out my theory that studying
the PRIMES is what is most important in musical
harmony.

...you said yourself that distortion will
tend to emphasize interpretation of the chord ratios
as harmonics over a fundamental.

Again, you are correct in that I overlooked the fact
that the 7th played by Jimi is sharper than the 7th
harmonic. We've all been listening to lots of music
this century that implies ratios of 7, but fudges their
correct intonation quite extensively. The bottom
line of my argument is that in order for us to
properly analyze what's already been done (tuning-wise)
by important artists whose legacy exists in recordings,
we need to pin down exactly what pitches they were
playing/ singing, before we can theorize about chords,
harmonics, etc. My work is mainly an attempt at a
notation that can be easily read by musicians, and
yet is precisely accurate as to the intonation. I hope
to get my book done soon -- the paper only gives the
foundation of what I'm doing; the book fleshes it out
a lot with historical examples.

-Monzo
=====================================================
from TUNING DIGEST # 1359

Date: Thu, 19 Mar 1998 15:40:46 -0500
From: "Paul H. Erlich"
To: "'tuning@eartha.mills.edu'"
Subject: Jimi Hendric chord

I must admit that with enough distortion,
8:10:14:19 is the preferable tuning for this chord,
since distortion creates high-order difference tones.
However, we're talking about so much distortion
that the individual notes cease to be audible
and all that remains is a harmonic series above 1.
I don't think even Jimi often took distortion to
that extreme. You would have to overdrive a transistor
amp to get an idea of the extreme distortion I'm
talking about. Think of the distortion sound in
"Revolution" (fast version) by the Beatles.
I think that's transistor distortion. Vacuum tubes
(or a simulation thereof) are the usual means of
effecting a distorted guitar sound and they have
a way of letting the individual notes ring through.

===========================================================
From: "Paul H. Erlich"
To: "'monz@juno.com'"
Date: Fri, 20 Mar 1998 16:30:22 -0500
Subject: RE: Hendrix and 7/3, accurate notation

> By the way, it's important to realize that utonal
> chords can also be analyzed as harmonic-series chords...

Yes, but which is more relevant for describing
the chord's effect/affect? I would argue that
any description employing such high numbers is
psychoacoustically meaningless. Do you hear the
missing "root"? Does it sound right as a bass
note for the chord?

> ...There are numerous arguments among
> "tuning people" about what the "blue notes" are.

Blue notes aside, I will admit that the 12-equal
minor triad in root position has a certain
stability that is lacking from the just minor
triad due to the former's proximity to 16:19:24,
which has the "root" in the right place.

> ...the emotional and acoustical effect
> and affect of that interval is determined by
> its prime-number constituents.

I have heard that theory from many people, but I
am convinced that it is a fallacy. The ear does
many things, but prime factorization is not one of
them. The only validity I will allow this theory
is a cultural conditioning-type process where
musicians will become accustomed to the result of
"stacking" low prime-number intervals before moving
on to the next prime number. However, to someone
brought up with 5-limit music, 13:11 isn't going
to bear more similarity to 13:8 or 11:8 than,
say, 9:7.

> ...what LOOKS like a note that's high in
> the harmonic series may actually be a much lower
> utonal harmonic. Of course, I realize that 19
> is quite a high prime -- to me, this bears out
> my theory that studying the PRIMES is what is most
> important in musical harmony.

I don't get it.

I was just saying that the relevant acoustical ratio
for explaining why the sharp 9 sounds good is the
4/3 between the sharp 9 and the 7th. Take away the
7th, and the sharp 9 sounds awful. Another common
chord in rock, especially alternative rock, is a
major triad or dominant seventh chord with an added
fourth. What do you think the fourth is, the 11th
harmonic? The 21st? The 43rd? No, it's just a 4/3
above the root, and luckily in 12-equal, a 4/3 below
the 7th. That's why the note sounds good there, and
how it should be tuned.

