Re: Strict JI considered undesirable

In a previous message I gave the following optimal simultaneous distribution of the 224:225 and 384:385 (or 4096:4125) (11-limit). Its errors are half those of 1/4-comma meantone. I'm not sure whether to call it quasi-just or wafso-just.

Intvl Error (cents)
2:3 -1.36
4:5 -2.71
5:6 1.36
4:7 -0.43
5:7 2.28
6:7 0.93
4:9 -2.71
5:9 0.00
7:9 -2.28
4:11 -2.71
5:11 0.00
6:11 -1.36
7:11 -2.28
8:11 -2.71
9:11 0.00

Here's a 22-tone 11-limit tuning that makes maximum use of this.

Fx------Cx-----Gx
/ \ /
/ \ E /
/ \ /
5 C#------G#------D#------A#
/ \ / \ / \ / \ /
/ 7 \ / Fx\ / Cx\ F / Gx\ C /
/ 11 \ / Ebb \ / Bbb \ / Fb \ /
4-------6-------9 A-------E-------B-------F#
otonal hexad / \ / \ / \ /
legend / D#\ / A#\ Db/ \ Ab/ utonal hexad
/ \ / \ / \ / legend
F-------C-------G-------D 1/9-----1/6-----1/4
/ \ Fx / \ Cx / \ Gx / \ 1/11/
/ B \Ebb/ F#\Bbb/ \ Fb/ \1/7/
/ \ / \ / \ / \ /
Db------Ab------Eb------Bb 1/5
/ \ D# /
/ G \ /
/ \ /
Ebb----Bbb------Fb

Let me know if you want me to make a Scala file of it.

Regards,

-- Dave Keenan
http://dkeenan.com

[Dave Keenan wrote...]
> If you are going to introduce deliberate mistunings into adaptive JI they
>might as well be useful ones,

Oh, come on, guys! Wouldn't the inharmonicity of most timbres be enough to
take care of this, even in monophonic music with identical instruments
tuned in exact JI? Even assuming perfectly harmonic, additive timbres, how
much beating do we actually need?

[Paul Erlich wrote...]
>>The only reason Barbershop blends so much is because it's
>>monophonic.
>
>???

As opposed to polyphonic. Barbershop is supposed to blend into one voice,
because it's written as one voice. Try counterpoint in JI and get back to
me, if there's even an instrument around with enough resolution to produce
the effect.

-Carl

Carl Lumma wrote,

>Oh, come on, guys! Wouldn't the inharmonicity of most timbres be enough to
>take care of this, even in monophonic music with identical instruments
>tuned in exact JI?

No, Carl, remember, brass, reed, and bowed string instruments, as well as
the human voice, have no inharmonicity (to the extent that is well-defined).

>Even assuming perfectly harmonic, additive timbres, how
>much beating do we actually need?

I would say at least one beat per quarter note.

>As opposed to polyphonic. Barbershop is supposed to blend into one voice,
>because it's written as one voice.

Unless the voice in question is the virtual fundamental, your assertion
makes no sense. What do you mean, it's written as one voice? The melody? If
it all blended into one voice, the melody would evaporate! Anyway, the best
barbershop ensembles don't stay as close to JI as the best brass ensembles.

I wrote,

>I would say at least one beat per quarter note.

I meant one quarter beat per whole note -- in other words, fast enough to
give a reasonable range of cancellation values during each count of harmonic
rhythm. One reason I dislike synthesizers is that the octaves are too
perfect -- you get an unpredictable timbre every time you hit an octave --
try it!

Dave Keenan wrote,

>Here's a 22-tone 11-limit tuning that makes maximum use of this.

Cool! I wonder, can this be considered a tempering of a 4-dimensional
periodicity block where two of the unison vectors are 224:225 and 384:385?

>> [Dave Keenan, TD 454.5]
>> In a previous message I gave the following optimal
>> simultaneous distribution of the 224:225 and 384:385
>> (or 4096:4125) (11-limit).
>>
>> <snip table and excellent lattice>
>>
>> Here's a 22-tone 11-limit tuning that makes maximum
>> use of this.
>>
>> <snip excellent lattice diagram>
>>
>> Let me know if you want me to make a Scala file of it.

Yes, Dave, please post one to the List.

>
> [Paul Erlich, TD 455.11]
> Cool! I wonder, can this be considered a tempering
> of a 4-dimensional periodicity block where two of the
> unison vectors are 224:225 and 384:385?

Sounds right to me.

How about you two hash out the mathematics of it?

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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>>Oh, come on, guys! Wouldn't the inharmonicity of most timbres be enough to
>>take care of this, even in monophonic music with identical instruments
>>tuned in exact JI?
>
>No, Carl, remember, brass, reed, and bowed string instruments, as well as
>the human voice, have no inharmonicity (to the extent that is well-defined).

Really? Is that just in theory, or has it been measured? If so, I'm
wondering why...

<1.> Wendy Carlos found that by slightly detuning the partials of additive
patches made them sound more realistic.

<2.> The strictly-harmonic patches on my K5000 additive synth sound
artificial.

>Unless the voice in question is the virtual fundamental, your assertion
>makes no sense. What do you mean, it's written as one voice? The melody? If
>it all blended into one voice, the melody would evaporate! Anyway, the best
>barbershop ensembles don't stay as close to JI as the best brass ensembles.

What assertion doesn't make sense? I'm simply stating that Barbershop is
monophonic music, in the most normal, classical sense. There are four
parts and one voice. As a 4-part fugue has four parts and four voices.

Your last statement. The best brass quintets are The Empire Brass, The
Canadian Brass, and the American Brass Quintet. None of them remain as
continuously in JI as the best Barbershop quartets, and none of them use
the 7-limit, whereas all Barbershop quartets are firmly 7-limit. That
said, if you compare accuracy at the 5-limit with accuracy at the 7-limit,
and only compare the maximum accuracy to JI, then you're absolutely right.

-Carl

>I meant one quarter beat per whole note -- in other words, fast enough to
>give a reasonable range of cancellation values during each count of harmonic
>rhythm. One reason I dislike synthesizers is that the octaves are too
>perfect -- you get an unpredictable timbre every time you hit an octave --
>try it!

Hmm. I have been annoyed that the octaves are too perfect on synths
(although I can't say I've ever been bothered by too much accuracy with
other intervals, despite the fact that many instruments do have the
resolution required to meet your definition). I don't, however, understand
the "unpredictable timbre" part.

-Carl

>>No, Carl, remember, brass, reed, and bowed string instruments, as well as
>>the human voice, have no inharmonicity (to the extent that is
well-defined).

>Really? Is that just in theory, or has it been measured? If so, I'm
>wondering why...

We went through this about a year or two ago. All the measurements (I
believe Gary Morrison did quite a few) confirmed the theory on this, which
is quite clear. The mechanisms producing the vibrations in these instruments
(lips, reed, bow sticking and slipping, vocal cords) are single-element
oscillators whose nonlinearities can only produce phase-locked harmonic
partials. Go back to those old messages (I believe they're pre-onelist). We
went into great detail.

><1.> Wendy Carlos found that by slightly detuning the partials of additive
>patches made them sound more realistic.

No doubt a bit of second-order beating helps to simulate the amplitude
variations in real instruments.

><2.> The strictly-harmonic patches on my K5000 additive synth sound
>artificial.

Ditto.

>>Unless the voice in question is the virtual fundamental, your assertion
>>makes no sense. What do you mean, it's written as one voice? The melody?
If
>>it all blended into one voice, the melody would evaporate! Anyway, the
best
>>barbershop ensembles don't stay as close to JI as the best brass
ensembles.

>What assertion doesn't make sense? I'm simply stating that Barbershop is
>monophonic music, in the most normal, classical sense.

Very strange. Monophonic means one voice, unaccompanied. You must mean
homophonic, meaning one melody, accompanied by subservient harmony. Yes?

>The best brass quintets are The Empire Brass, The
>Canadian Brass, and the American Brass Quintet

I was just listening to the American playing Keith Jarrett last night on the
"In The Light" CD. Very far from JI, though Jarrett calls the piece
"unplayable". But the brass quartet playing Christmas music in the lobby of
my office building sounds like it's using an amazing adaptive (5-limit) JI.
It probably depends a huge amount on what key the music is in.

Carl Lumma wrote,

>Hmm. I have been annoyed that the octaves are too perfect on synths
>(although I can't say I've ever been bothered by too much accuracy with
>other intervals, despite the fact that many instruments do have the
>resolution required to meet your definition).

Really?

>I don't, however, understand
>the "unpredictable timbre" part.

On both of the synths I've owned, a Casio CZ-1 and an Ensoniq VFX-SD, the
octaves are so perfect that every time you hit one, the small but random
difference in the start time for the two notes translates into a very
audible, unpredictable set of phase-cancellations and reinforcements among
the upper partials. It's actually kind of cool as an electronic effect, but
not what happens with real instruments.

>On both of the synths I've owned, a Casio CZ-1 and an Ensoniq VFX-SD, the
>octaves are so perfect that every time you hit one, the small but random
>difference in the start time for the two notes translates into a very
>audible, unpredictable set of phase-cancellations and reinforcements among
>the upper partials. It's actually kind of cool as an electronic effect, but
>not what happens with real instruments.

Huh. I owned an Ensoniq ESQ-1 back in the day. I never noticed anything
with anything that I can remember, but I was young or maybe I've forgotten.
I haven't noticed any trouble with the wavetable synths I've used (Proteus
and whatever Emu is putting in SoundBlasters now). However, on my Kawai
K5000 additive synth, polyphony is achieved with the same oscillators as
are used to generate the timbres -- IOW, when you play chords, you might as
well be making new patches. The octaves cancel like mad. On certain organ
patches, you can hardly tell an octave dyad from the lower note alone! But
I don't mind the other intervals, in JI or ET.

-Carl

>Those brass ensembles are all playing very broad range of repertoire, with
>much more variety than the three minute song format, constant seventh chord
>planing characteristic of barbershop music.

Well, you can split harmony into triadic and tetradic. Barbershop is jazz,
which is tetradic. Brass ensembles play both kinds, but I've never heard
any brass group consistently tune either in the seven limit. Have you?

>(By the way, the best brass ensembles today are the (amateur) brass bands in
>England.)

My statement was, of course, not meant to include every group in the world.
Rather, it applied to major performing groups that are in the public eye.
Also, my statement did not apply to brass bands, it applied to quintets.
Lastly, I doubt these groups are better than the ones I mention. If you
can recommend recordings...

>If you go back to WWII era recordings during the period when the musician's
>union refused to allow instrumentals, you can find some fantastic
>unaccompanied vocal ensemble performances with real intonational virtuosity,
>particularly by African-American groups: try the Golden Gate Quartet for
>starters. Also very much worth hearing are the "Madrigals" of William
>Brooks, especially as performed by Electric Phoenic; one of the four
>madrigals, a setting of Stephen Foster's _Nelly was a Lady_ , demonstrates
>just how far beyond the constraint of the barbershop style one might
>potentially go.

Thanks! I'll check these out.

-Carl

>I was just listening to the American playing Keith Jarrett last night on the
>"In The Light" CD. Very far from JI, though Jarrett calls the piece
>"unplayable". But the brass quartet playing Christmas music in the lobby of
>my office building sounds like it's using an amazing adaptive (5-limit) JI.
>It probably depends a huge amount on what key the music is in.

It doesn't really depend on the key, since the keys of the instruments are
usually chosen for the music or vice versa, and the intonation of the
instruments is quite flexible anyhow. The tuba would be the only concern,
and ABQ uses a bass trombone.

I think you'll find that while sustained chords can be closer to just in
brass quintets than Barbershop quartets, the reverse is true for moving
passages. The distribution of parts probably has something to do with this
(4 evenly, v. 5 unevenly) -- even the best quintets play sour notes in
moving passages, especially between the trumpets. Also, controling the
intonation of brass instruments seem to involve, in my experience anyway,
more of a feedback mechanism (or at least a slower one!) than does the voice.

As for the ABQ, I'm not familiar with that album. What vintage is it? You
may remember Jeff from Microthon, who was Ray Mase's student at the time.
He's got all their albums. I have "Premiere", on which they play
splendidly, although I think the music is crap. I also have "Fyre and
Lightning", which is their latest Renaissance album -- quite good. I need
to obtain, and I highly recommend, any album containing anything by
Brazilian composer Osvaldo Lacerda (I heard them play one of his quintets
(only one?) a few weeks ago).

