Vai

>>Oh,and about microtonality in popular/rock music,I believe Steve Vai >had
>>a piece on an album a
>>couple of years ago.I think it was 22-TET ?
>
>Although he plays 16- and 24-TET guitar, he only included some 16-TET synth
>stuff on his album. He said in an interview that he found a 10-note mode in
>16-TET and called it the Xavian mode.
>
>I have two Ibanez 7-string guitars (which Steve Vai designed), one of which I
>had refretted to 22-TET. I'm sure he would love to know about that, if I knew
>how to get in touch with him!

David First wrote:
> Ultimately, you have to use your ears. If you hear beating/phasing
> between a given pair of frequencies, then something's off.

My approach is that if you can hear a beat of about 0.5 Hz, then you
can in fact hear the underlying pure interval which is doing the phase
shifting, so the harmonic objective has not been compromised. Indeed,
synth players sometimes intentionally detune a pair of oscillators so
they'd slowly phase shift and create a "fat" sound that is richer and
less boring than a single oscillator. Beating faster than 0.5 Hz
becomes less desirable, however, and beyond vibrato rates begins
introducing actual dissonance.

Some time back I did some audio experiments to determine the extent to
which I could actually discriminate beating in just intervals.
Listening to rather loud synthesized sawtooth waves through headphones
at about 440Hz, I found that the limit of my discrimination was at the
19 limit -- for example I could manage to tune a 19/13 by eliminating
beats, but not a 21/13. So to my ear, a 21/13 a few tenths of a cent
sharp or flat makes no distinguishable difference in terms of consonance,
but does (just barely) for a 19/13.

Figuring that the 19th partial of A440 will beat at 0.5 Hz if detuned
by 0.1 cent, that becomes my desired accuracy for the goal of avoiding
audible dissonance in a conventional pitch range. I can imagine
musics that would want precision beyond that for particular effects
such as very sustained chords with rock-solid lack of phase shifting,
but that's another issue than dissonance.

Ken Wauchope

> From: Daniel Wolf
> Perhaps you misunderstood me. If the tuning resolution is, for example, 1

> Hz, in lower registers there is going to be good intonation for the
> harmonic series above 1 Hz, and all else will be rough approximations. I
> think that the contrast between the exact and approximate tunings would
be
> musically unacceptable (with a resolution of 1 Hz, try modulating from
the
> key of 80 Hz to its subdominant!). If I can't have a tuning with the
> accuracy of the Rayna, then I would like the potential deviation from
Just
> to be spread around as much as possible in a temperament that represents
> harmonic identities consistently. As long as you are not working with
> sustained textures that really require something like the Rayna, I think
> that absolute frequency resolution will be secondary to consistency. I
made
> the suggestion that equal-tempered division of the octave divisible by 12

