Yahoo Tuning Groups Ultimate Backup tuning-math The 22-note per octave scale is better than 19-tone equal temperament.

You guys seem to neglect 22 note per octave scale while promoting 19-
tone edo.

I propose you promote the 22 note system as a superior alternative
to 19-tone edo.

Indian music has long used 22 notes per octave. If tuned to equal
temperament this gives a 54 cent quarter tone and good Major 7th
chord as well as a good minor triad. This temperament is good for
most intervals with factors of 2,3,5 and 7. The subminor third and
Major 3rd are nearly perfect for equal temperament. The perfect
fifth and subminor 7th are very slightly stretched.

Indian masters do not use this scale for equal temperament but
instead prefer to use something between equal temperament and just
intonation. The best compromise depends on the exact keys and
chords you wish to use.

There is a movement in India promoting 24 notes per octave. The
advocates of this do not understand that the 22-note system is not
an arbitrary choice. As an equal tempered system it works far
better than other equal tempered systems for the 7-limit. When
judiciously tuned it can be better still. The 24 note system can
only be superior if it differs radically from equal temperament.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, "rodney_clownpuncher"
<rodney_clownpuncher@...> wrote:
>
> You guys seem to neglect 22 note per octave scale while promoting 19-
> tone edo.

It's a deliberate ploy to get Paul Erlich to come back to this list.
We are hoping word reaches him, forcing him to reply.

> I propose you promote the 22 note system as a superior alternative
> to 19-tone edo.

I propose to promote peaches as superior to plums.

> Indian music has long used 22 notes per octave.

And Bosanquet found that very interesting, but the intervals are
usually not tuned very regularly. Anyway, it's more of a theory thing
than a practice thing--old, venerated theory says 22 notes to the
octave, but people have forgotten what notes.

If tuned to equal
> temperament this gives a 54 cent quarter tone and good Major 7th
> chord as well as a good minor triad. This temperament is good for
> most intervals with factors of 2,3,5 and 7. The subminor third and
> Major 3rd are nearly perfect for equal temperament. The perfect
> fifth and subminor 7th are very slightly stretched.

It'd say they were distinctly stretched, but not to the degree of
unusability. You've also not noted the fact that 22 can sort of deal
with the 11-limit, which is something it has over 19.

By the way, I wrote a few 22-et things back in the seventies, and
created the Wikipedia article

http://en.wikipedia.org/wiki/22_equal_temperament

so I'm not hostile. Paul Erlich more than anyone has his name
associated to it, using a "pajara" approach, and Herman Miller is fond
of "porcupine", which is another way of organizing the notes of 22.
You might find the "musical examples" section in the 22 equal article
of interest.

Welcome aboard and I hope you will stick around.

🔗Keenan Pepper <keenanpepper@gmail.com>

I agree with everything Gene said. 22-equal is very interesting and
very different from both 12-equal and 19-equal, but debating which
one's "better" is kinda pointless. Why don't you write some music in
it and show us how great it is?

But since this is tuning-math, let's do a little comparison:

12 tempers out 36/35 (so 7/6 and 6/5 are the same), 50/49 (so 7/5 and
10/7 are both half an octave), 64/63 (so 9/8 and 8/7 are the same),
81/80 (so 9/8 and 10/9 are the same), 126/125 (so 6/5 is half of
10/7), 128/125 (so 5/4 is one third of an octave), and 225/224 (so 5/4
is half of 14/9).

19 tempers out 49/48 (so 8/7 and 7/6 are both half of 4/3), 81/80,
126/125, 225/224, and 245/243 (so 9/7 is half of 5/3).

22 tempers out 50/49, 64/63, 225/224, 245/243, and 250/243 (so 10/9 is
half of 6/5), and also 55/54, 100/99, and 121/120 (so 10/9, 11/10, and
12/11 are all the same).