Excessive distortion will make most chords sound
really bad unless the notes are very close to a
harmonic series. Unless all the notes of the chord
except the 7th were bent up, there is little chance
for the Jimi Hendrix chord to achieve this condition
as an 8:10:14:19 chord. It is extremely difficult
to achieve the finger independence necessary to bend
with all fingers except the ring finger (which would
be required here).

I think you'll be disappointed at how rough and
inconsistent Jimi Hendrix's intonation was. John
McLaughlin was sometimes even worse. But by and
large, it's 12-equal. Where do you go from there?

-Erlich
=======================================================
From: Joseph L Monzo
To: PErlich@acadian-asset.com
Date: Fri, 20 Mar 1998 19:38:08 -0500
Subject: Re: Hendrix and 7/3, accurate notation

Certainly, the description with the smallest
[proportional] numbers would be the most relevant.

> I would argue that any description employing such
> high numbers is psychoacoustically meaningless.

There may be other reasons for wanting to describe
a utonal chord as the equivalent otonal chord with
higher numbers. We are limited to graphing pitch
ratios in 3 dimensions, but there may be many more
than 3 prime numbers used in a scale or piece.
One solution to this problem is to think of all
those primes and odd-identites as rungs in the
overtone ladder.

> Do you hear the missing "root"? Does it sound
> right as a bass note for the chord?

No matter how small or large the numbers are,
when they come from an arithmetic series, they
will always emphasize the "fundamental" (otonal
or utonal), whether it's sounding or not. In
some cases, the missing "root" *can* be heard
as a difference tone. As far as sounding right
as a bass note, it depends on many factors: the
particular ratios used, the timbre of the
instruments, the octave registers of the ratios,
whether the notes are in close or open spacing,
etc.

Similarly (conversely), the lower notes of
a chord can produce summation tones which appear
as higher pitches than those that are actually
being played. These effects are apparent when
listening to the music of La Monte Young. Most
likely, the use of distortion (even small amounts)
would emphasize this effect. If you try this out
on your guitar, I'd be interested to know if this
is true. Try playing a 4:5:14 chord.
5 + 14 = 19, so if my theory is correct, you
should be able to hear the 19th harmonic.
If it's not apparent, see if adding distortion
makes it audible. If this occurs, then it may
be possible that in a regular 4:5:7 chord, the
octave overtone of the 7 (14) is reacting with
the 5 to produce 19.

> ...the 12-equal minor triad in root
> position has a certain stability that is lacking
> from the just minor triad...

It's interesting that you should say that. I've
always thought that 10:12:15 (the just minor triad),
6:7:9, and 16:19:24 ALL sounded good as stable
minor triads, just different.

> ...The ear does many things, but prime
> factorization is not one of them.

Well, we definitely part company here. This is
the foundation of my entire theory. After reducing
any number or collection of numbers as far as they
will go, what's left is the prime series. And
personally, I (and many others) distinctly sense
a difference in sound and feeling between different
prime numbers.

> ...13:11 isn't going to bear more similarity
> to 13:8 or 11:8 than, say, 9:7.

Your point is valid, but my theory isn't written from
the point of view of "someone brought up with 5-limit
music", it's written with an eye toward describing
ANY possible rational combination of pitches.

> ...[about analyzing chords as either utonalities
> or otonalities:] I don't get it.

I was simply saying that sometimes when high numbers
turn up which are multiples of several primes, many
of the primes can be factored out when the analytical
orientation (otonal vs. utonal) is reversed.

> ...Take away the 7th, and the sharp 9 sounds awful.

I disagree -- on my Rational Guitar (the acoustic one)
4:10:19 sounds fine -- maybe a little strange, but far
from awful. And again, it may be that the sharp 9th
sounds better when the 7th is added because 19 is a
summation tone of 5 and 14.

> ...[about the "major triad or dominant seventh
> chord with an added fourth":]it's just a 4/3......
> That's why the note sounds good there, and how it
> should be tuned.

My, aren't we being dogmatic here? If the performer's
intonation is not precise, or indeed if it IS precisely
12-equal, there are many possible rational
interpretations. I will grant that 4/3 is probably
intended most of the time, as it is a very simple
ratio, and has been used as a basis for music for a
couple of thousand years (at least). But surely it's
not the ONLY ratio that musicians use to fit this
particular harmonic slot.