> be considered because, MIDI standard or not, tuning tables seem to be
> expressed at the software level most often in terms of unit deviations
from
> 12tet pitch classes, not in terms of octave divisions alone, and
> manufacturers will probably be interested in supporting microtonal
capacity
> only when that does not eliminate an accurate 12tet. From my experience,
> 768tet and 1200tet are not good choices but there are ET of this
magnitude
> that would be better.
>
No, I understood you perfectly. The MIDI Tuning Standard has a maximum
resolution of 100 cents / 2^14 = .0061 cents. Only 768TET, 1536 TET,
3072TET, etc. fit within that standard. Trying to achieve some other nTET
with better consistency will have several problems from a practical
standpoint:
1. The MIDI Tuning Standard, while currently implemented by only three
manufacturers (Emu, Ensoniq and Turtle Beach), is the only standard we've
got. To propose some division of the semitone other than its 2^n divisions
would set back the little progress it has made. I know some list members
are disappointed over the lack of response to the MIDI Tuning Standard, but
this is to be expected given the design life cycles of these instrument
lines that amount to 3-8 years on average.
2. Division of the octave or semitone by other than binary multiples is
not likely to be accepted by manufacturers. They have no margin timewise
for anything more complicated than binary multiplication/division in
calculating pitch. Remember that they are using microprocessors like the
68000, 68340 (68020 comparible), or 68020 running at 16 to 25 MHz, not the
latest Pentium or RISC processor. While this situation will improve in the
future, the increase in voice count is likely to eat up any processing
speed improvements.
3. My impression is that instruments with "1 cent resolution" actually map
some 100 cent / 2^n division to the 1 cent value. The Ensoniq
EPS/16Plus/ASR-10 certainly do this. I believe Roland does it as well.
The actual scale is stored as 100 / 2^n divisions, usually 768TET, not 1
cent. This could explain why some list members have worse results with
these instruments since the conversion errors are not consistent. For
instance, if you want to adjust a pitch by +0.6 cents, you would round up
to 1 cent. The 1 cent value on your synth display is closest to 1.5625
cents, so now your pitch is adjusted by +1.5625 cents, instead of 0 cents
which would have been closer. Perhaps one request should be to show the
actual cents values to several decimal places, not roughly translated
values or hard to decipher tuning units.

John Loffink
jloffink@pdq.net

> I did the exact same experiment many years ago. I swept
> through all the intervals within an octave, and wrote down all the
> points at which beating stopped.

Ditto here. I had an old analog synthesizer with a leaky FET
transistor in the keyboard "hold" circuit, so I just tuned two
oscillators in unison, hooked one to the keyboard, and set 'er adrift.
I could adjust the keyboard gain to set the scan rate, and about 1-2
cents/sec made for an endurable 15 minute session spanning the octave.

> I found that the limit of my discrimination was at the
> 19 limit -- for example I could manage to tune a 19/13 by eliminating
> beats, but not a 21/13.

I should revise that statement somewhat, having just gone back and
looked at my results again ("Just Intonation from Leaky Keyboards,"
Interval I:4-5, 1979). During the scan I roughly classified the
intervals I could hear as "easy", "medium" and "hard" depending on how
distinctive they were, and around 17-19 was the upper limit to the
ones I found relatively "easy" to hear. From 21 up to about 29 I
could still hear cessations of beating, but requiring more "squinting
the ears". So 17-19 probably still represents the limit to which
precise tuning might make a difference to me in an actual piece of
recorded Just electronic music.

> Very close to my results! The most complex ratio I stopped at was 17/13.

That's one I classified as "medium", probably because it is only 10
cents higher than the 13/10 ("easy") whose beating partials would
still be interfering with the ear. Similarly, the next highest in
that region, 21/16, I found "hard" because it's getting so close to
the overpowering domain of the 4/3.

--Ken Wauchope

Paul H. Erlich says:
>
> For an interval in perfect just intontation, whatever phase difference
> was present at the beginning of the sound would persist for the duration
> of the sound. Let's say two instruments are producing tones, both of
> which have an equally loud partial at 1000Hz. For someone standing in a
> particular location with respect to the two instruments, there is some
> delay time between the beginning times of the two instruments for which
> the 1000Hz components will interefere constructively, resulting in four
> times the energy at that frequency than what one instrument alone would
> produce. For a delay time just 1/2000 of a second longer, you get
> destructive interference, or no energy at that frequency. Obviously no
> human instrumentalist can control their onset time to within 1/2000 of a
> second relative to the onset time of another instrument. So you end up
> with a random number between zero and four to describe the energy of the
> 1000Hz component -- probably not the musical effect you were looking
> for. Even an individual performer playing both instruments can't do it
> -- try tuning two synth keys to the same note and notice how repeatedly
> playing both keys "simultaneously" leads to random fluctuations in the
> loudness.
>
> Now if the onset times are controlled really accurately by MIDI, and
> let's say a 90-degree phase shift is chosen so that the energy at 1000Hz
> is twice that from one instrument, then you're doing okay, right? But
> for someone standing in a different location in the room, though, the
> differece in the path lengths from the two instruments, and thus the
> phase, will be different. There are nodes of destructive interference
> and antinodes of constructive interference in the room no matter what
> the original phase shift was. So some members of the audience may be
> receiving no energy at 1000Hz, while others may be receiving four times
> what they would from one instrument. Again, probably not the effect you
> were looking for.
>
> If both instruments are electronic, and their signals are mixed into one
> speaker, there will be no spatial nodes or antinodes. So finally you can
> control the musical effect. But other than monophonic electronic
> MIDI-sequenced music, perfect JI can lead to big problems. In the real
> world, thankfully, acoustic instruments are slightly out-of-tune with
> each other, so that whatever the onset delay and whatever the position
> in the room, one gets a signal that frequencies common to both tones
> have an average energy twice that of one instrument. There will not be
> noticeable, let alone startling, differences in the musical effect
> depending on minute variations in the onset delay or position in the
> room.
>