So, using ~ to represent equivalent intervals, we have in 12:

10/9~9/8~8/7 | 7/6~6/5 | 5/4~9/7 | 4/3 | 7/5~10/7

in 19:

10/9~9/8 | 8/7~7/6 | 6/5 | 5/4 | 9/7 | 4/3 | 7/5 | 10/7

and in 22:

12/11~11/10~10/9 | 9/8~8/7 | 7/6 | 6/5~11/9 | 5/4 | 14/11~9/7 | 4/3 |
11/8 | 7/5~10/7

The simplest (crudest? coarsest?) linear temperaments compatible with
12 are dominant, diminished, and augmented, which characterize a lot
of "common practice" music.

For 19, the simplest linear temperaments are semaphore, keemun, and
septimal meantone.

For 22, as Gene said, the simplest linear temperaments are pajara,
porcupine, and a related one called "hedgehog" which Herman Miller
also used.

The only linear temperament compatible with both 12 and 19 is
meantone. 12 and 22 make pajara, and 19 and 22 make a complex
temperament called "magic". The only temperament compatible with all
three is the planar temperament that tempers out 225/224 (what's that
one called again?).

Overall, I'd say 19 is an easier step from 12 for beginners, mostly
because it tempers out 81/80, by far the most important comma in
centuries of Western music, but 22 has lots to offer too. It's a
strange new musical world, and I think the works of Erlich, Miller,
and others, brilliant as they are, have only scratched the surface.

Keenan

🔗Carl Lumma <ekin@lumma.org>

🔗rodney_clownpuncher <rodney_clownpuncher@yahoo.com>

Thanks for your intelligent response. Usually when I tell people
about this stuff their eyes just glaze. I feel I've found a home.

I was mistaken to imply you were not familiar with 22-tone ET.
When I say the 22 note per octave scale seems neglected, I mean it
is not discussed much on the website.

CALCULATION OF CENT VALUES:
Buy the way, some readers may not have the formula for calculating
cent values from frequency ratios. When I started I could not find
it anywhere. I had to figure it out. Here it is

Z cents=log(X/Y)/log(2)*1200

It is sometimes useful to separate the effects of different factors.
We can use the basic laws of logarithms to do this.

log(X*Y)=log(X)+log(Y) and log(X/Y)=log(X)-log(Y)

Thus, to get the cent values for any frequency ratio, we can simply
sum the cent values of all factors in the numerator(top) and
subtract the cent values of all factors in the denominator (bottom).

Z cents=log( (X1*X2)/(Y1*Y2) )/log(2)*1200
=log(X1)/log(2)*1200 + log(X2)/log(2)*1200
-log(Y1)/log(2)*1200 - log(Y2)/log(2)*1200

CALCULATION OF ERRORS:
To find the error of any equal tempered note I sometimes simplify
things further by taking note of the difference between the cent
value of each individual factor and the nearest equal tempered note.
As before the error of factors in the numerator can be added and the
errors of factors in the denominator can be subtracted.

COMPARISON OF ERRORS IN 12, 19 and 22 EDOs:

Here is a more detailed comparison for yourself and other readers.
I find tables like the one below good for comparing equal
temperaments.

If the same factor is in the denominator the error size is the same
but the sign is negated.

Note that if two factors have errors of the same sign in the chart
above the errors will cancel when one factor is in the numerator and
the other is in the denominator. If the errors are of opposite signs
the errors will sum. Thus the large errors for factors 5,7 and 11
in 12-tone edo being of the same sign cancel out when one factor is
in the numerator and the other is in the denominator. The error for
factor 11 is a small 6 cents but it is the opposite sign of the
errors for 7 and 3 so these errors sum when used together as
numerator and denominator.

WHY I DID NOT MENTION THE 11 LIMIT IN 22-TONE EDO:
Ratios mixing 11 with prime factors 3 and 5 such as 12/11 and 11/10
are small enough to be within a critical band when inverted and
displaced by octaves to the smallest form. Thus even these
intervals are not harmonic even though 11/10 is very close to just
intonation. The ratio 11/8 seems to me to be the only ratio of
harmonic value in the 11 limit. This is why I did not mention the
11 limit.

WHY I THINK THE 7 LIMIT IS A PEACH FOR 22-EDO AND A LEMON FOR 19-EDO:
In 19edo the error in for factor 7 is 9 cents larger than the error
in 22edo. Thus these error are about 60% greater for 19-tone edo.
I will confess 19-EDO is better than 12-tone EDO.

