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compasspnt

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#41 [url]

Feb 12 17 1:01 PM


A while back...in another world...one of my classical pieces was played at a big concert at which several well known modern composers were in attendance.

Afterwards one of these 'major guys' came up to me, congratulating profusely about my piece. (Without getting into the technical composition elements), he went on and on about how I did a certain technical thing harmonically. I thanked him appreciatively of course, but I had no idea what he was talking about. To me, it was just melody.

 

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maarvold

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#42 [url]

Feb 12 17 1:12 PM

gtoledo3 wrote:
Relatedly, I don't remember a whole lot of "major sevenths" on the Beano record... maybe in one of the licks on "Little Girl".

Although it's not a textbook example of what I was referring to (though, in my recollection, the "Badge" reference is), you can hear it on 'Beano' on the IV chord in "Steppin' Out" at around 1:25.  The band is playing the repeating figure including a lowered 7th degree of the IV (C) chord and the figure includes a Bb... meanwhile EC is banging away on C-B-C-B-C-B-C in his solo: so TOTALLY works for me in a way that the title topic will never do.  I believe it's because EC heard what was right about it--instead of not hearing what was wrong with it.  It even works FAR better than a peanut butter cup for me (see earlier post): maybe because it comes from a different place.  I have always maintained that early Clapton solos are like a good term paper: concise, compelling, well-researched, demonstrating a well thought out point of view and often based on pre-existing proven information.  FWIW, Joe Walsh uses the exact same device in his solo on "Life's Been Good".  
 

Last Edited By: maarvold Feb 12 17 1:18 PM. Edited 1 time.

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gtoledo3

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#43 [url]

Feb 12 17 1:17 PM

maarvold wrote:

gtoledo3 wrote:
Relatedly, I don't remember a whole lot of "major sevenths" on the Beano record... maybe in one of the licks on "Little Girl".

Although it's not a textbook example of what I was referring to (though the "Badge" reference is), you can hear it on 'Beano' the IV chord in "Steppin' Out" at around 1:25.  The band is playing the repeating figure including a lowered 7th degree of the IV (C) chord and the figure includes a Bb... meanwhile EC is banging away on C-B-C-B-C-B-C in his solo: so TOTALLY works in a way that the title topic could never hope to.  Because EC heard what was right about it--instead of not hearing what was wrong with it.  It even works FAR better than a peanut butter cup for me (see earlier post): maybe because it comes from a different place.  I have always maintained that early Clapton solos are like a good term paper: concise, compelling, well-researched, demonstrating a well thought out point of view and often based on pre-existing proven information.  
 

Hah, yep, of course, good call! That was even my first thought, and then I dismissed it for some reason...shoulda thought through all of it.

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maarvold

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#44 [url]

Feb 15 17 10:48 AM

soapfoot wrote:
At the risk of a derail into what should be a whole 'nother thread... "blue" notes are another matter entirely.

"blue" notes aren't "in-between" or 'minor-over major"... they're tunable notes, they just require an entirely different conception of harmony that extends to the seventh partial, whereas Western music only extends to the fifth.

In our Western harmonic system, all twelve pitches of the equal-tempered scale "stand for" just, resonance-based pitches. And all twelve can be arrived at by reckoning only third and fifth partials in various directions. In Indian solfege, "mi", or the third (fifth partial) is known as "ga", and "sol" or the third partial is known as "pa." That will be helpful as we go on...

If I start with "C" as my tonic, the third partial ("pa") is G and the fifth partial ("ga") is E. (that's three of twelve)

The third partial above the third partial (G) is D. (four)
The fifth partial above the third partial (G) is B. (five)
The third partial above the fifth partial is G# (six)
The fifth partial above the fifth partial is A. (seven)
The tonic is the third partial of F --Mathieu would call this "pa below tonic" (eight)
The tonic is the fifth partial of Ab, or "ga below" (which duplicates the G# in equal temperament, but is a slightly different pitch in just tuning; this is the more commonly used of the two. Still eight)
"Pa below" (F) is the fifth partial of Bb, or "pa below pa below" (nine)
"Ga Below" (or Ab) is the fifth partial of Db, this is "pa below ga below" (ten)
"Ga below Pa below" is Db again (but a different Db!), as this is the fifth partial below F. (still ten)
"Ga below Ga below" is Fb (or E natural) again, but a different E natural by a few cents in just tuning (still ten)
"Pa below Pa" is redundant... that's just our "C" tonic (still ten).
But "Ga below Pa" is Eb, our minor third (eleven).