> ...It is extremely difficult to achieve the
> finger independence necessary to bend with all
> fingers except the ring finger (which would be
> required here).

OK -- to me this sounds logical. This is the best
argument you've given me as to why your
interpretation of this chord is better than mine.

I haven't done any thorough analysis of Hendrix's
intonation -- I was just making statements based on
what I've heard in more-or-less casual listening.
Perhaps I don't have the authority to discuss this
particular corner of microtonality (as I've already
told you several times, I am no guitarist).

-Monzo
=======================================================
From: "Paul H. Erlich"
To: "'Pat Missin'"
Cc: "'monz@juno.com'"
Date: Tue, 24 Mar 1998 15:05:03 -0500
Subject: RE: Reply to Joe Monzo

Pat, I hope you don't mind my including Joe Monzo in
this discussion.

Joe, read what Pat has to say below. Notice the P.S.,
which happens to be a point I just made to you in e-mail.

Happy tuning, fellas!

>-----Original Message-----
>From: Pat Missin [SMTP:patm@globalnet.co.uk]
>Sent: Friday, March 20, 1998 5:46 PM
>To: Paul H. Erlich
>Subject: Re: Reply to Joe Monzo
>
> You wrote:
>>
>> ...The 4/3 between the 7th and sharp 9th
>> is very solid, while a 19:14 in the same position
>> is too complex a ratio and just sounds out-of-tune.

> Recently, on the harmonica list to which I subscribe,
> the subject of Canned Heat's "On The Road Again"
> was raised. Several people were confused about
> how Al Wilson got certain notes in the harmonica
> solo. I checked it over......and found that
> he had tuned one of the draw notes a semitone sharp.
> A typical blues harp would be tuned with the draw chord:

> D G B D F A B D F A

> Wilson had raised the middle A (and maybe the top A,
> although he doesn't use it on this solo) to a Bb.
> I immediately retuned a harp that was lying around
> and figured that the best voicing was to use 4/5/6/7
> (which is how Wilson's harp would have been tuned at
> the factory) and tune the Bb a perfect 4th above
> the 7/4. Sounded pretty good, although the difference
> tone from the Bb and the B played together, was a bit odd
> - but certainly not unpleasant.

> A few days later, I read the post about Hendrix
> introducing the the 19th harmonic and was immediately
> tempted to take that Bb up to 19/4. I really like the
> sound of septimal intervals, but I have never been too
> happy with their association with the "blue notes" of
> jazz and blues. I always figured a blue third was a
> little wider than 7/6. So, having an interval that
> sits right between 18/8 and 20/8 seemed to make a
> lot of sense. I wish I hadn't bothered - sounds REALLY
> horrible, for all the reasons you gave above. The
> "fourth" being the main cause of its horribility.
> I tried tuning the F up to a perect fourth under the
> 19/16, but that wasn't any better. I thought about
> posting something to the list, but pressure of work....
> etc., etc...

> ...All the best,

> -- Pat.

> PS - If we have to credit someone with introducing the
> 19th harmonic, wouldn't it make more sense to credit
> whoever first played a minor triad in 12TET. I mean,
> this is a pretty good approximation of 16/19/24 isn't
> it? Closer than most rock guitarists ever get, anyway...
========================================================
[5 April 1998]

Considering that the Pythagorean "sharp 9th" or
"minor 3rd" with the ratio 32/27 [294 cents] is
only 4 cents lower than 19/16, perhaps some ancient
Greek or Chinese musician used 32/27 in this function,
and is the one who ultimately deserves the credit
for introducing 19 as a harmonic identitiy.

Incidentally, in the tuning Pat used on his harmonica,
G:B:D:F:Bb would have the proportion 12:15:18:21:28.
The "bit odd - but certainly not unpleasant" difference
tone between the B (15) and Bb (28) is 13 - a prime
number that never really entered our discussion in any
significant way, but one which Paul would no doubt also
consider to be too high a prime to be useful. It's one
that has a sound that I love (so does Ben Johnston).