THE 19-TONE SCALE IS A SIMPLE ALTERNATIVE TO 12-TONE SCALE FOR THE 5
LIMIT.
The great advantage of the 19 tone edo is the ease with which it can
be adapted to 12 tone EDO scores. In most cases we can adapt the 19-
tone EDO to 12-tone EDO scores by thinking of sharps and flats as
being different notes. If we think of the Major scale as being
white keys and all other notes as being black keys on a piano, we
can use the same key signature and notation we used in 12-tone ET.
In 19-tone EDO to change from one key to a key a Perfect Fifth
higher or lower we only need to change one accidental and this
accidental is the same accidental as in 12-tone EDO. Thus although
19-tone EDO has one more black key between each adjacent pair than
in 12-tone EDO. the same scores can be used as in 12-tone EDO often
with little or no change. THIS IS A GREAT WAY TO INTRODUCE
MICROTONALITY TO THOSE WHO FEAR IT.

The 22-note scale does not adapt so easily to 12-tone scores because
to play the same major scale a perfect fifth higher or lower we need
to change more than one accidental.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Carl Lumma <ekin@lumma.org>

At 10:27 PM 5/31/2006, you wrote:
>Thanks for your intelligent response. Usually when I tell people
>about this stuff their eyes just glaze. I feel I've found a home.

Welcome aboard! You might also check out the main tuning list

/tuning

and Making Microtonal Music

/makemicromusic

if you haven't already.

>WHY I DID NOT MENTION THE 11 LIMIT IN 22-TONE EDO:
>Ratios mixing 11 with prime factors 3 and 5 such as 12/11 and 11/10
>are small enough to be within a critical band when inverted and
>displaced by octaves to the smallest form. Thus even these
>intervals are not harmonic even though 11/10 is very close to just
>intonation. The ratio 11/8 seems to me to be the only ratio of
>harmonic value in the 11 limit. This is why I did not mention the
>11 limit.

11:4, 11:5, 11:6, 11:7, and 11:9 are all useful intervals in
the context of the complete 4:5:6:7:9:11 chord ("otonal hexads"
to use Partch's terms), subsets of that chord (7:9:11 is but one),
and sometimes in other contexts (some like 18:22:27 triads).
Also in higher-limit chords (13:11, etc.).
Meanwhile, the 9-limit (and the 5-prime-limit) includes 10:9,
which is within the critical band throughout much of the audio
spectrum.
What we find out with extended JI is that octave equivalence
isn't perfect (we already knew that, though, from trying to
play close-position triads in the bass).

>WHY I THINK THE 7 LIMIT IS A PEACH FOR 22-EDO AND A LEMON FOR 19-EDO:
>In 19edo the error in for factor 7 is 9 cents larger than the error
>in 22edo. Thus these error are about 60% greater for 19-tone edo.
>I will confess 19-EDO is better than 12-tone EDO.

It's better than in 12, and the 7-limit is already being used
in 12 (from my point of view, at least), so why not upgrade to
19? The thing about 22, as Keenan said, is that it's hard to play
existing music in it. But many existing musical forms can easily
be cast into 19, so it's a more sellable tuning for professional
musicians. For those wishing to shock audiences a bit more and/or
work a bit harder, 22 is a superb choice.

>THE 19-TONE SCALE IS A SIMPLE ALTERNATIVE TO 12-TONE SCALE FOR THE 5
>LIMIT.
>The great advantage of the 19 tone edo is the ease with which it can
>be adapted to 12 tone EDO scores.

Oops, should have read further before replying!

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> 11:4, 11:5, 11:6, 11:7, and 11:9 are all useful intervals in
> the context of the complete 4:5:6:7:9:11 chord ("otonal hexads"
> to use Partch's terms), subsets of that chord (7:9:11 is but one),
> and sometimes in other contexts (some like 18:22:27 triads).

5:8:11 is one chord possibility; it's in pretty good tune.