The next few in the sequence are redundant in equal temperament (but are actually distinct pitches in just tuning), so let's skip ahead to the first place we get the twelfth equal-tempered pitch class--

Ga above Pa above Pa, or the fifth partial above the third partial above the third partial above tonic (F# above D above G above C).

Now we've got all of our twelve equal-tempered pitch classes, using only third partials and fifth partials. Western harmony is thusly referred to as a "five limit" system, as we can limit the harmonic series to five partials and still find all the pitches we need.

Now for the blues...

The blues is descended in part from some West African musical traditions that utilize the harmonic series up through the seventh partial, which in C is that slightly out-of-tune (to the Western ear) Bb harmonic.

And these origins are what inform the "blue notes". The blue notes are tunable pitches that are by degrees either in-tune or out-of-tune, sure as a perfect fifth or major third. They're not just random places in-between major and minor. And it's also not random that the I, IV, and V chords are commonly played as dominant seventh chords, and that this isn't well-explained by Western music theory. The lowered seventh is nothing more arcane than an imperfect approximation of the seventh partial which is so integral to the music's tuning and harmonic conception.

The "blue" minor third is the seventh partial below the fifth partial below the tonic. In C, "pa" below is F. And F is the seventh partial of a pitch lying somewhere between Eb and E. The blue minor seventh is, of course, just the seventh partial itself (which Mathieu extrapolates Indian solfege to call, appropriately "blu").

So... not really "bitonal"... a whole other system of harmony.

Now all the reckonings we did above in the five-limit system, imagine doing all of those in the seven-limit system. That is a massive amount of tunable, singable pitches. To try and explain those via our Western system of harmony and solfege is reductive, and doesn't really "work" so well to explain the blues.

Of course, Western hegemony being what it is, the accepted 20th century explanation was that blues was "primitive" music and was somehow naive to our "sophisticated" Western concepts of harmony.

Hardly.

Anything but, really.

 
As I said I would, I revisited this: it's DEEP.  And the second half seems by far the most interesting to me.  The first, when viewed in light of the simple and elegant "circle of 5ths" (maybe the Apple Campus 2 of its day?), seems a bit like a search for a different way to explain the 12 tone western scale.  

As John LaPorta taught so well at Berklee when I was there:
"This... (plays a G7 chord on the piano) ...is a V (five) chord: it wants to go home; This... (plays a C chord on the piano) ...is a I (one) chord: it's home".  

So, for those who don't know, the circle of fifths is based on general the idea that the basic engine of western harmony--the tritone interval--in the Dominant chord (G7 for example) wants to go home to C Major; then if we add a lowered 7th degree to the C triad, our ear hears that C7 as Dominant and it's wanting it to go home yet again: this time an F triad brings it home.  And so on.  

Following this motion through until it begins to repeat encompasses all of the keys and all of the key signatures; spelled out in terms of the flat keys:
C
F
Bb
Eb
Ab
Db
Gb
B
E
A
D
G
and then completing the circle with a return to C.  

Spelled out in terms of the sharp keys:
C
F
A#
D#
G#
C#
F#
B
E
A
D
G
and then completing the circle with a return to C.  

On the piano, Bb and A# (B flat and A sharp) are enharmonic--meaning two different ways of referring to the same note.  

There are subtle ramifications to the tuning differences between the way untempered orchestral instruments (instruments which don't have exact, pre-determined tuning) might play a D# vs Eb, but for simplicity's sake, let's not get into that here.  

Last Edited By: maarvold Feb 15 17 10:54 AM. Edited 2 times.

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gtoledo3

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#45 [url]

Feb 15 17 12:39 PM

The circle of fifths is elegant in its simplicity. Also, a useful tool for providing jumping points to break out of major key melody/harmony, or modulate key, etc.

The point about the emphasis on the feeling that various degrees want to "go home", e.g. "resolve" in differing amounts, is also very appealing to me. It eventually led me to a much more chromatic thought process when creating, whether it be melody or harmony, as opposed to being as caught up in modes, or even inner harmonic relationships.

I may know that I'm tending towards a given mode, or that a certain harmonic relationship exists that lends itself towards various chord substitutions, etc...but I feel like a highly chromatic view, and also viewing each note as a voice of one big thing regardless of instrument....is like walking far away from a big image, and being able to see all of it at a glance, as opposed to being up close and seeing only a bit of detail.

I like to keep in that headspace. So it's very interesting to read the various thought processes and tendencies among others.

 

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maarvold

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#46 [url]

Feb 15 17 1:08 PM

It did occur to me, as I was out walking my dogs, that "the basic engine of western harmony" (the tritone) was meant to read: "the basic engine of MODERN western harmony", but I forgot to include that in my editing process. So with that in mind:

In the formative years of more sophisticated western music, 2 simultaneous pitches separated by 3 whole steps (F and B, for example)--the tritone--were referred to as "the devil's interval" and its use was definitely unwelcome, if not downright forbidden.  Speaking without heavily researched casuallness, in Bach's time, although I'm quite sure he would have fully explored the qualities of the tritone in the dead of night when nobody was around, a big part of his world was pipe organs--with the ability to replicate the notes of the harmonic series of middle C upwards through quite a few partials--and I think what Brad took the time to spell out about the harmonic composition of single notes and that quality's relationship to intonation and western harmony's version of 'acceptable' and 'unacceptable' is possibly more valid than I might have thought at first.  

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soapfoot

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#48 [url]

Feb 15 17 1:53 PM

Mike, Thanks for looking it over!

A couple of points of discussion--

The circle of fifths is NOT "simple and elegant"-- in fact, it's anything but, as soon as you step outside the confines of the artificially-constructed world of equal temperament! Its apparent logic was back-engineered by Europeans; it's a square peg that's been forced into the round hole that exists naturally in physics.

An equal-tempered "perfect" fifth is narrow by 2 cents compared to a true 3:2 perfect fifth. 2 cents isn't very much, and the ear tolerates it well. But if you were to stack twelve "true" fifths on top of one another... C, G, D, A, E, B, F#, etc. etc. on and on down the line... by the time you got back to "C," it would be 24 cents sharp from the C you started out with (or 23.46 cents, to be precise). This discrepancy is known as the "Pythagorean comma."

Obviously this won't do... and physicists, composers, and instrument-makers wrestled with this for centuries before coming up with the highly inelegant solution of just making every fifth narrow by 2 cents to compensate. Fourths and fifths still play acceptably in-tune; thirds and sixths are quite sour in equal temperament (we've just grown accustomed to tolerating it).

But the thing is... and this is the important part... the twelve-tone scale has never been "arrived" at through the circle of fifths. The temperament came much later. The fact that you arrive at something close to an "E" by stacking true Pythagorean fifths above C is nothing more than coincidence-- the "E-like object" you get from stacking true fifths is VERY different from the E you arrive at when taking the fifth partial of C-- and the third of a major triad was never based on the Pythagorean E.

To put it another way: the fifth partial is so strong that anything out of tune with it will cause beating against the harmonic. You can hear this on a piano... many tuners actually tune in part by counting beats against a timing reference. This is a good time to emphasize that arriving at these pitches is not merely theoretical-- it's physical, and the physics causes audible artifacts in the form of beats against the harmonic. That's how we know "where" each pitch comes from in relation to tonic.

The fifth partial of C gives you an "E" which is 14 cents narrower than an equal-tempered "E." The "Pythagorean" E you arrive at from stacking fifths a la "circle of fifths" (the third partial of A above D above G above C) is 8 cents wider than the equal-tempered E... a full 22 cents sharp of the C natural's fifth partial. This will sound very, very sour indeed, with audible and fast beating against the C's fifth harmonic.

As a side note, we should acknowledge that there are other tuning commas as well-- for instance, if I were to stack three fifth partials on top of one another (C to E to Ab back to C), when we arrived back at that C we would be 41.06 cents flat... this is known as the "Diesis." Four minor thirds gives the "greater diesis," which would leave us sharp by 62.57 cents. There are many other commas... the "didymic comma," the "diaschisma"... you can research them. They all have names and were all studied and documented as composers, tuners, and instrument makers sought better solutions to the temperament question from the dawn of polyphony.

Eventually, we just kind of "gave up" and accepted the sour thirds of equal temperament because it makes modulation so damn convenient. But I do want to stress that the "circle of fifths" is not as simple and elegant as it's presented in first semester harmony classes... in fact, it's really nothing more than a fantasy that we've tried to will into existence, consequences be damned.

As you alluded to earlier, all of this is the reason why we have different names for the same pitch class. If we're in the key of C and have a secondary dominant II7, the F# in that chord is in actuality a VERY different pitch from the Gb in a C diminished that decorates tonic. On a piano keyboard, one key "stands" for both notes. But a violist with a good ear would play each differently!

brad allen williams

Last Edited By: soapfoot Feb 15 17 1:58 PM. Edited 3 times.

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gtoledo3

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#49 [url]

Feb 15 17 3:19 PM

It strikes me that there is probably a good analogy that could be drawn between the tempered system, and what has to be done to create a 3D perspective matrix with a "field of view".

Both have the values...weighted, to put it very crudely...in order to create a functional system that appears to be able to create a representative reality, while not necessarily functioning 1:1 with true natural phenomena.

Figured I'd go ahead and share that oddball thought.

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maarvold

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#52 [url]

Feb 15 17 10:02 PM

soapfoot wrote:
...You can hear this on a piano... many tuners actually tune in part by counting beats against a timing reference. This is a good time to emphasize that arriving at these pitches is not merely theoretical-- it's physical, and the physics causes audible artifacts in the form of beats against the harmonic. That's how we know "where" each pitch comes from in relation to tonic...

...Eventually, we just kind of "gave up" and accepted the sour thirds of equal temperament because it makes modulation so damn convenient. But I do want to stress that the "circle of fifths" is not as simple and elegant as it's presented in first semester harmony classes...

As you alluded to earlier, all of this is the reason why we have different names for the same pitch class. If we're in the key of C and have a secondary dominant II7, the F# in that chord is in actuality a VERY different pitch from the Gb in a C diminished that decorates tonic. On a piano keyboard, one key "stands" for both notes. But a violist with a good ear would play each differently!

 
When I took the class on piano tuning at Berklee (one semester, with just enough knowledge to be a 'serviceable tuner') they handed out a sheet with beating values in Hz and fractions of Hz (called CPS back then) for the various intervals, as you stated.  (I find it ironic that both you and I are guitar players who are generally forced to live in this 'tempered world'.)  I have always been pretty focussed on how various piano tuner's tunings feel to me.  I won't name names, but some guys don't really work that well for me and some guys work spectacularly well for me.  For example, one tuner who did a lot of work in L.A. had a slightly 'sparkling' quality to his tunings.  A guy that my super-fussy friend has tune his concert grand has a tuning that feels very relaxed and consonant and vibration-free.  And so it goes for me.  

The part of the circle of 5ths that feels elegant to me is that you keep using the most basic 'modulation motor'--tonic becomes the new dominant--and you end up right back where you were after 'touring the countryside'.
Being that I'm pretty fussy about pitch stuff, I don't know why it is that I'm not bothered by tempered instruments (pianos, vibes, glocks, guitars, marimbas, etc) 'living' in the same composition as non-tempered... except that maybe the skill of orchestrators and musicians who know how to make the tempered instruments at their disposal 'put their best foot forward in that world' is why it doesn't bother me.  Also, the total thrill of hearing low brass playing beatless triads when they are really nailing the pitch may offset a lot of other things.  And it may be that the amazing variations in tone colors outweighs any rubbing for me... or maybe even makes the rubbing more interesting.  