-Carl

>It doesn't really depend on the key, since the keys of the instruments are
>usually chosen for the music or vice versa

I don't think Keith Jarrett's piece concentrated on one key-area in
particular, and if it did, it likely wasn't the right one for the
instruments.

>The distribution of parts probably has something to do with this
>(4 evenly, v. 5 unevenly)

Don't understand.

>As for the ABQ, I'm not familiar with that album. What vintage is it?

Keith Jarrett, _In the Light_, (c)1974

>>It doesn't really depend on the key, since the keys of the instruments are
>>usually chosen for the music or vice versa
>
>I don't think Keith Jarrett's piece concentrated on one key-area in
>particular

Then key _really_ wasn't the problem.

>>The distribution of parts probably has something to do with this
>>(4 evenly, v. 5 unevenly)
>
>Don't understand.

I think it's easier to tune a 5/2 than it is a 5/4 or a 5/1. In
barbershop, you have, more or less, the classic SATB-type arrangement. In
a quintet, the range of the instruments all overlap, and in particular you
have two of the same instrument playing in the sensitive sopranino range
with a very bright sound. The voicing just isn't as clear, even
music-wise. There's only a small catalog of accepted barbershop voicings
-- one or two for each chord type -- which have been carefully chosen for
stability and clarity of a tonic. In a quintet, you're doing all sorts of
stuff, and there're simply more parts to have to control.

-Carl

As requested by Monz. Here's a Scala file of the 11-limit wafso-just tuning I gave earlier. I expect that, within 2.72 cents, it is a superset of about a zillion strict-JI scales, and in a sense, renders all such scales obsolete (except for the advantage of having fewer than 22 tones and being able to tune them by ear). I'd like to find all those in Manuel's archive, but it looks difficult.

-- cut here --
!
! keenan5.scl
!
11-limit with distrib 224:225 and 384:385, max err 2.7c, Dave Keenan 24-Dec-99
22
! classic septimal undecimal
66.60106526 ! C# 25/24 28/27 512/495
115.8026469 ! Db 16/15 15/14 275/256
151.994179 ! Cx 1125/1024 35/32 12/11
201.1957607 ! D 9/8 28/25 1536/1375
231.6052938 ! Ebb 256/225 8/7 55/48
267.7968259 ! D# 75/64 7/6 64/55
316.9984075 ! Eb 6/5 135/112 2475/2048
383.5994728 ! E 5/4 56/45 1024/825
432.8010544 ! Fb 32/25 9/7 165/128
499.4021197 ! F 4/3 75/56 1375/1024
584.7952335 ! F# 45/32 7/5 384/275
651.3962987 ! Fx 375/256 35/24 16/11
700.5978803 ! G 3/2 675/448 12375/8192
767.1989456 ! G# 25/16 14/9 256/165
816.4005272 ! Ab 8/5 45/28 825/512
18/11 ! Gx 3375/2048 105/64 852.592c
883.0015925 ! A 5/3 375/224 6875/4096
932.2031741 ! Bbb 128/75 12/7 55/32
968.3947062 ! A# 225/128 7/4 96/55
9/5 ! Bb 1017.596c 405/224 7425/4096
1084.197353 ! B 15/8 28/15 512/275
2/1 ! C
-- cut here --

Here's the lattice again (with a few more tones repeated).

Fx------Cx-----Gx
/ \ /
A / \ E / B
Db / Ab \ / Eb
5 C#------G#------D#------A#
/ \ / \ / \ / \ /
/ 7 \ / Fx\ / Cx\ F / Gx\ C /
/ 11 \ / Ebb \ / Bbb \ / Fb \ /
4-------6-------9 A-------E-------B-------F#
otonal hexad / \ / \ / \ /
legend / D#\ / A#\ Db/ \ Ab/ utonal hexad
/ \ / \ / \ / legend
F-------C-------G-------D 1/9-----1/6-----1/4
/ \ Fx / \ Cx / \ Gx / \ 1/11/
/ B \Ebb/ F#\Bbb/ \ Fb/ \1/7/
/ \ / \ / \ / \ /
Db------Ab------Eb------Bb 1/5
G# / \ D# / A#
C / G \ / D
/ \ /
Ebb----Bbb------Fb

The Scala file above gives the minimum maximum-absolute-error over all 11-(odd)limit itervals. This maximum error of 2.71 cents is actually 2/9 of the 4096:4125 undecimal semicomma (= 224:225 * 384:385). This distribution was found using the algorithm I give in
http://dkeenan.com/Music/DistributingCommas.htm

The comma could instead be distributed to minimise the maximum beat rate in the 4:5:6:7:9:11 chord. Let me know if you want that.