-Monzo
=========================================================
From: "Paul H. Erlich"
To: "'monz@juno.com'"
Date: Tue, 24 Mar 1998 15:24:14 -0500
Subject: RE: Hendrix and 7/3, accurate notation

> There may be other reasons for wanting to describe
> a utonal chord as the equivalent otonal chord with
> higher numbers...

I don't see utonal chords as presenting any particular
problems in graphing pitch ratios, particularly when
using triangular lattices as Erv Wilson and I do.

>> Do you hear the missing "root"? Does it sound
>> right as a bass note for the chord?

> No matter how small or large the numbers are,
> when they come from an arithmetic series, they
> will always emphasize the "fundamental" (otonal
> or utonal), whether it's sounding or not...

Yes, but in the example we are discussing,
15:21:35:105, do you hear the note represented
by 1 or 16 "emphasized" in any way? What do your
ears tell you?

I noticed that you have ignored a couple of the
other listening experiments I suggested to you.
I would love to continue this discussion, but
it is not really too relevant to the field of
music unless we ground ourselves in actual sounds.
Let's take that as our reference point -- try to
describe what you really hear.

> Similarly (conversely), the lower notes of a chord
> can produce summation tones which appear as higher
> pitches than those that are actually being played
> ......it may be possible that in a regular
> 4:5:7 chord, the octave overtone of the 7 (14) is
> reacting with the 5 to produce 19.

Distortion produces many high-order difference and
summation tones which obscure the issue. Just playing
loud will cause the ears to produce low-order combination
tones, such as those of th 2a+b type you are referring
to here, and so will be a better way to bring this out.

>> to someone brought up with 5-limit music, 13:11 isn't
>> going to bear more similarity to 13:8 or 11:8 than,
>> say, 9:7.

> ...my theory isn't written from the point of
> view of "someone brought up with 5-limit music",
> it's written with an eye toward describing ANY
> possible rational combination of pitches.

I don't see a contradiction, nor do I see a
counterargument. Just what do you hear in the interval
13:11 as regards prime qualities? Do you hear 13-ness
and 11-ness? Is this the same effect you get from 143:128?
=============================================================
[5 April 1998]

13:11 = 11^-1 * 13^1 (289 cents)
143:128 = 2^7 * 11^1 * 13^1 (192 cents)

Paul's point is that both ratios contain 11 and 13 as
factors. I'm listening to both of these frequently now,
to hear exactly what the effects are. No comment yet.

-Monzo
=============================================================
From: "Paul H. Erlich"
To: "'monz@juno.com'"
Date: Tue, 24 Mar 1998 16:06:19 -0500
Subject: psychoacoustical aside

Are you aware that, when the ear is presented with
tones forming a not-quite-harmonic series, such as
1020, 1520, 2020, 2520 Hz, it will hear a
fundamental at a sort of "best-fit" frequency,
around 507 Hz, even though the difference tones
would all be 500 Hz? See "Musical Acoustics" by
Hall for an easy-to-read summary of this and other
important phenomena.

-Erlich
===========================================================
[5 April 1998]

1020:1520:2020:2520 can be reduced to 51:76:101:126.
The cents values are 0, 691, 1183, (octave +)366.

In JustMusic notation, the prime factors are:

[2520] 3^2 * 7^1
[2020] 101^1
[1520] 19^1
[1020] 3^1 * 17^1

Calculated as harmonics above 507, they fall extremely
close to 2:3:4:5. Cents values calculated from 507 are:
(2 octaves +) 376
(octave +)1193
(octave +) 701
(octave +) 10.

Cents values of 2:3:4:5 would be 0, 702, 1200, 386.

The fact that the ear interprets the fundamental
as a "best fit" of 507 Hz, despite actual
difference tones of 500 Hz, only emphasizes
to me the psychological power of the overtone
series as a harmonic archtype into which
we will try to fit what we hear, even when it
disagrees with the true acoustics and mathematics
of the sounds.