Last Edited By: maarvold Feb 15 17 10:05 PM. Edited 1 time.

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soapfoot

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#53 [url]

Feb 15 17 10:08 PM

Yes, and also our ears have come to accept a lot of inharmonicity... inharmonicity is in fact natural in some contexts.

You're obviously acquainted with piano tuning, so you know that at the extremes, the thickness (at the bass end) and the shortness (at the treble end) cause the strings to behave less like "strings" and more like "chimes," which means the harmonics begin to progressively deviate from the harmonic series somewhat.

And though it's not natural... the harmonics produced by stops or drawbars on electronic organs (tonewheel or transistor) are actually equal-tempered! This inharmonicity is part of the characteristic nasal timbre of electronic organs.

So we're no strangers to it... we've been exposed enough that it begins to just be "a sound" to our ear, as opposed to "wrong."

brad allen williams

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gtoledo3

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#54 [url]

Feb 16 17 10:03 AM

We could concentrate on the stated "imperfection" of the intervals with equal temperament...

But aren't there harmonic combinations, especially various commonly used complex chords which use non-major key intervals, that would sound very off using a tuning based on the overtone series? I'd have to really contrive a test, but I am fairly sure this would be true, and even a partial reason for equal temperament to begin with as opposed to just modulation.

 

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soapfoot

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#55 [url]

Feb 16 17 10:29 AM

gtoledo3 wrote:
We could concentrate on the stated "imperfection" of the intervals with equal temperament...

But aren't there harmonic combinations, especially various commonly used complex chords which use non-major key intervals, that would sound very off using a tuning based on the overtone series? I'd have to really contrive a test, but I am fairly sure this would be true, and even a partial reason for equal temperament to begin with as opposed to just modulation.





 

The short answer to your question is "no."

The medium-length answer is "not as long as you pick the right notes--and the right notes are available to you on the instrument."

The exception might be serialism. Serial music is arguably the only music I can think of in which equal temperament is sonically preferable. And that's because it willfully denies and subverts these natural physical relationships, and goes to great length to deny hearing any pitch as tonic (well, Schöenberg, anyway... some of his successors kind of weren't as opposed to faking tonic within the serial paradigm, but that's drifting off-topic...)

If you give me an example of a harmonic construction, I might be able to explain to you the most consonant intervals... what our ear "wants" to hear.

You said "non major-key," so let's just do a basic one-- the minor triad.

C minor triad, C, Eb, G. 

C is tonic. The G should vibrate at a 3:2 ratio with respect to tonic. It is the third partial.

Eb is the fifth partial below the 3:2 G... in other words, the G should vibrate at a 5:4 with respect to the Eb (or, it might be easier to say that the Eb should vibrate at a 4:5 ratio with respect to this G). This makes it a 6:5 ratio with respect to tonic.

The Eb should be 316 cents above tonic. The G is 702 cents above tonic.

This will give a very "sweet" and true sounding minor triad much unlike what you'd hear on a piano keyboard.

A piano keyboard gives you 0, 300, and 700 cents... 16 cents flat on the third, 2 cents flat on the fifth.

When I say "as long as you pick the right notes" above, I mean that there are other notes "represented" by the black key between D and E... and if you pick the wrong one, your minor triad will sound LESS in-tune. For instance, there's a D# which is 274 cents above tonic, which is the fifth partial of the seventh scale degree (major third above B), and also the third partial above the third partial above the fifth partial (same 274 cents above tonic). Both are "just intonation," both are "arrived at through low-prime ratios," but one is the "right note" in a minor triad and one is the "wrong note." The wrong note gives you an out-of-tune chord.

Again, this is why we have both D# and Eb in our notation... that dates from a time BEFORE instruments were tempered!