In addition, it has been my experience that when creating JI related sample
loops, that it is not enough that the frequency relationships be exact - the
period of the wave cycles involved all must have a simple divisible integer
relation with one common compound period/sample length. Further, I believe
that it is best to have a simple divisible relationship with the sampling rate
one is employing. As an example, for a recent set of 21 JI pitch relationships
I developed, I used a common 1/1 period number/sample base length of 21870. I
then doubled this number to achieve the sampling rate of 43740.

21870 divided by 1/1 = 21870
21870 " " " 9/8 = 19440
21870 " " " 6/5 = 18225
21870 " " " 5/4 = 17496
21870 " " " 27/20 = 16200
21870 " " " 45/32 = 15552
21870 " " " 3/2 = 14580
21870 " " " 27/16 = 12960
21870 " " " 9/5 = 12150
21870 " " " 15/8 = 11664
21870 " " " 81/40 = 10800
21870 " " " 135/64 = 10368
21870 " " " 729/320 = 9600
21870 " " " 1215/512 = 9216
21870 " " " 81/32 = 8640
21870 " " " 729/256 = 7680
21870 " " " 243/80 = 7200
21870 " " " 405/128 = 6912
21870 " " " 2187/640 = 6400
21870 " " " 3645/1024 = 6144
21870 " " " 243/64 = 5760

Obviously, this base number gets larger with each added interval and can very
quickly go out of bounds even when one is using the simplest of source
materials. I, myself, have found it most unwieldy, if not impossible, to
create viable samples using anything besides sawtooth, square, etc. More
complex waveforms, or anything remotely resembling the physical model of an
acoustic instrument would take more sample RAM than I,for one, currently have!

At any rate, if I read (the slightly truncated) explanation above correctly,
JI is an impossible goal in the real, 3D/air based world we reside in. That
makes perfect sense to me, but one can only try...

David F.

> From: "Paul H. Erlich"
> For a delay time just 1/2000 of a second longer, you get
> destructive interference, or no energy at that frequency.
>
> Now if the onset times are controlled really accurately by MIDI, and
> let's say a 90-degree phase shift is chosen so that the energy at 1000Hz
> is twice that from one instrument, then you're doing okay, right?
>
..otherwise known as phase cancellation.

A MIDI note on message is 20 bits at 31250 Kbaud, so the best case timing
between each consecutive note on the same MIDI cable is 640 us. Actual
results would be much worse due to processing time by your sequencer and
especially your MIDI synthesizer. Keyboard's MIDI Timing Slop tests in the
July 1996 issue showed average timing irregularities of 0.29 to 1.95 ms for
a single MIDI note and 0.30 to 3.36 ms for 1 voice alternating with 12
voices, except the Roland XP-10 which showed 19.14 ms of timing variation.
Instruments tested included some of the common microtonal ones: Ensoniq
MR-Rack, Korg M1 and Trinity, Kurzweil K2500RS, Roland XP-50 and Yamaha
TX802.

