RE: Positive Linear Temperaments

From: PAULE

John C.,
Perhaps you didn't see what I was trying to do with the optimal
positive temperaments I proposed. The idea is that we are optimizing the
_diatonic_ dominant seventh chord with respect to a tuning of 4:5:6:7 by
minimizing the weighted or unweighted mean of the squared errors of all six
intervals in that chord. No schismatic thirds, augmented sixths, etc. are
used.
There is a discussion currently about whether the 4:5:6:7 is the
classical dominant seventh chord and whether it is too consonant. In most
cases I would say that 4:5:6:7 should not be used for the dominant seventh
and therefore a negative tuning such as meantone is best, as only the triads
should be rendered as consonant as possible. But there are other styles for
which diatonic dominant sevenths or minor 6th (that means minor with an
added major 6th) chords are tonics, and should be tuned accordingly. My
results indicate that mildly positive tunings are better for such styles.
12-tET is a compromise that leans slightly toward the latter; in fact I
think it unlikely that mixolydian and dorian would have achieved their
current popularity had a negative system become standard in Western music.
-Paul E.

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Thu, 13 Mar 1997 00:03 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04955; Thu, 13 Mar 1997 00:03:56 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA04952
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id PAA15168; Wed, 12 Mar 1997 15:02:17 -0800
Date: Wed, 12 Mar 1997 15:02:17 -0800
Message-Id: <009B12C606EBAEF3.6A55@vbv40.ezh.nl>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

Aline Surman wrote:

>The discussion about whether or not our ears can hear certain
>frequencies, and the implications thereof, is interesting to me. Simply,
>just because we cannot see or hear certain vibrations does not
>necessarily mean that they have no effect on us...

Low frequencies (beyong hearing) in particular can resonate with body
parts and organs and affect us in that way. This is a fast track way to
many emotions, feelings and even pleasure and pain.

It was discovered not so long ago that although people could not hear
frequencies above 22kHz which is the maximum CD frequency that some
could tell the difference between (say) 15kHz waves with different
timbre or different overtones. Some CD manufacturers now add overtones
to the high frequencies because that normally produces a more natural
sound. This little trick does fool those individuals who could tell the
difference before.

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Fri, 14 Mar 1997 05:28 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA13534; Fri, 14 Mar 1997 05:28:09 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA13531
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id UAA08996; Thu, 13 Mar 1997 20:26:19 -0800
Date: Thu, 13 Mar 1997 20:26:19 -0800
Message-Id: <199703132324_MC2-1292-664F@compuserve.com>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

-------------------- Begin Original Message --------------------

Message text written by Neil H.

" Unfortunately, many people seem to
think if you can't see or hear something, it has no effect on us...I
don't believe this for a moment. We just don't KNOW the effect
yet..."


-------------------- End Original Message --------------------

(The following is mostly a general comment; it's a reply to Neil's
statement here only in a very loose, high-level way.)

Mankind would certainly be very unwise to draw the clearly ludicrous
conclusion that we understand everything going on around us. But on the
other hand, when it comes to things we don't know, we would also be unwise
to discount ideas widely accepted to be true for no specific reason.

Example: It would be very unwise assume that Relativity will NEVER be
replaced as the basis for the study of macroscopic mechanics. But it would
be equally unwise to suggest real validity to a UFO visitation hypothesis
that depends upon faster-than-light-speed travel.

We must build our view of the world around us upon the best model of the
universe we know, and not ignore parts of that model frivolously.

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Sat, 15 Mar 1997 02:44 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA19874; Sat, 15 Mar 1997 02:44:25 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA19869
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id RAA26554; Fri, 14 Mar 1997 17:42:52 -0800
Date: Fri, 14 Mar 1997 17:42:52 -0800
Message-Id: <199703142041_MC2-129A-89E4@compuserve.com>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

Ray Tomes wrote:

>It was discovered not so long ago that although people could not hear
>frequencies above 22kHz which is the maximum CD frequency that some

22kHz is the highest frequency that can be recorded on a CD, but I
believe the threshold of human hearing to be significantly higher
-- from memory, 60kHz, declining with age. As I only have digital
synths, I can't actually verify this!