Regards,

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>>> Here's a 22-tone 11-limit tuning that makes maximum
>>> use of this.

I wrote,

>> Cool! I wonder, can this be considered a tempering
>> of a 4-dimensional periodicity block where two of the
>> unison vectors are 224:225 and 384:385?

Joe Monzo wrote,

>Sounds right to me.

I think the answer is no. The tuning contains a gap of 116 cents between B
and C, while small intervals of 30 cents occur between D and Ebb and between
Gx and A.

Dave, why did you stop at 22 tones?

> [Paul Erlich, TD 460.19]
> Dave Keenan wrote,
>>>>
>>>> Here's a 22-tone 11-limit tuning that makes maximum
>>>> use of this.
>
> I [Paul] wrote,
>>>
>>> Cool! I wonder, can this be considered a tempering
>>> of a 4-dimensional periodicity block where two of the
>>> unison vectors are 224:225 and 384:385?
>
> Joe Monzo wrote,
>>
>> Sounds right to me.
>
> [Paul]
> I think the answer is no. The tuning contains a gap of
> 116 cents between B and C, while small intervals of 30 cents
> occur between D and Ebb and between Gx and A.

Oh well, I was just guessing... and not an educated guess, at
that. More like a shot in the dark.

I will readily accept your more carefully considered
characterization.

(hmmm... nice alliteration...)

(hmmm... nice rhyme... I must be lapsing into poetry :) ... )

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

[Paul Erlich, TD 457.13:]
[Paul:]
>>>No, Carl, remember, brass, reed, and bowed string instruments, as
>>>well as the human voice, have no inharmonicity (to the extent that is
>>>well-defined).

[Carl Lumma:]
>>Really? Is that just in theory, or has it been measured? If so, I'm
>>wondering why...

[Paul:]
>We went through this about a year or two ago. All the measurements (I
>believe Gary Morrison did quite a few) confirmed the theory on this,
>which is quite clear. The mechanisms producing the vibrations in these
>instruments (lips, reed, bow sticking and slipping, vocal cords) are
>single-element oscillators whose nonlinearities can only produce
>phase-locked harmonic partials. Go back to those old messages (I
>believe they're pre-onelist). We went into great detail.

Very interesting... this is closely related to a discussion on
resonators in September of this year. I'm a bit skeptical with it comes
to bowed instruments, since the timing of sticking and slipping can be
influenced by the string and its intimate characteristics to some
extent.

Clearly, pianos are not bound by this limitation: the single hammer
pulse will excite whatever partials are inherent in the stretched
string, harmonic or not (and they are, in fact, slightly inharmonic).

What about a pipe organ? I tend to think that, like the bowed string,
the constraint would largely but perhaps not completely apply.

JdL

John A. deLaubenfels wrote,

>I'm a bit skeptical with it comes
>to bowed instruments, since the timing of sticking and slipping can be
>influenced by the string and its intimate characteristics to some
>extent.

Well, reed, brass, and bowed string instruments _can_ produce chaotic
multiphonics under special circumstances. Other than that, the sticking and
slipping will tend to produce a periodic waveform, and all periodic
waveforms have exact harmonic partials.

>What about a pipe organ? I tend to think that, like the bowed string,
>the constraint would largely but perhaps not completely apply.

The mechanics of airflow over flute or pipe openings is not fully
understood, so we must let measurement be our guide. This is very tricky --
it involves segregating the noise and the amplitude fluctuations of the
partials from the signal to be Fourier transformed -- a process which
involves some subjective judgment (remember, a Fourier transform will give
you a set of pure tones that reproduce any original signal, but these pure
tones always have constant amplitude).