I would like to become familiar with the math used
to describe this phenomenon and calculate this
"best-fit" frequency. I have seen something similar
to this in the writings of Ernst Terhardt (published
a couple of years ago; I believe it was in the
Journal of Music Theory).

-Monzo
===========================================================
From: Joseph L Monzo
To: PErlich@acadian-asset.com
Date: Tue, 24 Mar 1998 16:20:05 -0500
Subject: Re: Hendrix and 7/3, accurate notation

> ...this discussion...is not really too
> relevant to the field of music unless we ground
> ourselves in actual sounds...

I agree totally -- theorizing on paper without hearing
*is* pretty much irrelevant, unless you happen to
stumble onto a useful new way of describing something.
It's not that I've *ignored* your experiments, just
that I don't have much equipment left on which to listen.
As I told you, all my instruments were stolen......
And I *did* describe a few things to you that I'd tried
out on my Rational Guitar. Specifically, the chord
4:10:19 sounds fine on this (acoustic) guitar, even
without the 7th. And to my ears, 4:10:14:19 sounds
even better. Other than that, I have to do everything
on my computer with pitch-bend -- a time-consuming
process...

-Monzo
=====================================================
[5 April 1998]

I have since tried Paul's chord on my acoustic guitar.
It sounds wonderful, but certainly very different from
4:10:14:19. In my opinion, it's neither better nor
more consonant (nor Jimi), just different.

It's important to realize that Paul's analysis of
this chord can *also* be described as harmonic
proportions over a fundamental. If F# is 1/1, then
the fundamental would be the missing B with the
ratio 4/3. His chord would have the proportions
6:15:21:28. Obviously these numbers are larger
than my interpretation as 4:10:14:19, but the
highest prime in his chord is 7, while the highest
in mine is the much higher 19. Analyzed in my
JustMusic notation, we get:

Erlich Monzo
---------- -----
A 7^1 19^1
E 3^1 * 7^1 7^1
A# 3^1 * 5^1 5^1
F# 3^-1 n^0

There are aspects to both chords which influence
the feeling of consonance and dissonance
simultaneously. Both chords have the same strong
2:5:7 proportion in the lowest three notes, which
gives both of them quite a strong feeling of
consonance.

The aspect which makes Paul's chord more
consonant is obviously that the factors in
his ratios are all such small primes, and thus
the internal interval-structure is also more
consonant.

The aspects which makes mine more consonant are
that the notes are piled up as harmonics over a
fundamental which is both present and in the bass,
and they are single-prime identities of the chord.

Of course, the factors which make either chord
dissonant are the converse of those which make
the other consonant: Paul's is only obliquely
related to a missing fundamental, and the two
middle ratios are combinations of two primes,
while mine has the high prime 19.

Our entire debate is centered around the fact that
(as both Ben Johnston and Harry Partch pointed out
long ago) consonance-dissonance is determined not
only by how large the primes are, but also how
large their exponents are, and also how many of
them are combined in particular ratios.

I think the most important thing to realize is --
again raising Partch's ghost -- "the ability of
the human ear is vastly underestimated". A tiny
difference in the pitch of a note can give it an
enormously different sound and feeling; I believe
that studying the prime-number relationships among
the tones is what will disclose these effects.

Another important distinction in our different methods
of analysis is the dichotomy between the analysis of
a chord as one harmonic complex and analysis of it as
the sum of individual constituent interval ratios. My
interpretation of the chord emphasizes the harmonic
complex view, while Paul's emphasizes the interval
view.

There it is. All comments and debates are welcome
and appreciated.

-Monzo

Joseph L. Monzo
monz@juno.com
4940 Rubicam St., Philadelphia, PA 19144-1809, USA
phone 215 849 6723

_____________________________________________________________________
You don't need to buy Internet access to use free Internet e-mail.
Get completely free e-mail from Juno at http://www.juno.com
Or call Juno at (800) 654-JUNO [654-5866]

Monzo-Erlich debate over the Hendrix chord

🔗monz@juno.com (Joseph L Monzo)

🔗Paul Hahn <Paul-Hahn@...>