Graham

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Mon, 17 Mar 1997 03:34 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00295; Mon, 17 Mar 1997 03:34:56 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA00289
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id SAA18109; Sun, 16 Mar 1997 18:33:18 -0800
Date: Sun, 16 Mar 1997 18:33:18 -0800
Message-Id: <199703162129_MC2-12B0-6FDA@compuserve.com>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

-------------------- Begin Original Message --------------------

"22kHz is the highest frequency that can be recorded on a CD, but I
believe the threshold of human hearing to be significantly higher
-- from memory, 60kHz, declining with age. As I only have digital
synths, I can't actually verify this!
"

-------------------- End Original Message --------------------

That would be nice, but unfortunately that's probably wishful thinking.
Our upper limit of hearing as children isn't much higher than 22KHz, and it
declines depressingly with age.

I couldn't find the usual set of curves for showing decline with age,
but according to Psychology of Music by Carl Seashore (Dover), a
60-year-old typically experiences an 8dB loss at 1KHz, 24dB at 2KHz, 44dB
at 4KHz, and 60dB at 8KHz.

Considering that there's about 120dB between the threshold of hearing
and theshold of pain, it's probably pretty safe to say that a 60-year-old
is barely able to hear anything above about 12KHz. And based on this
information, I doubt if even a 30-year-old is likely to hear much at 22KHz.


Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Mon, 17 Mar 1997 06:16 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01036; Mon, 17 Mar 1997 06:16:36 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA01032
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id VAA25370; Sun, 16 Mar 1997 21:15:06 -0800
Date: Sun, 16 Mar 1997 21:15:06 -0800
Message-Id: <199703162129_MC2-12B0-6FE0@compuserve.com>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

>Date: Sun, 16 Mar 97 18:47 GMT0
>From: gbreed@cix.compulink.co.uk (Graham Breed)
>To: tuning
>Subject: Re: unheard frequencies
>Message-ID:
>
>Ray Tomes wrote:
>
>>It was discovered not so long ago that although people could not hear
>>frequencies above 22kHz which is the maximum CD frequency that some
>
>22kHz is the highest frequency that can be recorded on a CD, but I
>believe the threshold of human hearing to be significantly higher
>-- from memory, 60kHz, declining with age. As I only have digital
>synths, I can't actually verify this!

This is constantly being argued on rec.audio.pro. However, no-one
has ever produced any documentary evidence of hearing much above
20kHz. I have no idea where this 60kHz number comes from but it
seems quite out of the ballpark!

/t

Tom Ritchford tom@mvision.com

Verge's "Little Idiot" -- Music for the mentally peculiar.
1-800-WEIRDOS http://www.weirdos.com/verge

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Mon, 17 Mar 1997 19:13 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA03583; Mon, 17 Mar 1997 19:13:16 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA03582
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id KAA06035; Mon, 17 Mar 1997 10:11:26 -0800
Date: Mon, 17 Mar 1997 10:11:26 -0800
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

Ray Tomes wrote:

>It was discovered not so long ago that although people could not hear
>frequencies above 22kHz which is the maximum CD frequency that some
>could tell the difference between (say) 15kHz waves with different
>timbre or different overtones. Some CD manufacturers now add overtones
>to the high frequencies because that normally produces a more natural
>sound. This little trick does fool those individuals who could tell the
>difference before.

I'm afraid I'm not following. How could CD manufacturers add overtones? If
you mean that they are adding harmonics to the recorded sound, then the
highest pitch that they could add a harmonic to would be 11 kHz, since its
second harmonic (first "overtone") would be at 22kHz. All CD players
include an anti-imaging filter that effectively eliminates all frequencies
higher than 22k anyway (often with significant attenuation above about
18k).