I wouldn't recommend MIDI for anyone needing Rayna/Kyma/CSOUND type
resolution.

John Loffink
jloffink@pdq.net

Ken Wauchope wrote:

> synth players sometimes intentionally detune a pair of oscillators so
> they'd slowly phase shift and create a "fat" sound that is richer and
> less boring than a single oscillator. Beating faster than 0.5 Hz
> becomes less desirable, however, and beyond vibrato rates begins
> introducing actual dissonance.

This is standard with the dual play mode on my DX21, although
that's slightly different because the two notes will have
different timbres. Even with identical timbres, you don't have
to hear the phase shifting for the sound to be changed. I usually
set a 1 or 2 cent detuning so that it doesn't interfere with the
harmony.

Doubling a waveform with detuning halves the polyphony. That means
there's no additional problem with 16 channels and pitch bend
tuning on an AWE64.

Incidentally, General MIDI includes a voice called "fifths". In
the Sound Fonts that came with my AWE64, this is two waveforms a
fifth apart. Except the "fifth" is really 700 cents, so there's
effectively a 2 cent detuning. Correcting this considerably
changes the sound. So, 12-equal functions as a microtonal scale.

While I'm plugging the AWE64... You can set a Sound Font with
instruments in 88CET fairly easily. You need one zone for each
run of notes with equal steps. Other scales are more difficult,
but I'm hoping for supporting software by the end of the year.


Anyway, after that diversion, down to business.

> Figuring that the 19th partial of A440 will beat at 0.5 Hz if detuned
> by 0.1 cent, that becomes my desired accuracy for the goal of avoiding
> audible dissonance in a conventional pitch range

I make this almost exactly 0.1 cents. There's obviously a greater
force out there shaping our destiny ...

You may want to stop the tuning getting _too_ accurate so that
the phasing becomes audible. To aim the beats between 0.4 and 0.5
cents you need about 20 millicent precision. Higher precision
means the beats can be set in tune with the tempo or harmony.
That complicates the tuning process, of course. Fast, accurate
pitch bends may be required.


What are the practical problems with implementing this precision
on a synth? I'm assuming the errors creep in when you convert
from a cents value to a frequency ratio. This looks like the
simplest algorithm that works:

if (n<0)
r = 1+ n*5.8454e-4
else
r = 1+ n*5.707e-4
endif

Where n is the tuning in cents relative to 12-equal, and -50 <= n
<= 50. I'm assuming the ratios for 12-equal will be in a lookup
table. Multiply the relevant one by r.

The worst intervals using my algorithm are 0.25 cents out.
Maybe this is why we're only given 1 cent steps at the moment. A
larger lookup table with third-tone ratios should give enough
accuracy for 0.1 cent steps.

This should be enough for wavetable synths, given the inherent
inaccuracies in the sampling process. It's also the minimum
precision for additive synthesis if it's going to reproduce small
inharmonicities.

A quadratic approximation in the cents to ratio conversion is
accurate to within 3.5 millicents. This should be enough for a
full implementation of the MIDI Tuning Standard. For a software
synth running with a Pentium chip, a cubic approximation
shouldn't be a problem. That gives a very accurate implementation
of MTS. This would be suitable for an algorithmic synth, where
the partials will usually be harmonic so that you can play with
phasing.

>While I'm plugging the AWE64... You can set a Sound Font with
>instruments in 88CET fairly easily. You need one zone for each
>run of notes with equal steps.

I'm glad to hear you're using 88CET! :-)

No, seriously now, what matters most to me is just that we explore great
new musical possibilities in whatever new tuning.

But I did want to ask something kinda embarrassing: I can't for the
life of me seem to remember what "AWE64" refers to! It seems like I used
to know that, but I can't place it. Is that a PC sound card perhaps?
(Maybe I'm just zarked after a long day!)

Gary Morrison wrote:

>> While I'm plugging the AWE64... You can set a Sound Font with
>> instruments in 88CET fairly easily. You need one zone for each
>> run of notes with equal steps.
>
> I'm glad to hear you're using 88CET! :-)

Hang on, I said you _can_ get 88CET, not that I was doing this!
Actually, before I got pitch bend tuning working with the sample
editor, i was playing with 88CET and the best approximation to
29-equal, on the "anything rather than 12" principle.

> But I did want to ask something kinda embarrassing: I can't for the
> life of me seem to remember what "AWE64" refers to! It seems like I used
> to know that, but I can't place it. Is that a PC sound card perhaps?

To your last question, yes. It stands for "Advanced Wave Effects
64 note polyphony." It uses an EMU8000 chip, so has all the
advantages of that family, but unfortunately no tuning tables and
no real time control of filters.

Sound Fonts are an Emu standard for wave samples thayt they hope
to become an industry standard. There's a "Scale Tune" parameter
that sets the interval between notes, and this can be used to get
88CET. Unfortunately, it can only take cent values so it's
useless even for a piano tuner's octave. If you think this is
stupid, try writing to soundfont@emu.com and telling them so.

The one like Dave Rybarczyk plans to write software to retune a
Sound Font to a whole keyboard scale with cent step precision.
I'm hoping he'll find the time by the end of the year. Otherwise,
I might have to do it myself!


John Loffink wrote:

> Synths frequency resolution is ultimately limited by hardware, which is
> specified in Hertz rather than cents due to the nature of the phase
> accumulator used in all wavetable/DSP synths and samplers. A lookup table
> is used by the software that converts cents to Hertz (actually the phase
> increment), either one master octave or the whole keyboard. This table
> might be stored in the synth manufacturer's custom integrated circuits or
> in their software code ROM. Due to the speed requirements and low math
> capabilities of the main microprocessors it is unlikely that an algorithm
> would be used.

I'm sure a 16-bit processor can get 0.1 cent precision with
frequencies in an efficient way. If there's enough memory for a
1536-TET tuning table, linear interpolation should only require an
8-bit multiplication and addition. If this takes a sizeable chunk
out of the 0.3ms response time, I see no problem with doing the
interpolations in a quiet moment after all the notes have been
turned on. Or, accurate frequencies could be calculated when the
table is loaded, at the expense of pitch bends.

The real reason we only get 1 cent precision must be that the
manufacturers think this is all we want. The Sound Font blurb
implies this.


Graham Breed
gbreed@cix.co.uk www.cix.co.uk/~gbreed/

>Hang on, I said you _can_ get 88CET, not that I was doing this!

Ah. But, no matter; as long as you're doing cool stuff in whatever
tuning, then I be grinnin'!





>There's a "Scale Tune" parameter
>that sets the interval between notes, and this can be used to get
>88CET. Unfortunately, it can only take cent values so it's
>useless even for a piano tuner's octave. If you think this is
>stupid, try writing to soundfont@emu.com and telling them so.

I get the impression that Emu's stand-alone MIDI synthesizers, however,
can do arbitrary tuning tables. Is that true?

mr88cet@texas.net (Gary Morrison) writes:

> I get the impression that Emu's stand-alone MIDI synthesizers,
> however, can do arbitrary tuning tables. Is that true?

Being an E-mu owner, I'd say "yes with limitations".

The module I have (Classic Keys, part of the Proteus series, I
believe) does have a user definable tuning table, and it can
be used to tune individual MIDI notes arbitrarily.

The limitation? It uses 768TET. Courtesy of the highly enlightening
discussion that was/is going on, I know now why this is less than
ideal. As a practical matter, when I tuned up 22TET on the thing, I
had some errors to distribute equitably around the octave, some of
which are as much as .7 cents (1/2 of a tuning unit). It's good
enough for my purposes, but I suspect that others would find it
otherwise.

Steve