NOW is a good time to make a very important point--

This is all a description of something very intuitive. When the tuning "locks" in and the ensemble rings like a bell... it's a feeling on a perceptual level. This is all very instinctive and intuitive. A good ear, and a musician with good intonation, INSTINCTIVELY will seek these low-prime resonances. It's wired into us.

This is part of the reason that a college jazz choir which tunes up with a piano sounds SO different from a southern gospel choir. The tone sounds different, the delivery sounds different, the projection and power is different... part of this is a different concept of tuning that's more aligned with the physics. The "beating" you hear when two notes are almost (but not exactly) in tune is phase cancellation just the same as the phase cancellation we try to avoid in our listening rooms. It's literal destructive interference.

This is also a big part of why "sound/tone" and "performance" are inextricable from one another, and a big part of why a better player can pick up the exact same instrument and make it sound better than a lesser player. A good, well-in-tune PERFORMANCE will seem to have a better tone, because the tuning isn't fighting against the power and projection.

brad allen williams

Last Edited By: soapfoot Feb 16 17 10:42 AM. Edited 3 times.

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soapfoot

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Feb 16 17 11:16 AM

more than 13! Actually more than 13 by far

I could cite no fewer than 57 tunable, individual pitches within one octave of the key of "C."

And that's just as they relate to C... if we make tonic any other pitch, it would have its own set of 57 (at least).

Now we should say... "sound very off" is cultural... our Western ear has grown so accustomed to temperament that we actually kind of appreciate the chorusing of the fundamentals against the overtones as "rich" or "brilliant," in some contexts. But this is "nurture," and not "nature." Sort of how we like the sound of a whole section of slightly out-of-tune violins more than if they were all playing perfect unison (which is almost literally impossible, anyway).

The video below, while not GREAT (and actually a commercial for some process or another) has kind of a decent example, with oscilloscope traces, of some intervals and then some music both justly tuned and then "twelfth-root-of-2" equal tempered. Check out the tempered tuning at 1:16 and then the same progression justly tuned at 1:44, and imagine you had to place each in a mix. How easy or difficult would it be for each one to 'place' and get to 'speak'? Does one actually sound "muddier" while the other actually sounds more brilliant?



Side note-- beware of clicking recommended videos. A lot of the "just intonation" stuff on YouTube is kind of a casual understanding, and sometimes the way they arrive at the tuning for their "examples" is suspect. This is an area where VERY MUCH "not everyone is an expert" and it's extremely possible to have "just enough information to be dangerous." Treat "just versus equal tuning" on youtube kind of the same as you'd treat "mixing tutorials" on youtube.

brad allen williams

Last Edited By: soapfoot Feb 16 17 11:23 AM. Edited 2 times.

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gtoledo3

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#58 [url]

Feb 16 17 11:46 AM

"more than 13! Actually more than 13 by far"

I stated, that shifts around. So sure, it's a matter of perspective.

But I think you have just illustrated how:

"The short answer to your question is "no.""

Is incorrect... unless you want to add countless notes to the system.

You can't have 12 notes, use a "just intonation", and then voice the commonly used complex chords, or chord changes, that are often done in western pop or elsewhere. You would inevitably end up with much worse out of tune combos.

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soapfoot

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#59 [url]

Feb 16 17 12:18 PM

George, I'd say that's not "just intonation."

That's "random use of notes that can be arrived at through low-prime ratios."

"Tuning" is not a thing you do once before you play, unless you're playing a keyboard instrument. It's something you do CONSTANTLY as you play. It's a skill. It's done by ear.

All I've described above is a theoretical description of what happens when good musicians play together, or sing together. They do NOT instinctively tune to "piano pitches."

And there ARE "countless notes" in the system... infinite ones, actually. We don't have to "add" any. We choose the appropriate one at all times, to the best of our ability... that's what singing is. We do not choose from a selection of twelve notes when we sing. We create a pitch out of literally infinite possibilities to fit the situation.

On MOST instruments, this same thing is possible (to a degree), we just happen to have "starting points" that approximate where we want to end up.

brad allen williams

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