> The thing about 22, as Keenan said, is that it's hard to play
> existing music in it. But many existing musical forms can easily
> be cast into 19, so it's a more sellable tuning for professional
> musicians. For those wishing to shock audiences a bit more and/or
> work a bit harder, 22 is a superb choice.

If you want to sample some shocking existing music in 22-et, you could
try Night on Porcupine Mountain, Mahler's Pet Porcupine, or even
Symphony Fantastique in pajara on my Mad Science page:

http://66.98.148.43/~xenharmo/mad.html

🔗Keenan Pepper <keenanpepper@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

At 11:32 PM 5/31/2006, you wrote:
>On 6/1/06, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
>> Hemifourths = semifourths = (from the sound of it) semaphore. I've
>> called it godzilla, because it is a part of the Japanese movie monster
>> gang with 8/7 generators. You take a meantone fourth, and slice it in
>> two, to get a compromise 8/7~7/6. It's basically a system just for
>> 19-et, though since you can slice an 81-et fourth in half also you can
>> use it with that. You can call it 19&24 if you like. Anyway, if you
>> want to force 19 to step up to the plate as a septimal system, this is
>> a good way to proceed.
>
>Yup, that's semaphore. The 9-note MOS is on my list of intriguing
>scales, along with keemun[7], pajara[8], negrisept[9], and a bunch of
>decatonic scales.
>
>Keenan

Care to share your list? Mine's at

http://lumma.org/tuning/CKL-InterestingScls.zip

-Carl

🔗Keenan Pepper <keenanpepper@gmail.com>

On 6/1/06, rodney_clownpuncher <rodney_clownpuncher@yahoo.com> wrote:
> I was mistaken to imply you were not familiar with 22-tone ET.
> When I say the 22 note per octave scale seems neglected, I mean it
> is not discussed much on the website.

Which website?

> COMPARISON OF ERRORS IN 12, 19 and 22 EDOs:
>
> Here is a more detailed comparison for yourself and other readers.
> I find tables like the one below good for comparing equal
> temperaments.

There's just one thing you're ignoring here: tempered octaves. They
really help in certain situations.

> WHY I DID NOT MENTION THE 11 LIMIT IN 22-TONE EDO:
> Ratios mixing 11 with prime factors 3 and 5 such as 12/11 and 11/10
> are small enough to be within a critical band when inverted and
> displaced by octaves to the smallest form. Thus even these
> intervals are not harmonic even though 11/10 is very close to just
> intonation. The ratio 11/8 seems to me to be the only ratio of
> harmonic value in the 11 limit. This is why I did not mention the
> 11 limit.

Of course, this strongly depends on the timbre and range of the
instrument. Even a 5/4 sounds muddy on the lowest strings of a double
bass.

> The 22-note scale does not adapt so easily to 12-tone scores because
> to play the same major scale a perfect fifth higher or lower we need
> to change more than one accidental.

Exactly. In fact, the diatonic scale as we know it doesn't exist in
22-equal, because 81/80 is not tempered out. So the basic scale has to
be something else, like Paul Erlich's decatonic:
http://lumma.org/tuning/erlich/

Keenan

🔗Keenan Pepper <keenanpepper@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

🔗rodney_clownpuncher <rodney_clownpuncher@yahoo.com>

> On 6/1/06, rodney_clownpuncher <rodney_clownpuncher@...> wrote:
> > I was mistaken to imply you were not familiar with 22-tone ET.
> > When I say the 22 note per octave scale seems neglected, I mean
it
> > is not discussed much on the website.
>
<keenanpepper@...> wrote:
> Which website?
I mean the tuning-soft dictionary website.

> There's just one thing you're ignoring here: tempered octaves. They
> really help in certain situations.
>
Tempered octaves! Thats an interesting idea. Is this similar to
the out-of-tune piano effect?

This list is not meant to be comprehensive. For example, the 18
and 20 edos are also interesting. The 18 gives good "blues" 7/6
subminor thirds and good minor thirds and the same Major 3rds as
12edo. The 20 gives a theoretically interesting blues third on the
boundry between consonance and dissonance (240 cents) and the same
diminished 7th chord has 12-tone ET. The hyperdiminished pentad
based on 0, 240, 480, 720, 960 cents and the out of tune fourths and
fifths make this an interesting scale experimental scale for
harmonic quasiatonality.

> > The 22-note scale does not adapt so easily to 12-tone scores
> > because to play the same major scale a perfect fifth higher or
> > lower we need to change more than one accidental.

> Exactly. In fact, the diatonic scale as we know it doesn't exist in
> 22-equal, because 81/80 is not tempered out. So the basic scale
> has to be something else, like Paul Erlich's decatonic:
> http://lumma.org/tuning/erlich/
>
> Keenan

I am glad to see I am not the only one who noticed this. Yasser did
not mention it in his book.

Rodney

🔗rodney_clownpuncher <rodney_clownpuncher@yahoo.com>

> > The 22-note scale does not adapt so easily to 12-tone scores
> > because to play the same major scale a perfect fifth higher or
> > lower we need to change more than one accidental.

I am glad to see I am not the only one who noticed this. Yasser did
not mention it in his book.

Rodney

🔗rodney_clownpuncher <rodney_clownpuncher@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "rodney_clownpuncher"
> <rodney_clownpuncher@> wrote:
> >
> > Thanks for your intelligent response. Usually when I tell
people
> > about this stuff their eyes just glaze. I feel I've found a
home.
>
> Where did you learn tuning theory, and why do you pu8nch clowns?
>
I am mostly self taught. I' ve studied Helmholtz, Partch, Yasser
(though I do not agree with all his theories), Plomp and Ottman,
Kiang, Fourier, Acoustical Society of America--whatever I can get
my claws on. On tuning mostly I've done the math and done self-
exploration. I have had trigonometry, calculas and differential
equations. I believe trigonometry and fourier analysis are the
richest lines of study for the development of music theory.
Trigonometry is far too rich to be neglected. Trigonometry and
math generally are untapped gold mines for the parameterization of
sound, but we need to look deeper into trigonometry than standard
texts.

I'd like to add a new word to the tonalsoft dictionary -
the "eigenclownpuncher"...

🔗rodney_clownpuncher <rodney_clownpuncher@yahoo.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Keenan Pepper <keenanpepper@gmail.com>

On 6/1/06, Carl Lumma <ekin@lumma.org> wrote:
> The reason I asked is because I don't know what half
> of these are. Do you have scl files?

All right, here goes Keenan's List-O-Scales:

!
TOP Dominant
7
!
203.5
406.9
495.9
699.4
902.8
991.8
1195.2
!

!
TOP Keemun
7
!
68.2
317.8
386.0
635.7
703.9
953.5
1203.2
!

!
TOP Dimisept (symmetric form)
8
!
101.5
298.5
400.0
597.1
698.5
895.6
997.0
1194.1
!

!
TOP Pajara (symmetric form)
8
!
106.6
385.3
491.9
598.5
705.0
983.8
1090.3
1196.9
!

!
TOP Pajara (asymmetric form)
8
!
278.7
385.3
491.9
598.5
705.0
983.8
1090.3
1196.9
!

!
TOP August (symmetric form)
9
!
185.4
292.7
400.0
585.4
692.7
800.0
985.4
1092.7
1200.0
!

!
TOP Semaphore
9
!
193.8
387.5
446.2
640.0
698.7
892.5
951.2
1144.9
1203.7
!

!
TOP Negrisept
9
!
124.8
249.7
374.5
499.4
703.8
828.7
953.5
1078.4
1203.2
!

!
TOP Augene (symmetric form)
9
!
92.5
306.6
399.0
491.5
705.6
798.0
890.5
1104.6
1197.0
!

!
TOP Blacksmith (symmetric form)
10
!
155.4
239.2
394.6
478.4
633.8
717.5
873.0
956.7
1112.2
1195.9
!

!
TOP Pajara (symmetric form)
10
!
106.6
213.1
385.3
491.9
598.5
705.0
811.6
983.8
1090.3
1196.9
!

!
TOP Pajara (asymmetric form)
10
!
106.6
213.1
385.3
491.9
598.5
705.0
877.2
983.8
1090.3
1196.9
!

!
TOP Injera (symmetric form)
10
!
93.6
187.2
280.8
374.4
600.9
694.5
788.1
881.7
975.3
1201.8
!

!
TOP Injera (asymmetric form)
10
!
93.6
187.2
280.8
374.4
468.1
694.5
788.1
881.7
975.3
1201.8
!

!
TOP Negrisept
10
!
124.8
249.7
374.5
499.4
579.0
703.8
828.7
953.5
1078.4
1203.2
!

!
TOP Lemba (symmetric form)
10
!
90.9
230.9
370.8
461.7
601.7
692.6
832.6
972.5
1063.4
1203.4
!

!
TOP Lemba (asymmetric form)
10
!
140.0
230.9
370.8
461.7
601.7
692.6
832.6
972.5
1063.4
1203.4
!

!
TOP Magic
10
!
263.0
321.9
380.8
439.7
702.7
761.6
820.5
1083.5
1142.4
1201.3
!

!
TOP Doublewide (symmetric form)
10
!
54.6
109.3
327.0
381.6
599.3
653.9
708.6
926.2
980.9
1198.6
!

!
TOP Doublewide (asymmetric form)
10
!
54.6
272.3
327.0
381.6
599.3
653.9
708.6
926.2
980.9
1198.6
!

!
TOP Keemun
11
!
68.2
136.3
317.8
386.0
567.5
635.7
703.9
885.4
953.5
1021.7
1203.2
!

!
TOP Sensisept
11
!
131.1
262.2
393.3
443.2
574.3
705.3
755.2
886.3
1017.4
1148.5
1198.4
!

!
TOP Myna
11
!
40.7
81.5
309.9
350.6
391.4
619.8
660.5
701.2
929.7
970.4
1198.8
!

🔗Keenan Pepper <keenanpepper@gmail.com>

On 6/1/06, rodney_clownpuncher <rodney_clownpuncher@yahoo.com> wrote:
> Tempered octaves! Thats an interesting idea. Is this similar to
> the out-of-tune piano effect?

Well, it's for a different purpose. Piano tuners often stretch the
octaves, especially in the low register, because the strings are thick
and stiff and the overtones aren't exactly harmonics, as they would be
for an ideally flexible string.

I'm talking about tempering octaves in order to improve other
intervals. According to the Fundamental Theorem of Arithmetic, every
rational number can be expressed as a product of prime numbers to
integer powers. Of course no product of prime powers is exactly one,
but temperament means we adjust the prime factors until a certain
product does equal one. Take for example meantone, in which the
syntonic comma 81/80 = 2^-4 * 3^4 * 5^-1 is tempered out. We make the
prime 3 a little bit smaller (temper the fifths flat) or the prime 5 a
little bit larger (temper the major thirds sharp), and then pitches
that differ by 81/80 become the same, so things like the diatonic
scale are possible.

But wait! We forgot about the prime 2! If we make the prime 2 a little
larger (stretch the octaves), then we don't have to stretch 3 and 5 as
much, and it sounds better overall. Meantone works better with
stretched octaves, but some other temperaments work better with
squeezed octaves.

> I am glad to see I am not the only one who noticed this. Yasser did
> not mention it in his book.

Well, Yasser took the disappearence of 81/80 for granted, so obviously
he only got equal temperaments compatible with meantone: 12, 19, and
31.

Keenan

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, "rodney_clownpuncher"
<rodney_clownpuncher@...> wrote:

> I am mostly self taught. I' ve studied Helmholtz, Partch, Yasser
> (though I do not agree with all his theories), Plomp and Ottman,
> Kiang, Fourier, Acoustical Society of America--whatever I can get
> my claws on.

You might be interested in read Paul Erlich's paper on 22-et. He also
has a forthcoming one on the sort of "middle way" or "new paradigm"
theory done on tuning-math, for which an advance copy could probably
be OKed with Paul. An introduction to key concepts for people who know
a lot of math is on my website, http://www.xenharmony.org. More
digestible to most people would be Graham Breed's tuning page:

http://x31eq.com/start.htm

And Herman Miller's notes on tuning:

http://www.io.com/~hmiller/music/index.html

On tuning mostly I've done the math and done self-
> exploration. I have had trigonometry, calculas and differential
> equations. I believe trigonometry and fourier analysis are the
> richest lines of study for the development of music theory.

You are overlooking the extreme usefulness of algebra. I'd say very
important is linear algebra, group theory, elementary number theory,
and multilinear algebra. Trig, calculus, differential equations and
Fourier analysis are all quite relevant to music theory, but not so
much to tuning theory.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, "rodney_clownpuncher"
<rodney_clownpuncher@...> wrote:
>
> I must say I find the ET scales just an exercise. Since we have
> computers, to me it just makes sense to tune notes exactly to the
> tuning or mistuning you really want.

But even with computers, an et might be precisely the tuning that we
want. Suppose we decide we want 50/49 tempered out, so that 7/5~10/7.
Suppose also we want dominant seventh harmony equated to utonal 7
harmony, so that we want 8/7~9/8, and 64/63 is tempered out. So far we
have the tank two temperament "pajara". However, suppose finally that
we want to be able to use 5-limit porcupine as our organizing
principle in the 5-limit, rather than pajara, so that 250/243 is
tempered out. That last one means that instead of two minor tones
giving a major third, as in meantone, they give a minor third.

Now, all you've done is given specs which say first, how 7-limit
tetrads are structured, and second, how triads connect together. But
the conditions are equivalent to being in 7-limit 22-et. If we take
the same structure of 7-limit tetrads and decide the 5-limit is
meantone, we get 12-et instead, whereas if we decide to use the
hanson/kleismic system, we get (one version of) 34-et. So you can be
forced into an equal temperament just by making structural choices.

> I am new to this site. You guys seem to be connected to sources of
> information I have searched for but not found. Where do you find your
> information? The library is very inadequate. Do you have books you
> recommend?

The single best resource for tuning theory is this Yahoo group. You
should certainly read Helmholtz and Partch if you haven't already, though.

🔗Carl Lumma <ekin@lumma.org>

>> Care to share your list? Mine's at
>>
>> http://lumma.org/tuning/CKL-InterestingScls.zip
>
>Looks like there's a lot of overlap. Mine's basically:
>
>7 dominant keemun
>8 dimisept pajara
>9 august semaphore negrisept augene
>10 blackwood pajara injera negrisept lemba magic doublewide
>11 keemun sensisept myna
>
>I started with a list of all the MOSs of good linear temperaments,
>then took out the ones I didn't like because they didn't have enough
>chords or they had big gaps in them.

That's exactly what I've been doing! Except I haven't been doing
that much of it lately, and most of these things hadn't been
discovered when I put together the above.

What do you think about non-MOS subsets of some of these
temperaments? One of my favorite examples is hanson[8], but
I'm sure there are more good examples out there (perhaps
filling in the gaps you complained about...).

-Carl

🔗Graham Breed <gbreed@gmail.com>

Gene Ward Smith wrote:

> But even with computers, an et might be precisely the tuning that we
> want. Suppose we decide we want 50/49 tempered out, so that 7/5~10/7.
> Suppose also we want dominant seventh harmony equated to utonal 7
> harmony, so that we want 8/7~9/8, and 64/63 is tempered out. So far we
> have the tank two temperament "pajara". However, suppose finally that
> we want to be able to use 5-limit porcupine as our organizing
> principle in the 5-limit, rather than pajara, so that 250/243 is
> tempered out. That last one means that instead of two minor tones
> giving a major third, as in meantone, they give a minor third.
> > Now, all you've done is given specs which say first, how 7-limit
> tetrads are structured, and second, how triads connect together. But
> the conditions are equivalent to being in 7-limit 22-et. If we take
> the same structure of 7-limit tetrads and decide the 5-limit is
> meantone, we get 12-et instead, whereas if we decide to use the
> hanson/kleismic system, we get (one version of) 34-et. So you can be
> forced into an equal temperament just by making structural choices.

No. You missed the zeroth condition: that you want a fixed, regular temperament. If you use porcupine it only has to be tuned to 22-equal for those sections where you make use of the 7:5~10:7 approximation. You could even try an adaptive tempering to 7-limit just intonation.

Graham

🔗Herman Miller <hmiller@IO.COM>

rodney_clownpuncher wrote:
> > Tempered octaves! Thats an interesting idea. Is this similar to > the out-of-tune piano effect?

The effect is similar but more subtle. On an electronic keyboard with perfectly tuned octaves, and a sound without vibrato, you may notice that two notes an octave apart tend to blend together to form a single sound. This can be an interesting effect, but often it just sounds artificial. Very slightly detuning the octave won't sound out of tune, but it makes the individual notes sound more distinct for a more natural effect. Detuning the octave more introduces beating, which may or may not be a desired effect. But the effect of tempering the octave reduces beating in some of the other intervals of the tuning (at the expense of others).

Take the example of meantone; if the octave is just and the fifth is 1/7 comma flat, the major third will be 3/7 of a comma sharp. But if you also temper the octave by adding 1/7 of a comma, the major third is reduced by 2/7 of a comma, so it ends up only 1/7 of a comma sharp. The drawback is that some of the other intervals are not as well in tune as they would be without stretching the octave. But you can optimize the tuning of the octave and the fifth so that the error of the simpler intervals is given more weight than the error of the more complex intervals.

🔗monz <monz@tonalsoft.com>

You might be interested in taking a look at a couple of
pages on my website, which describe the error of approximation
to JI ratios and prime-factors for various equal-temperaments:

http://tonalsoft.com/enc/e/edo-11-odd-limit-error.aspx

http://tonalsoft.com/enc/e/edo-prime-error.aspx

The first link shows the error of approximation for all
ratios within the 11-odd-limit. The second link shows
the error for all prime-factors up to 43. The second page
has a mouse-over comparative graph at the bottom which works
the same way as the one on the first page.

The difference with your approach is that instead of giving
the amount of error in absolute terms (i.e., cents values),
i give the error in relative terms as a percentage of the
equal-temperament's step-size. Thus, for example, if the
error is close to 50%, that ratio or prime-factor falls
nearly midway between the two closest degrees of that
equal-temperament, and therefore that equal-temperament
does a terrible job of approximating that ratio or factor.

My point in showing you this is that the description of
one tuning as "better" than another, depends on the criteria
you are using to make the comparison. 22-edo and 19-edo both
have fairly similar error for prime-factors 3 and 5, but 22-edo
is clearly better than 19-edo at approximating primes
11, 17, 23, and 29. However, 19-edo is much better at
approximating prime-factor 23 -- in fact, 23 is nearly midway
between two degrees of 22-edo, so if you want to compose
music which features 23 as a factor, using an equal-temperament,
22-edo will be useless and 19-edo will do a good job.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗rodney_clownpuncher <rodney_clownpuncher@yahoo.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, "rodney_clownpuncher"
<rodney_clownpuncher@...> wrote:

> What I find most interesting from your demos is the way some
> temperaments polarize the intervals.
>
> For example, 18,23,24,29,33,34,36.

I presume you mean the way they have lower error on subgroups. Looking
at 18, you can see it breaks into five separate layers. The middle
layer is the {2,9/8,7/6} subgroup, which shares the tuning of 72-et.
The 9/8 has the 200-cent tuning of 12-et, but it doesn't come from a
700-cent tuning for the fifth. The excellent 7/6 can be seen as the
result of error cancelation between the very sharp 3s and 7s, or you
could just say it's inherited from 9-et, which has the limma circle of
27/25s, two of which are a 7/6 modulo 4375/4374.

If instead of 18 steps of size 34 sk, (sk=step of 612), you alternate
steps of 25sk with steps of 43sk, you get Ennealimma[18], which still
has the superb 7/6s, but now has some very fine 6/5s and 7/5s also,
and hence has nine 1-6/5-7/5 and nine 1-7/6-7/5 diminished triads in
effectively just (or whatever we decided to call it) tuning. So,
subgroups can be fun.

Ennealimmal[18] is also equipped with neutral thirds, and also has
7/6-6/5-7/6-60/49 chords, so actually you could do a lot in a
near-just sense with it. Nobody has, so it's another wide-open
opportunity.