Some harmonics do occur because of distortion created in the analog
circuits as well as the "squaring-off" of waves close to the digital
resolution in either amplitude or frequency. However, audio companies work
very hard to MINIMIZE these harmonics, as listeners find them very
objectionable. Any digital harmonic distortion of tones above 11kHz should
be eliminated by the anti-imaging filter.

Harmonic distortion of waves close to the lowest representable amplitude is
generally "smoothed out" by the introduction of dither, a small amount of
noise, which effectively distributes the energy within the spurious
harmonics throughout the frequency spectrum. Listener tests have shown that
the extra noise is much less objectionable than the equivalent amount of
harmonic distortion.

It is very clear that very few people can consciously detect sine waves at
20kHz. Many people, in fact, have a hard time hearing above 15 or even
12kHz, depending on their age, sex, and lifetime exposure to loud sounds. I
have heard that some people can detect timbral differences in very high
sounds, but I have not yet seen a reference. I do not deny the possibility
that ultrasonic sounds can have some kind of significant physiological
effect, but I've yet to see any evidence to support that, much less that
this effect is somehow perceptible.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)621-8360 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^




Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Mon, 17 Mar 1997 20:15 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA03590; Mon, 17 Mar 1997 20:15:15 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA03641
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id LAA12277; Mon, 17 Mar 1997 11:13:31 -0800
Date: Mon, 17 Mar 1997 11:13:31 -0800
Message-Id: <332D9DB5.6276@dnvr.uswest.net>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

alves@Orion.AC.HMC.Edu (Bill Alves) wrote:

>Ray Tomes wrote:
>>... Some CD manufacturers now add overtones
>>to the high frequencies because that normally produces a more natural
>>sound. This little trick does fool those individuals who could tell the
>>difference before.

>I'm afraid I'm not following. How could CD manufacturers add overtones? If
>you mean that they are adding harmonics to the recorded sound, then the
>highest pitch that they could add a harmonic to would be 11 kHz, since its
>second harmonic (first "overtone") would be at 22kHz. All CD players
>include an anti-imaging filter that effectively eliminates all frequencies
>higher than 22k anyway (often with significant attenuation above about
>18k).

I'm sorry, I have confused things by leaving out a word. I should have
said CD-player manufacturers. The harmonics are added during the play
of the CD because obviously they cannot be recorded as you say.

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Tue, 18 Mar 1997 21:10 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04970; Tue, 18 Mar 1997 21:10:14 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA04967
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id LAA23623; Tue, 18 Mar 1997 11:37:08 -0800
Date: Tue, 18 Mar 1997 11:37:08 -0800
Message-Id: <3405e71d.1563949473@kcbbs.gen.nz>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu

>I'm sorry, I have confused things by leaving out a word. I should have
>said CD-player manufacturers. The harmonics are added during the play
>of the CD because obviously they cannot be recorded as you say.

That clarifies things somewhat, but I'm still interested to know the source
of this information. I find it amazing that they would ADD harmonic
distortion to the audio, and after going to all the trouble of removing
frequencies in that range with the anti-imaging filter. If we are indeed
talking about harmonics in the >20khz range, I also wonder if most people's
stereos are able to reproduce them. If the player manufacturers are
measuring their specs in a fair and standard way (admittedly an iffy
proposition), then it would be easy to see if they are adding any
harmonics, because the harmonic distortion rating on player specs would be
higher now than in the past.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)621-8360 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^




Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Wed, 19 Mar 1997 05:23 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05387; Wed, 19 Mar 1997 05:23:49 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA05390
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id UAA13367; Tue, 18 Mar 1997 20:21:50 -0800
Date: Tue, 18 Mar 1997 20:21:50 -0800
Message-Id: <199703182243_MC2-12CC-97BD@compuserve.com>
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu