Yahoo Tuning Groups Ultimate Backup tuning A 17-note scale in an alternative Western History

Dear George,

Intriguingly, you have given a date which marks the era just after Avicenna and Averroes. The prominent 13th century Maqam theorist was, of course, Safiyuddin Urmawi of Baghdad. Interestingly enough, he did propose a 17-note per octave system with the following ratios:

Safiyuddin's 17-tone pythagorean scale
|
0: 1/1 0.000 unison, perfect prime
1: 256/243 90.225 limma, Pythagorean minor second
2: 65536/59049 180.450 Pythagorean diminished third
3: 9/8 203.910 major whole tone
4: 32/27 294.135 Pythagorean minor third
5: 8192/6561 384.360 Pythagorean diminished fourth
6: 81/64 407.820 Pythagorean major third
7: 4/3 498.045 perfect fourth
8: 1024/729 588.270 Pythagorean diminished fifth
9: 262144/177147 678.495 Pythagorean diminished sixth
10: 3/2 701.955 perfect fifth
11: 128/81 792.180 Pythagorean minor sixth
12: 32768/19683 882.405 Pythagorean diminished seventh
13: 27/16 905.865 Pythagorean major sixth
14: 16/9 996.090 Pythagorean minor seventh
15: 4096/2187 1086.315 Pythagorean diminished octave
16: 1048576/531441 1176.540 Pythagorean diminished ninth
17: 2/1 1200.000 octave

Having read your comment about a possible super-pythagorean diversion in Western history, I must make mention of the fact that a cycle thru the relative frequencies of the perdes Saba - Şehnaz - Hisar - Segah - Evc must be based on fifths that are super-pythagorean, or else, the mapping of the pitches will be incorrect.

Thus, employing a highly flexible quarter-tone notation, the aforementioned pitches can only be notated for transposing instruments as shown below:

Sol b - Re d - La d - Mi d - Si

The 79 MOS 159-tET proposition of mine where Si is a natural tone without any accidentals, does indeed have the property of starting such a cycle at any degree:

Saba (710) Şehnaz (710) Hisar (710) Segah (710) Evc

In reality, these perdes should be understood as pitch-clusters, where there is a possible 30-40 cent variation for each note depending on the Maqam. The above-mentioned cycle is a theoretical necessity though, where Evc may start from B or B/ according to a 79-tET notation.

Now I'm very curious as to your suggestion which comprises 11 and 13 limit ratios.

Cordially,
Ozan

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 26 Eylül 2005 Pazartesi 21:51
Subject: [tuning] Re: scales formed by alternating thirds/dual generators

Margo Schulter and I speculated about an alternate historical path
for *western Europe* beginning around the 13th century, but in a
superpythagorean rather than a schismic framework. Following the
idea that the requirements for effective melody (i.e., small
semitones via wide fifths) might have taken precedence over consonant
harmony (i.e., more consonant triads via narrow fifths), we concluded
that a tempered 6:7:9 would have eventually become the most consonant
tonic triad in a septimal non-5 harmonic system, with 11 (as
6:7:9:11, e.g.) entering the picture as harmonic development
progressed. This would have resulted in a closed system of either 17
or 22 tones. In our alternate path we chose 17 over 22 for several
reasons: its melodic properties, a fifth less heavily tempered, our
desire to maintain non-5 harmony, and the possibility of interpreting
neutral intervals as ratios of 13 (as well as 11). In this there are
striking parallels (as well as notable differences) with the
development of 5-limit harmony in a meantone framework (as explained
in my neo-medieval paper, which should appear some day in
Xenharmonikon 18). For example, for the 17-division we expressed a
preference for a well-temperament over 17-ET, but had we chosen the
22-division, it would have probably been 22-equal, in which an
interval of 4 degrees is very close to a super-meantone 2nd (midway
between 8:9 and 7:8), making 22 analogous to 31.

In our private discussion we wondered whether this might have any
relationship to Turkish or Arabic tuning. Perhaps others would like
to comment.

--George

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🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>
> Intriguingly, you have given a date which marks the era just after
Avicenna and Averroes. The prominent 13th century Maqam theorist was,
of course, Safiyuddin Urmawi of Baghdad. Interestingly enough, he did
propose a 17-note per octave system with the following ratios:
>
> Safiyuddin's 17-tone pythagorean scale
> |
> 0: 1/1 0.000 unison, perfect prime
> 1: 256/243 90.225 limma, Pythagorean minor
second
> 2: 65536/59049 180.450 Pythagorean diminished third
> 3: 9/8 203.910 major whole tone
> 4: 32/27 294.135 Pythagorean minor third
> 5: 8192/6561 384.360 Pythagorean diminished
fourth
> 6: 81/64 407.820 Pythagorean major third
> 7: 4/3 498.045 perfect fourth
> 8: 1024/729 588.270 Pythagorean diminished
fifth
> 9: 262144/177147 678.495 Pythagorean diminished sixth
> 10: 3/2 701.955 perfect fifth
> 11: 128/81 792.180 Pythagorean minor sixth
> 12: 32768/19683 882.405 Pythagorean diminished seventh
> 13: 27/16 905.865 Pythagorean major sixth
> 14: 16/9 996.090 Pythagorean minor
seventh
> 15: 4096/2187 1086.315 Pythagorean diminished
octave
> 16: 1048576/531441 1176.540 Pythagorean diminished ninth
> 17: 2/1 1200.000 octave
>
> Having read your comment about a possible super-pythagorean
diversion in Western history,

But completely hypothetical

> I must make mention of the fact that a cycle thru the relative
frequencies of the perdes Saba - Þehnaz - Hisar - Segah - Evc must
be based on fifths that are super-pythagorean, or else, the mapping
of the pitches will be incorrect.
>
> Thus, employing a highly flexible quarter-tone notation, the
aforementioned pitches can only be notated for transposing
instruments as shown below:
>
> Sol b - Re d - La d - Mi d - Si
>
> The 79 MOS 159-tET proposition of mine where Si is a natural tone
without any accidentals, does indeed have the property of starting
such a cycle at any degree:
>
> Saba (710) Þehnaz (710) Hisar (710) Segah (710) Evc
>
>
> In reality, these perdes should be understood as pitch-clusters,
where there is a possible 30-40 cent variation for each note
depending on the Maqam. The above-mentioned cycle is a theoretical
necessity though, where Evc may start from B or B/ according to a 79-
tET notation.

That's quite a variation in pitch! :-)

> *
>
> Now I'm very curious as to your suggestion which comprises 11 and
13 limit ratios.
>
> Cordially,
> Ozan

I've already written a 26-page paper about how this all comes
together in my 17-tone well-temperament (which I was hoping would be
published in XH18 by this time).

If you want to try the tuning, here it is:

! secor_17wt.scl
!
George Secor's well temperament with 5 pure 11/7 and 3 near just
11/6
17
!
66.74120
144.85624
214.44090
278.33864
353.61023
428.88181
492.77955
562.36421
640.47925
707.22045
771.11819
849.23324
921.66136
985.55910
1057.98722
1136.10226
2/1

The fifths are in two different sizes. If C 0 cents, then the five
best keys for building 6:7:9:11:13 chords are on F, C, G, D, and A.
I suggest starting with the following heptatonic scale using this as
a tonic chord:

C Dv Eb F G Av Bv C
0 2 4 7 10 12 15 17 degrees of 17
1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1

Interestingly, this scale requires no tempering at all, so you could
also try it in JI.

Another approach is to use a conventional diatonic scale (except, of
course, that it's tuned super-pythagorean), introducing the 11's and
13's as chromatically or enharmonically altered tones.

Also possible are MOS scales:

1) 9-tone scale using 2-deg generator (comprising three 6:7:9:11:13
chords)
2) 11-tone scale using 6-deg generator (comprising five 6:7:9:11
chords)

That's the answer to your question in a nutshell.

Best,

--George

🔗Ozan Yarman <ozanyarman@superonline.com>

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 27 Eylül 2005 Salı 21:43
Subject: [tuning] Re: A 17-note scale in an alternative Western History

>
> Saba (710) Şehnaz (710) Hisar (710) Segah (710) Evc
>
>
> In reality, these perdes should be understood as pitch-clusters,
where there is a possible 30-40 cent variation for each note
depending on the Maqam. The above-mentioned cycle is a theoretical
necessity though, where Evc may start from B or B/ according to a 79-tET notation.

That's quite a variation in pitch! :-)

I should have said that each perde is the center of a cluster whose width is about 30-40 cents.

I've already written a 26-page paper about how this all comes
together in my 17-tone well-temperament (which I was hoping would be published in XH18 by this time).

If you want to try the tuning, here it is:

The fifths are in two different sizes. If C 0 cents, then the five best keys for building 6:7:9:11:13 chords are on F, C, G, D, and A.

Why, it's a fabulous suggestion! I already have located many maqams in this beautiful tuning.

I suggest starting with the following heptatonic scale using this as a tonic chord:

C Dv Eb F G Av Bv C
0 2 4 7 10 12 15 17 degrees of 17
1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1

That's nothing other than the fundamental scale for Maqam Huseini.

Interestingly, this scale requires no tempering at all, so you could also try it in JI.

I already know how beautiful it sounds. It's most gratifying to know that your tuning incorporates it by default.

Another approach is to use a conventional diatonic scale (except, of course, that it's tuned super-pythagorean), introducing the 11's and 13's as chromatically or enharmonically altered tones.

What is the difference between the two alterations?

Also possible are MOS scales:

1) 9-tone scale using 2-deg generator (comprising three 6:7:9:11:13
chords)
2) 11-tone scale using 6-deg generator (comprising five 6:7:9:11
chords)

Why the need to specify MOS scales out of so few pitches? You could define these as MOS modes instead, is that not so?

That's the answer to your question in a nutshell.

Best,

--George

Thank you very much.

Cordially,
Ozan

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> ----- Original Message -----
> From: George D. Secor
> To: tuning@yahoogroups.com
> [gs:] ...
> I've already written a 26-page paper about how this all comes
> together in my 17-tone well-temperament (which I was hoping would
be published in XH18 by this time).
>
> If you want to try the tuning, here it is:
>
> ! secor_17wt.scl
> !
> George Secor's well temperament with 5 pure 11/7 and 3 near just
> 11/6
> 17
> !
> 66.74120
> 144.85624
> 214.44090
> 278.33864
> 353.61023
> 428.88181
> 492.77955
> 562.36421
> 640.47925
> 707.22045
> 771.11819
> 849.23324
> 921.66136
> 985.55910
> 1057.98722
> 1136.10226
> 2/1
>
> The fifths are in two different sizes. If C 0 cents, then the
five best keys for building 6:7:9:11:13 chords are on F, C, G, D, and
A.
>
> Why, it's a fabulous suggestion! I already have located many maqams
in this beautiful tuning.
>
> [gs:]
> I suggest starting with the following heptatonic scale using this
as a tonic chord:
>
> C Dv Eb F G Av Bv C
> 0 2 4 7 10 12 15 17 degrees of 17
> 1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1
>
> [oy:]
> That's nothing other than the fundamental scale for Maqam Huseini.

Aha! I was experimenting with this scale in a more Western manner,
with full harmony -- very different from a maqam style, no doubt. :-)

I also found that occasional chromatically or enharmonically altered
tones had a nice effect.

> ...
> [gs:]
> Another approach is to use a conventional diatonic scale (except,
of course, that it's tuned super-pythagorean), introducing the 11's
and 13's as chromatically or enharmonically altered tones.
> [oy:]
> What is the difference between the two alterations?

In 17 a chromatic alteration (using a sharp or flat) is 2 degrees,
whereas enharmonic pairs of sharps and flats (e.g., D# vs. Eb) differ
by a single degree. So an enharmonic alteration would be one degree
(using a semisharp or semiflat). An example of an enharmonic
progression (taken from my paper) would be:

Bv B C
G------
F---- Eb
Dv Db C

This is the resolution of a 13:16:18:22 chord to a subminor
(6:7:9:12) triad with passing tones introduced between the two. The
first and final chords are all members of the fundamental scale I
gave above.

> Also possible are MOS scales:
>
> 1) 9-tone scale using 2-deg generator (comprising three
6:7:9:11:13
> chords)
> 2) 11-tone scale using 6-deg generator (comprising five 6:7:9:11
> chords)
>
> Why the need to specify MOS scales out of so few pitches? You could
define these as MOS modes instead, is that not so?

How do you reckon 9 or 11 tones to be "so few" pitches for a scale.
I thought that 11 tones might even be too many to be comprehended
easily.

Of did you mean that a 17-tone tuning is "so few" pitches? Even that
is too many tones for a scale, if one defines a scale as a set of
tones which one can use to write a melody. A scale may or may not be
a subset of a tuning (and it may or may not be MOS).

I wouldn't call the above MOS scales "modes." As I understand it,
modes of a scale consist of all of the members of that scale, being
distinguished from one another only by which tone is considered the
tonic or starting tone.

BTW, Jacob Barton rediscovered the above 11-tone MOS scale earlier
this year:
/makemicromusic/topicId_9132.html#9141
Shortly thereafter I proposed the name "barton" for the scale, since
he seems to have been the first to use it in a composition:
/tuning-math/message/11700

Best,

--George

🔗George D. Secor <gdsecor@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

Dear George,

>
> C Dv Eb F G Av Bv C
> 0 2 4 7 10 12 15 17 degrees of 17
> 1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1
>
> [oz:]
> That's nothing other than the fundamental scale for Maqam Huseini.

[gs]
Aha! I was experimenting with this scale in a more Western manner,
with full harmony -- very different from a maqam style, no doubt. :-)

[oz]
Really now, I would not have thought that one could practice with such a scale without discovering elements of Maqam Music by chance, or something akin to it.

> What is the difference between the two alterations?

[gs]
In 17 a chromatic alteration (using a sharp or flat) is 2 degrees,
whereas enharmonic pairs of sharps and flats (e.g., D# vs. Eb) differ by a single degree. So an enharmonic alteration would be one degree (using a semisharp or semiflat). An example of an enharmonic progression (taken from my paper) would be:

Bv B C
G------
F---- Eb
Dv Db C

[oz]
You mean to say, that a chromatic alteration is an augmented prime, which is "the difference between the `whole-tone` and the `diatonic semitone` (that is otherwise 5 `fifths` up minus 3 `octaves` subtracted from the fundamental tone)"?

Then a chromatic semitone of your 17 tone scale would be:

214.441 - 63.898 = 150.543 cents

However, at other locations in your scale, the size of the chromatic semitone seems to diminish as far down as 130.639 cents. Correct?

As for the enharmonic semitone, I have read a little from Monz's "Speculations on Marchetto of Padua's `Fifth-tones`" (©, 1997-8-9) at http://www.sonic-arts.org/monzo/marchet/marchet.htm
that an enharmonic semitone is as wide as 2 diesis within the whole-tone.

But I understand that you mean those enharmonic pairs of tones which include, e.g., Eb and D#, both of which, in a closed cyclic system of tuning such as your temperament extra-ordinaire, produce the same pitch. There is obviously no enharmonic equivalance of tones in, say, a 1/4 comma meantone. Likewise, your 17-tone temperament (circular as it is) also has no flat-sharp equivalance, although perhaps `flat-doublesharp`, or `sharp-doubleflat` equivalance at certain locations. Correct?

[gs]
How do you reckon 9 or 11 tones to be "so few" pitches for a scale. I thought that 11 tones might even be too many to be comprehended easily.

Or did you mean that a 17-tone tuning is "so few" pitches? Even that is too many tones for a scale, if one defines a scale as a set of tones which one can use to write a melody. A scale may or may not be a subset of a tuning (and it may or may not be MOS).

[oz]
I, for one, consider 79 MOS 159-tET to be insufficient for Maqam Music, let alone 11. One may use as many as 30-40 pitches in a rolling Fasl that modulates through 14 Maqams and Terkibs within a single hour.

For expressing all Maqamat adequately through all Ahenks (which requires a change of diapason based practically on 12-EQ) with all the demanded microtonal articulations, vibratos, tremolandos, trills, etc... I might need to consider all the tones of 193-EQ which gives us 3 uninterrupted cycles of fifths whose sizes are 708.8, 702.6 and 696.4 cents respectively.

We are only able to categorize the subtle-nuances of pitch in Maqam Music as pitch-clusters, which obviously need to be sub-divided into microtonal compartments for better clarity. On the other hand, my highly crowded MOS suggestions are, I reluctantly admit, insufficient to express the complex tonal patterns of a live Maqam Music performance. Such is the scope of the tonal resource that Maqam Music contains!

[gs]
I wouldn't call the above MOS scales "modes." As I understand it,
modes of a scale consist of all of the members of that scale, being
distinguished from one another only by which tone is considered the
tonic or starting tone.

[oz]
I imagine that not all would share your definition of `mode` here, which you clearly restrict to the octave-species of a single diatonical framework. In Fractal Tune Smithy, Robert uses the term Arpeggio for picking the desired tones from a larger scale, in Scala, Manuel uses the term mode instead.

[gs]
BTW, Jacob Barton rediscovered the above 11-tone MOS scale earlier
this year:
/makemicromusic/topicId_9132.html#9141
Shortly thereafter I proposed the name "barton" for the scale, since
he seems to have been the first to use it in a composition:
/tuning-math/message/11700

[oz]
Do we have a chance to hear it online?

Best,

--George

Cordially,
Ozan

🔗Ozan Yarman <ozanyarman@superonline.com>

Carl,

If we shall constrain the usage of mode to Byzantine liturgical modes (and the Greek modes prior to that) where the only meaning maintained is `octave-species` in a diatonical framework, then I approve entirely, even if it means drastic revisions in our current understanding of the `Orient`.

But then, Maqams are not simply scales. They are `keys` or `change of keys` that set the tonality of the composition. For example, a composition in the Key (Maqam) of Rast can modulate to Segah and back, without any significant damage to the character of Rast, or rather, the Rast-ness of the piece, since they use, more or less, the same scale.

Even when there is considerable diversion when modulating from one Maqam to the next, the concept of key is still in place. Furthermore, there need be no change in the key signature for a modulation to take place in Maqam Music.

Cordially,
Ozan

----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 29 Eylül 2005 Perşembe 21:52
Subject: [tuning] Re: A 17-note scale in an alternative Western History

Hi George & Ozan,

> I wouldn't call the above MOS scales "modes." As I understand it,
> modes of a scale consist of all of the members of that scale, being
> distinguished from one another only by which tone is considered the
> tonic or starting tone.

That's an important issue I've been considering bringing up.
As I think you both know, somebody seems to have translated
Maqam music theory such that "modes" is used where "scales"
is meant. Some of this terminology even made it into Scala.
The usage as George undestands it, though, seems the better.
What does everyone think? Can we agree to a standard?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> BTW, Jacob Barton rediscovered the above 11-tone MOS scale earlier
> this year:
> /makemicromusic/topicId_9132.html#9141
> Shortly thereafter I proposed the name "barton" for the scale, since
> he seems to have been the first to use it in a composition:
> /tuning-math/message/11700

I think it might be better to use "barton" as a temperament name.
The 11/31 generator is on the boundry of two linear temperaments.
One is what we've been calling "squares", the 31&45 temperament
with wedgie <<4 16 9 16 3 -24|| and TM basis {81/80, 2401/2400}. The
other I propose could be called "barton", it is the 17&31 temperament
with wedgie <<4 -15 9 -33 3 63|| and TM basis {1728/1715, 3645/3584}.
The latter, since you mention a 17-et MOS, may be more what you have
in mind.

A copop generator for barton is 94/265, this gives MOS of sizes
5, 8, 11, 14, 17, 31, 48, 79 etc. 94/265 is 9.42 cents flat from a
pure 9/7 at 425.66 cents. A copop generator for squares is 104/293;
this gives MOS of sizes 5, 8, 11, 14, 17, 31, 45, 76 etc. 104/293 is
9.15 cents flat from a pure 9/7 at 425.94 cents. Obviously, in
practice for smaller MOS these are conflated, but theory wants to
treat them seperately. This sort of thing sometimes happens when there
is an equal temperament which serves as the boundry between them and
which may as well be used for both. In this case, that equal
temperament is 31-et, which has a 9/7 which is 9.28 cents flat at
425.81 cents. It is reasonable therefore to treat these as the same
and consider the MOS to be 31-et MOS, which is what Barton did.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗microtonalist <mark@equiton.waitrose.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

Carl, it's been tried for certain, by Rauf Yekta foremost of all. If you are able, I recommend that you go find a 1922 Edition of Albert Lavignac's Encyclopedia of Music and Conservatory Dictionary. Look for the lenghty article on `Turkish Music` within pages 2945-3064 in French. This section has been prepared by Yekta personally. I don't agree with his zeal or his boorish attempts to prove that Pythagorean tuning is supreme for Maqams. Nevertheless, he DID try to introduce Western-oriented (excuse the pun) ears to the world of Maqamat by composing little pieces in the manner of waltz according to his own system of microtonal notation.

Nevertheless, 24-Pythagorean does not suffice to explain the delicate nature of this genre. His shortcomings can be summarized as follows:

1. Disregarding the importance of temperament as a necessary tool for easy&correct modulation and transposition.

2. Misunderstanding Western staff notation and the function of accidentals (apotome sharp being 7 fifths up minus 4 octaves without exception, tempered or not).

3. Misrepresenting the pitches of a Pythagorean system of tuning by using the sharps and flats the wrong way and hampering transposability as a result.

4. Forsaking a strict JI approach entirely.

5. Notating Rast as D first, then as G according to Mansur Ahenk in standard diapason with an octave error.

6. Leaving behind him a musical heritage that has no bearing of the concept of Key Transposition and what it entails.

Of course I will continue to use the term Maqam for the Maqamat. But when it comes to explaining what this really is, which do you think is a better term? Mode or Key?

Granted, some maqams function as modes of a common scale, but that is not all there is to it. Segah is not just a Rast scale with the tonic at 3rd degree, because unlike rast, it requires a 13/11 (or 32/27) as a leading tone, which you will not find it Rast. And certain notes of Rast can alterate radically, such as chargah (4th degree) becoming hisar (7/5 to 10/7), and evdj becoming adjem (16/9)

For a better explanation, I ask for your patience as I prepare the framework of my dissertation in the English language.

Cordially,
Ozan
----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 30 Eylül 2005 Cuma 12:35
Subject: [tuning] Re: A 17-note scale in an alternative Western History

> But then, Maqams are not simply scales. They are `keys` or
> `change of keys` that set the tonality of the composition. For
> example, a composition in the Key (Maqam) of Rast can modulate
> to Segah and back, without any significant damage to the
> character of Rast, or rather, the Rast-ness of the piece, since
> they use, more or less, the same scale.

Then let's use the word Maqam for Maqams. But I think our
Western readers (including myself) could benefit greatly from
a 'for dummies' explanation of this kind of modulation, and
exactly what sort of information a Maqam conveys, with examples
and entirely in Western terminology. Does such a thing exist?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>
> >
> > C Dv Eb F G Av Bv C
> > 0 2 4 7 10 12 15 17 degrees of 17
> > 1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1
> >
> > [oz:]
> > That's nothing other than the fundamental scale for Maqam Huseini.
>
> [gs]
> Aha! I was experimenting with this scale in a more Western manner,
> with full harmony -- very different from a maqam style, no
doubt. :-)
>
> [oz]
> Really now, I would not have thought that one could practice with
such a scale without discovering elements of Maqam Music by chance,
or something akin to it.

Elements, perhaps -- but the style I would expect to be quite
different, since I was using mostly 4-part harmony.

> > What is the difference between the two alterations?
>
> [gs]
> In 17 a chromatic alteration (using a sharp or flat) is 2 degrees,
> whereas enharmonic pairs of sharps and flats (e.g., D# vs. Eb)
differ by a single degree. So an enharmonic alteration would be one
degree (using a semisharp or semiflat). An example of an enharmonic
progression (taken from my paper) would be:
>
> Bv B C
> G------
> F---- Eb
> Dv Db C
>
> This is the resolution of a 13:16:18:22 chord to a subminor
> (6:7:9:12) triad with passing tones introduced between the two. The
> first and final chords are all members of the fundamental scale I
> gave above.
>
> [oz]
> You mean to say, that a chromatic alteration is an augmented prime,
which is "the difference between the `whole-tone` and the `diatonic
semitone` (that is otherwise 5 `fifths` up minus 3 `octaves`
subtracted from the fundamental tone)"?

Yes.

> Then a chromatic semitone of your 17 tone scale would be:
>
> 214.441 - 63.898 = 150.543 cents
>
> However, at other locations in your scale, the size of the
chromatic semitone seems to diminish as far down as 130.639 cents.
Correct?

Yes.

> As for the enharmonic semitone, I have read a little from
Monz's "Speculations on Marchetto of Padua's `Fifth-tones`" (©, 1997-
8-9) at http://www.sonic-arts.org/monzo/marchet/marchet.htm
> that an enharmonic semitone is as wide as 2 diesis within the whole-
tone.

The single degree in 17-WT ranges from about 64 to 78 cents, and a
diesis is 1/5 tone (~40 cents). So 2 dieses for an enharmonic
semitone is in the right ballpark. (I also see that Marchetto's
chromatic semitone is given as 4 dieses.)

> But I understand that you mean those enharmonic pairs of tones
which include, e.g., Eb and D#, both of which, in a closed cyclic
system of tuning such as your temperament extra-ordinaire, produce
the same pitch.

That statement would apply only to a closed cyclic system of *12*
tones.

> There is obviously no enharmonic equivalance of tones in, say, a
1/4 comma meantone. Likewise, your 17-tone temperament (circular as
it is) also has no flat-sharp equivalance, although perhaps `flat-
doublesharp`, or `sharp-doubleflat` equivalance at certain locations.
Correct?

No; flat-semisharp and sharp-semiflat would be equivalent in 17.
Your statement above would be true for the 19-division of the octave.

> [gs]
> How do you reckon 9 or 11 tones to be "so few" pitches for a scale.
I thought that 11 tones might even be too many to be comprehended
easily.
>
> Or did you mean that a 17-tone tuning is "so few" pitches? Even
that is too many tones for a scale, if one defines a scale as a set
of tones which one can use to write a melody. A scale may or may not
be a subset of a tuning (and it may or may not be MOS).
>
> [oz]
> I, for one, consider 79 MOS 159-tET to be insufficient for Maqam
Music, let alone 11. One may use as many as 30-40 pitches in a
rolling Fasl that modulates through 14 Maqams and Terkibs within a
single hour.

But what one commonly thinks of as a "scale" (see my definition,
above) would exclude that many pitches. If a melody modulates from
one set of tones to another, then it is used in more than one scale,
inasmuch as it is being "transported" from one scale to another. I
would consider those 30-40 pitches to be a "tuning", or at least a
subset of a tuning. And I would not feel compelled to consider non-
scale tones used as ornamental embellishments _de facto_ members of
the scale, just as a chromatically altered tones in Western diatonic
music are not considered members of the (heptatonic) major or minor
scale in which a composition is written.

> For expressing all Maqamat adequately through all Ahenks (which
requires a change of diapason based practically on 12-EQ) with all
the demanded microtonal articulations, vibratos, tremolandos, trills,
etc... I might need to consider all the tones of 193-EQ which gives
us 3 uninterrupted cycles of fifths whose sizes are 708.8, 702.6 and
696.4 cents respectively.
>
> We are only able to categorize the subtle-nuances of pitch in Maqam
Music as pitch-clusters, which obviously need to be sub-divided into
microtonal compartments for better clarity. On the other hand, my
highly crowded MOS suggestions are, I reluctantly admit, insufficient
to express the complex tonal patterns of a live Maqam Music
performance. Such is the scope of the tonal resource that Maqam Music
contains!
>
> [gs]
> I wouldn't call the above MOS scales "modes." As I understand it,
> modes of a scale consist of all of the members of that scale, being
> distinguished from one another only by which tone is considered the
> tonic or starting tone.
>
> [oz]
> I imagine that not all would share your definition of `mode` here,
which you clearly restrict to the octave-species of a single
diatonical framework. ...

There was further discussion about this, and I would agree that using
the term "Maqam" instead of "mode" would eliminate all sorts of
confusion -- once its meaning was explained.

> [gs]
> BTW, Jacob Barton rediscovered the above 11-tone MOS scale earlier
> this year:
> /makemicromusic/topicId_9132.html#9141
> Shortly thereafter I proposed the name "barton" for the scale,
since
> he seems to have been the first to use it in a composition:
> /tuning-math/message/11700
>
> [oz]
> Do we have a chance to hear it online?

/makemicromusic/topicId_9132.html#9132
/makemicromusic/topicId_9358.html#9358

Best,

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 30 Eylül 2005 Cuma 22:59
Subject: [tuning] Re: A 17-note scale in an alternative Western History

I am making a conscious effort to understand. So I got these right so far:

Diatonic Semitone:
1/1 minus (5 fifths - 3 octaves)

Chromatic Semitone / Augmented Prime:
(2 fifths - 1 octave) - D.s.

Enharmonic Semitone:
2 enharmonic dieses (two 128/125)

Is there any easier formula to define these in a glance?

That statement would apply only to a closed cyclic system of *12*
tones.

Exactly, your 17 does not have enharmonic equivalance of apotome sharps and flats. Right?

> There is obviously no enharmonic equivalance of tones in, say, a
1/4 comma meantone. Likewise, your 17-tone temperament (circular as it is) also has no flat-sharp equivalance, although perhaps `flat-
doublesharp`, or `sharp-doubleflat` equivalance at certain locations. Correct?

No; flat-semisharp and sharp-semiflat would be equivalent in 17.
Your statement above would be true for the 19-division of the octave.

I meant to say apotome flat-apotome sharp equivalance. Yes, you are right about 19 of course.

> I, for one, consider 79 MOS 159-tET to be insufficient for Maqam
Music, let alone 11. One may use as many as 30-40 pitches in a
rolling Fasl that modulates through 14 Maqams and Terkibs within a
single hour.

But what one commonly thinks of as a "scale" (see my definition,
above) would exclude that many pitches. If a melody modulates from one set of tones to another, then it is used in more than one scale, inasmuch as it is being "transported" from one scale to another. I would consider those 30-40 pitches to be a "tuning", or at least a subset of a tuning. And I would not feel compelled to consider non-scale tones used as ornamental embellishments _de facto_ members of the scale, just as a chromatically altered tones in Western diatonic music are not considered members of the (heptatonic) major or minor scale in which a composition is written.

Ah, I see that I've been talking about `scale` in the broadest sense. I agree with your definitions. I shall henceforth refer to voluminous sound systems as `tunings`.

There was further discussion about this, and I would agree that using the term "Maqam" instead of "mode" would eliminate all sorts of confusion -- once its meaning was explained.

Right. My definition for the Maqamat is `Tonality`. The Rast Maqam is simply the Rast Tonality, or the Key of Rast. Certain maqams are known to modulate between each other in the same Key. That's all there is to it.

Best,

--George

Cordially,
Ozan

🔗Carl Lumma <clumma@yahoo.com>

> Carl, it's been tried for certain, by Rauf Yekta foremost of all.
> If you are able, I recommend that you go find a 1922 Edition of
> Albert Lavignac's Encyclopedia of Music and Conservatory
> Dictionary. Look for the lenghty article on `Turkish Music` within
> pages 2945-3064 in French. This section has been prepared by Yekta
> personally. I don't agree with his zeal or his boorish attempts to
> prove that Pythagorean tuning is supreme for Maqams. Nevertheless,
> he DID try to introduce Western-oriented (excuse the pun) ears to
> the world of Maqamat by composing little pieces in the manner of
> waltz according to his own system of microtonal notation.

Sounds interesting, however I don't read French. :(

> Nevertheless, 24-Pythagorean does not suffice to explain the
> delicate nature of this genre. His shortcomings can be summarized
> as follows:
>
> 1. Disregarding the importance of temperament as a necessary tool
> for easy&correct modulation and transposition.
>
> 2. Misunderstanding Western staff notation and the function of
> accidentals (apotome sharp being 7 fifths up minus 4 octaves
> without exception, tempered or not).
>
> 3. Misrepresenting the pitches of a Pythagorean system of tuning
> by using the sharps and flats the wrong way and hampering
> transposability as a result.
>
> 4. Forsaking a strict JI approach entirely.
>
> 5. Notating Rast as D first, then as G according to Mansur Ahenk
> in standard diapason with an octave error.
>
> 6. Leaving behind him a musical heritage that has no bearing of
> the concept of Key Transposition and what it entails.

It sounds like the drawbacks to Yekta's treatment are severe
enough that I should not regret too much that I can't read
French.

> Of course I will continue to use the term Maqam for the Maqamat.
> But when it comes to explaining what this really is, which do
> you think is a better term? Mode or Key?

I don't really know. It sounds like they most closely resemble
scales. Though we have a very precise definition of scales on
tuning-math that would not include things like Maqam, conventional
Western theory does use the term "scale" for things like the
melodic minor, in which the pitch classes depend on which direction
you're playing in the scale!

> Granted, some maqams function as modes of a common scale, but
> that is not all there is to it. Segah is not just a Rast scale
> with the tonic at 3rd degree, because unlike rast, it requires a
> 13/11 (or 32/27) as a leading tone, which you will not find it
> Rast. And certain notes of Rast can alterate radically, such as
> chargah (4th degree) becoming hisar (7/5 to 10/7), and evdj
> becoming adjem (16/9)

This is the sort of thing that I need to understand more of
before I can be of any use. My preferred way of learning this
stuff would be to have audio examples annotated as you have
here.

> For a better explanation, I ask for your patience as I prepare
> the framework of my dissertation in the English language.

Of course!

-Carl

🔗Ozan Yarman <ozanyarman@superonline.com>

Hello again Carl,
----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 02 Ekim 2005 Pazar 19:57
Subject: [tuning] Re: A 17-note scale in an alternative Western History

SNIP!

It sounds like the drawbacks to Yekta's treatment are severe
enough that I should not regret too much that I can't read
French.

He was highly literate, yet he did not grasp the function of temperament in its own right.

> Of course I will continue to use the term Maqam for the Maqamat.
> But when it comes to explaining what this really is, which do
> you think is a better term? Mode or Key?

Could we perhaps agree that a scale is an assortment of pitches within an identity interval? This would either mean that the melodic minor is made up of two scales instead of one, or that it uses a 9-note scale just as the Rast Maqam uses Bb on its way back. Still, I hesitate to include this Bb in the main scale of Rast.

This is the sort of thing that I need to understand more of
before I can be of any use. My preferred way of learning this
stuff would be to have audio examples annotated as you have
here.

I aim to make a CD recording with audio examples as well.

> For a better explanation, I ask for your patience as I prepare
> the framework of my dissertation in the English language.

Of course!

-Carl

Cordially,
Ozan

🔗Carl Lumma <clumma@yahoo.com>

>>> Of course I will continue to use the term Maqam for the
>>> Maqamat. But when it comes to explaining what this really
>>> is, which do you think is a better term? Mode or Key?
>>
>> I don't really know. It sounds like they most closely
>> resemble scales. Though we have a very precise definition
>> of scales on tuning-math that would not include things
>> like Maqam, conventional Western theory does use the term
>> "scale" for things like the melodic minor, in which the
>> pitch classes depend on which direction you're playing in
>> the scale!
>
> Could we perhaps agree that a scale is an assortment of
> pitches within an identity interval?

I believe that is close to how Gene has defined them.

> This would either mean that the melodic minor is made up of
> two scales instead of one, or that it uses a 9-note scale
> just as the Rast Maqam uses Bb on its way back.

Either way, it includes the concept of direction. So it is
more than the tuning-math definition allows for. But my guess
is that "scale" is probably the best translation of Maqam in
English. However, if I were writing a text explaining Turkish
music to a Western audience, I would carefully explain what a
Maqam is, and then use that term throughout the rest of the
text.

>> This is the sort of thing that I need to understand more of
>> before I can be of any use. My preferred way of learning this
>> stuff would be to have audio examples annotated as you have
>> here.
>
> I aim to make a CD recording with audio examples as well.

Wonderful!

-Carl

🔗klaus schmirler <KSchmir@online.de>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗microtonalist <mark@equiton.waitrose.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > Some scales in 17EDO
> >
> > 21212121221
> > 3313331 :)
> > 2212212221
> > 333332
> > 2232323
> >
> > Mark
>
> Heya Mark,
>
> Care to explain how you arrived at these or what they're
> supposed to be good for?
>
> -Carl

In the first instance by writing out every generator of 17EDO, then
applying my basic rules for scales:

transposition by the generator interval results in only one pitch
being changed (kinda obvious), then ensuring that the movement from
the discarded pitch to the new pitch forms an interval less than the
larger of the scale step intervals.

Alternatively I go by the scale formation rules:

... aaab aab ab abb abbb ...

then assign a to some value such as Ls or LLs or sL or ssL, then set
b is equal to a plus one additional L or s step. I posted the
beginning of my 'Ls' collection a wekk or so ago. If you reverse the
Ls to sL, then the sL collection appears. I have been given to
believe that this is the result of the 'scale tree' whatever that is.

If, for a given scale, the generator can be bisected or trisected
then 'harmony' results, otherwise, the generator is the only stable
harmony of the scale. Of course, everything needs to be tried out and
checked by ear. So what they're good for, it's anyone's guess...

I repeat the scale collection for Ls here, ellipses being for the
purpose of saying, it goes on and on...
... ... ...
LsssLsssLsss LsLsLs LLLsLLLsLLLs
LsssLsss LsLs LLLsLLLs
... Lsss Lss Ls LLs LLLs ...
LssLss LLsLLs
LssLssLss LLsLLsLLs
... ...

You read this by taking one of the groups along the center line, then
adding one of the groups in one of the adjacent columns. These all
result in scales made from a generator. Of course, there is a second
and higher order scale structuring by taking adjacent pairs of scales:

LsLsLLs/LsLLs = Yasser's 12 from 19 scale, if L=2 and s=1.

Of course, you need to then set L and s to some values. The sum
results in the EDO. I merely had a search for 17EDO.

HTH

Mark

🔗Ozan Yarman <ozanyarman@superonline.com>

Dear Carl and Klaus,

----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 03 Ekim 2005 Pazartesi 21:02
Subject: [tuning] Re: A 17-note scale in an alternative Western History

> Could we perhaps agree that a scale is an assortment of
> pitches within an identity interval?

I believe that is close to how Gene has defined them.

How has he defined it?

> This would either mean that the melodic minor is made up of
> two scales instead of one, or that it uses a 9-note scale
> just as the Rast Maqam uses Bb on its way back.

Either way, it includes the concept of direction. So it is
more than the tuning-math definition allows for. But my guess
is that "scale" is probably the best translation of Maqam in
English.

In the broadest sense, no. Scale is just scale. It doesn't imply any modulation. The Key of C Major implies modulation to fifths closer in the chain.

However, if I were writing a text explaining Turkish
music to a Western audience, I would carefully explain what a
Maqam is, and then use that term throughout the rest of the
text.

I say unto you, that the best English word for `Maqam` is `Key`.

A piece written in Rast Maqam = a piece written in the Key of an Harmonic Major Scale.

> I aim to make a CD recording with audio examples as well.

Wonderful!

-Carl

I can't look up the names of the maqams now, but how would the concept of scale do to differentiate the two maqams that look like F major,
with the tonic of one being the bottom F, of the other the top F?

klaus

Good point Klaus. That is why I defer to the term `Key`.

Cordially,
Ozan

🔗klaus schmirler <KSchmir@online.de>

🔗Ozan Yarman <ozanyarman@superonline.com>

Dear Klaus,
----- Original Message -----
From: klaus schmirler
To: tuning@yahoogroups.com
Sent: 04 Ekim 2005 Salı 12:54
Subject: Re: [tuning] Re: A 17-note scale in an alternative Western History

Actually, this is why I would call maqams a modal system, since "mode" for me implies a) a catchall for whatever methods and rules you have to turn a set of notes into music, and b), in its narrower sense, concerns treatments of a scale, not of harmony. "Key" to me seems bound up exclusively with western practice, dominants or even mere transposition. Looking at the classical sonata form from this point of view, pieces in the major mode tend to modulate to transpositions of the major mode, whereas pieces in the minor mode, when the modulate, alos change into the major mode. This is very different from, say, the tendency of a piece in a key with many accidentals not to modulate towards a key with even more accidentals. And no western key tells you in which octave your final note will be.

You agree to limiting the term "scale" within a historically accurate context, and not the term "mode"? Mode is only the changing of the tonic in a given economical scale, which does not at all describe what Maqams are, let alone the melodic and harmonic minor genera in Western common practice. I understand that you wish mode could be interpreted in such a way as to include all possible alterations from a larger pitch-set, in which case I would have little to object against when defining the Maqamat. However, the concept "genera" would be a more exact term if we are talking about joined tetrachords and their divisions. Still, all these say nothing about how the SEYIR (melodic excursion) of a Maqam should be, which degrees should be emphasized, which modulations are more desirable, which transpositions are allowed without destroying the character of a Maqam, etc...

Key, on the other hand, is not bound up exclusively in Western practive as you assume, because there is an inherent harmony as implied by the usage of a particular Maqam. This clandestine harmony exists in the Ney, Kemencha, Tanbur, Oud, and even the Qudum (bowl sized timpani made from animal skin). For sure, tonics and dominants exist in the Maqamat. It's just that they are not always a pure fifth apart! Mere transpositions also exist, even when scales are not tempered, as some Maqams are carried up by a pure fourth or a fifth, and even an octave.

You assume that a key with many accidentals should modulate to keys with more accidentals in Classical sonatas? How about:

C# B A
E# E E
C# D C#
G# G# A
C# E A
?

It is equally easy to modulate in both directions, and I see this done in many pieces from the Classical and Baroque eras. So, a key does not specify an exact direction, it merely includes the element of direction, modulation, transposition, alteration, etc... Also, you neglect the role of tonics an octave apart since your ear is conditioned by `octave equivalance`. In the world of Maqamat, the pitches of the fundamental gamut are classified all the way up three octaves from Kaba Rast to Tiz Gerdaniye, and these are all relative frequencies with Rast at 1/1 and they are all produced from the Ney! It does matter at which relative frequency one comes to rest in Maqam Music, even when (tempered) octaves of a piano register the same to Western ears. You overlook the fact that a Maqam cannot be called a scale or mode for that very reason. Notice the concept `Key of C Major`. Likewise take note of the concept `Key of Huzzam`. Their tonal resources are entirely different and there is no implication of octave equivalance with Huzzam, while there indeed IS with C Major. C Major must end with C no matter where it may be, a composition in Huzzam must end with perde Segah (hence, 3rd place/diatonical degree in Persian), and it is only to be found at 27/22 to 5/4 from Rast, whose absolute frequency changes with the Ahenk (diapason). As for mode, a scale starting on the 8th degree is not different from a scale starting on the 1st degree of a diatonical scale, hence the term `mode` makes no distinction between these.

On a microtonal level and in a context of well temperaments, you have a point, because exact transposition doesn't exist, However, I hold against that that an aria in an opera was readily transposed to the range of whoever happenend to sing it, and to hell with key
characteristics.

klaus

Key color only exists in tempered instruments obviously! And an opera cannot be transposed without preparing related parts for standard diapason instruments such as the violins and harpsichords. All other instruments whose variation in proportion won't affect finger-to-pitch correlation (hence, the linear shift) are secure with the practice of Key Transposition. One does not need to re-write a score for the Clarinets or Trumpets or French Horns or the NEYs, as they can be, and ARE produced in any key.

Cordially,
Ozan

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

Paul, according to my Harvard's Dictionary of Music 4th Edition, it says that the key in tonal music is pitch relationships that establish a single pitch-class as a tonal center or tonic (or key note), with respect to which the remaning pitches have subordinate functions.

While the definition is restricted to Western tonal practice and its various areas of application, nevertheless, this is an equally adequate description of what a "Maqam" is.

It says, furthermore, that the notion of scale (mode) of a key embodies not only the selection of seven pitch-classes (per instance I might add) from the available twelve (or more in the case of Maqams), however, but also the organization of the seven in a hierarchy around the one that serves as tonic.

Again, we find a parallel with the Maqamat.

Also, it says here that although a tonal piece is usually described as being in a single key, it may incorporate passages in other keys before returning finally to the principal key. The process of moving from one key to another in the course of a piece is called modulation. Often, though not always, the key signature of a work is left unchanged throughout one or more modulations, the necessary changes in pitch being specified with accidentals. Pieces consisting of more than one movement may include one or more movements in a different key altogether. Such movements will have the appropriate key signature.

Like the modes before them, keys have sometimes been associated with ethical or emotional qualities. The most enduring of these associations, with the roots in the 16th century, is that of major keys with happiness or brightness and minor keys with sadness and darkness.... Keys have sometimes also been associated with colors. All such associations are learned or rest on convention.

Once again, we detect irrefutable similarities with Maqam Music.

Often the association of a particular key is not so much with some emotion or abstract quality as with a type of melody, meter, or tempo.

This is obviously the final and fatal blow to those who think that key is nothing but an absolute pitch designation.

I feel you confuse key in the keyboard pitch-level sense, with Key in the Tonality sense Paul. We most certainly do not understand the same thing.

Cordially,
Ozan
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 05 Ekim 2005 Çarşamba 0:05
Subject: [tuning] Re: A 17-note scale in an alternative Western History

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> I say unto you, that the best English word for `Maqam` is `Key`.
>
> A piece written in Rast Maqam = a piece written in the Key of an
>Harmonic Major Scale.

In my experience, musicans say "the Key of A" or "the key of g# minor" -
- in other words, a "key" is a reference to a *specific absolute pitch
to be used as the tonic, along with the mode to be used (major is
default)*. The statement "a piece written in the Key of an Harmonic
Major Scale" seems to be a redundant way of saying "a piece written in
an Harmonic Major Scale" and in fact doesn't tell you the key at all.

Or am I taking you too literally again? Forgive me if I am.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

Ozan, let's put our dictionaries down for a moment. I'm a practicing
musician who's played with hundreds of practicing musicians. If
someone asks me what key a piece is in, at least part of my answer
has to specify a letter-name which designates the absolute pitch of a
keynote. Otherwise, I will not have answered the question. If they
ask me what mode something was in, it's a different story, I don't
have to specify a keynote (but I can). Your experience may be
different, as may that of others on this list, and I'd love to hear
about any and all such experiences. I just know that *some* musicians
have had the same experience I have with this term, which is why I
saw fit to step in and clarify when I saw the word "Key" used in a
different way.

Your references to Harvard's Dictionary of Music 4th Edition leave me
with questions: how can a singular ("key") be defined as a plural
("pitch relationships . . .")? I'm curious . . . anyway, it looks
like this dictionary is trying to give you a lot of information,
perhaps enough to understand the meaning of an unfamiliar passage,
but isn't really going to help you use these words to communicate
with other musicians if you haven't used them before. And this
dictionary is not infallible -- just try looking up just about any
tuning or microtonal concept in there and see what you find!

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Paul, according to my Harvard's Dictionary of Music 4th Edition, it
>says that the key in tonal music is pitch relationships that
>establish a single pitch-class as a tonal center or tonic (or key
>note), with respect to which the remaning pitches have subordinate
functions.
>
> While the definition is restricted to Western tonal practice and
its various areas of application, nevertheless, this is an equally
adequate description of what a "Maqam" is.
>
> It says, furthermore, that the notion of scale (mode) of a key
embodies not only the selection of seven pitch-classes (per instance
I might add) from the available twelve (or more in the case of
Maqams), however, but also the organization of the seven in a
hierarchy around the one that serves as tonic.
>
> Again, we find a parallel with the Maqamat.
>
> Also, it says here that although a tonal piece is usually described
as being in a single key, it may incorporate passages in other keys
before returning finally to the principal key. The process of moving
from one key to another in the course of a piece is called
modulation. Often, though not always, the key signature of a work is
left unchanged throughout one or more modulations, the necessary
changes in pitch being specified with accidentals. Pieces consisting
of more than one movement may include one or more movements in a
different key altogether. Such movements will have the appropriate
key signature.
>
> Like the modes before them, keys have sometimes been associated
with ethical or emotional qualities. The most enduring of these
associations, with the roots in the 16th century, is that of major
keys with happiness or brightness and minor keys with sadness and
darkness.... Keys have sometimes also been associated with colors.
All such associations are learned or rest on convention.
>
> Once again, we detect irrefutable similarities with Maqam Music.
>
> Often the association of a particular key is not so much with some
emotion or abstract quality as with a type of melody, meter, or tempo.
>
> This is obviously the final and fatal blow to those who think that
key is nothing but an absolute pitch designation.
>
> I feel you confuse key in the keyboard pitch-level sense, with Key
in the Tonality sense Paul. We most certainly do not understand the
same thing.
>
> Cordially,
> Ozan
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 05 Ekim 2005 Çarþamba 0:05
> Subject: [tuning] Re: A 17-note scale in an alternative Western
History
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
>
> > I say unto you, that the best English word for `Maqam` is `Key`.
> >
> > A piece written in Rast Maqam = a piece written in the Key of
an
> >Harmonic Major Scale.
>
> In my experience, musicans say "the Key of A" or "the key of g#
minor" -
> - in other words, a "key" is a reference to a *specific absolute
pitch
> to be used as the tonic, along with the mode to be used (major is
> default)*. The statement "a piece written in the Key of an
Harmonic
> Major Scale" seems to be a redundant way of saying "a piece
written in
> an Harmonic Major Scale" and in fact doesn't tell you the key at
all.
>
> Or am I taking you too literally again? Forgive me if I am.

🔗klaus schmirler <KSchmir@online.de>

Ozan Yarman wrote:

> Dear Klaus, ----- Original Message ----- From: klaus schmirler To:
> tuning@yahoogroups.com Sent: 04 Ekim 2005 Salı 12:54 Subject: Re:
> [tuning] Re: A 17-note scale in an alternative Western History
>
>> Actually, this is why I would call maqams a modal system, since
>> "mode" for me implies a) a catchall for whatever methods and rules
>> you have to turn a set of notes into music, and b), in its narrower
>> sense, concerns treatments of a scale, not of harmony. "Key" to me
>> seems bound up exclusively with western practice, dominants or even
>> mere transposition. Looking at the classical sonata form from this
>> point of view, pieces in the major mode tend to modulate to
>> transpositions of the major mode, whereas pieces in the minor mode,
>> when the modulate, alos change into the major mode. This is very
>> different from, say, the tendency of a piece in a key with many
>> accidentals not to modulate towards a key with even more
>> accidentals. And no western key tells you in which octave your
>> final note will be.
>
>
> You agree to limiting the term "scale" within a historically
> accurate context, and not the term "mode"?

No, I didn't talk about scales at all. I do find "mode" useful as a
general term to talk about specific modal traditions like Gregorian
chant, ragas, of maqams. When you happen to walk by a New Grove, try
looking at the mode article there. You'll be surprised how many
traditions, possibilities of rules, ... and pages there are.

Mode is only the
> changing of the tonic in a given economical scale,

No, it's not. The popular terms for the rotations of the white note
scale are derived from Henricus Glarean's Dodekachordon, which treated
12 modes, two each on C, D, E, F, G, and A. The two kinds were
authentic and plagal, with their finals at the bottom of the compass
or in the middle. The modes on C and A were Glarean's invention,
acknowledging what would become the Major and Minor modes of Western
music (he included the B modes for completeness' sake, but called them
unusable for their lacking fifth/fourth). The modes on D, E, F, and G
had their own traditional melodic turns, finals and recitation tones
(sometimes called "dominants").

The current popular use of Glarean's mode names is probably due to
Joseph Schillinger, who suggested "transposition by modes" to
introduce variation or to arrive at new ideas. Note that he would use
the interval structure of one mode with melodies that owed their
existence to the melodic rules of a different one. Schillinger's
teachings are the grounds on which the BerkLee College was built, and
from there it obviously spread throughout the world. The popular
notion of Dorian, Myxolydian, Ionian derives from playing around with,
perverting, if you want, modes on a superficial level, but modes are a
far richer concept.

which does not
> at all describe what Maqams are, let alone the melodic and harmonic
> minor genera in Western common practice. I understand that you wish
> mode could be interpreted in such a way as to include all possible
> alterations from a larger pitch-set, in which case I would have
> little to object against when defining the Maqamat. However, the
> concept "genera" would be a more exact term if we are talking about
> joined tetrachords and their divisions. Still, all these say
> nothing about how the SEYIR (melodic excursion) of a Maqam should
> be, which degrees should be emphasized, which modulations are more
> desirable, which transpositions are allowed without destroying the
> character of a Maqam, etc...

see above

>
> Key, on the other hand, is not bound up exclusively in Western
> practive as you assume, because there is an inherent harmony as
> implied by the usage of a particular Maqam. This clandestine
> harmony exists in the Ney, Kemencha, Tanbur, Oud, and even the
> Qudum (bowl sized timpani made from animal skin). For sure, tonics
> and dominants exist in the Maqamat. It's just that they are not
> always a pure fifth apart!

Then they are modal recitation tones, not the dominants of functional
harmony.

Mere transpositions also exist, even
> when scales are not tempered, as some Maqams are carried up by a
> pure fourth or a fifth, and even an octave.
>
> You assume that a key with many accidentals should modulate to keys
> with more accidentals in Classical sonatas?

I said the opposite, that a piece in a key with many sharps may have a
tendency to modulate into keys with less sharps rather than advancing
towards the double sharps. This was my attempt to make a key determine
the fate of the music; it has nothing to do with what I think.

How about:
>
> C# B A E# E E C# D C# G# G# A C# E A ?
>
> It is equally easy to modulate in both directions, and I see this
> done in many pieces from the Classical and Baroque eras. So, a key
> does not specify an exact direction, it merely includes the element
> of direction, modulation, transposition, alteration, etc... Also,
> you neglect the role of tonics an octave apart since your ear is
> conditioned by `octave equivalance`. In the world of Maqamat, the
> pitches of the fundamental gamut are classified all the way up
> three octaves from Kaba Rast to Tiz Gerdaniye, and these are all
> relative frequencies with Rast at 1/1 and they are all produced
> from the Ney! It does matter at which relative frequency one comes
> to rest in Maqam Music, even when (tempered) octaves of a piano
> register the same to Western ears. You overlook the fact that a
> Maqam cannot be called a scale or mode for that very reason.

No, it can't be called a key, because the key of C ends on a C - any C. Call it a mode, and the necessary question about what defines this
mode will have to follow.

Notice
> the concept `Key of C Major`. Likewise take note of the concept
> `Key of Huzzam`.

But with a grain of salt: the Major mode and the Huzzam mode.

Their tonal resources are entirely different and
> there is no implication of octave equivalance with Huzzam, while
> there indeed IS with C Major.

Because their modal systems are concerned with different things.

C Major must end with C no matter
> where it may be, a composition in Huzzam must end with perde Segah
> (hence, 3rd place/diatonical degree in Persian), and it is only to
> be found at 27/22 to 5/4 from Rast, whose absolute frequency
> changes with the Ahenk (diapason).

But the rules for the Major mode are the same from the key of Dbb to
B#, just as you can adjust the pitch of a makam to the range of the
singer and the music.

As for mode, a scale starting on
> the 8th degree is not different from a scale starting on the 1st
> degree of a diatonical scale, hence the term `mode` makes no
> distinction between these.
>

I noticed that my dictionary gives "makam" as the only translation of
Tonart (key), whereas Tongeschlecht (the major-minor distinction)
doesn't even appear. My immediate reaction is to blame the translation
on someone like Hindemith or rather Donizetti who appropriated the
next available Turkish term to teach them Turks some real music - it
certainly doesn't fit the matter.

klaus

🔗Carl Lumma <clumma@yahoo.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

I am pleased that you are a practicing musician in the field Paul. As for my insignificant self who has never deemed it appropriate to climb the stage due to a ridiculous fear of appearing in public, I can only argue with you on terms and concepts as written on pages and scores.

For one thing, I never disagreed with the fact that a fundamental frequency of a given set-of pitches is implied with the Key. In fact, the "Key (Maqam) of Huzzam" does contain specific pitch-level information in terms of relative frequencies! You on the other hand think that key has to be an absolute frequency. Granted! The existance of transposable instruments of the orchestra justify a standard diapason to determine the reference key. But you seem to think that the standard was always in place, which wasn't until at least two centuries ago according to my knowledge. Hypothetically, the key of A could be played anywhere in the pitch-continuum! Seeing as Maqam Music similarly can be performed in any Ahenk (diapason), it stands to reason that one needs to consider "natural perdes" diatonical degrees (relative frequencies), which happen to coincide with the standard diapason in Sipurde Ahenk with perde Rast at C4.

I will let this be the Key of Sipurde for your satisfaction as the definition of key without capital.

My point was that you implied no other information could be contained in the key, whereas I have shown you that my claims are supported by a published reference.

Thus I will let Huzzam, Rast, Usshaq, Hijaz, Saba, Nihavend, Kurdi et al. be the Key with the capital implying all that you read in my previous post.

In short, you can retain your simplified meaning, I shall retain my definition which is included with the 4th Edition of Harvard's Dictionary of Music in accordance with historical practice and my experience.

Cordially,
Ozan
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 05 Ekim 2005 Çarşamba 2:16
Subject: [tuning] Re: A 17-note scale in an alternative Western History

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

You assume that Harmonic Major has no relative frequency which can be called a tonic! You overlook the fact that 1/1 or 0 cents IS the tonic. You want absolute frequencies denoted by pitch-letters in order to be able to define a key. I say unto you that absolute frequencies are redundant when defining a key. What you need instead are chromatic degrees within a 12-tone framework. The letter names are thusly important. They denote degrees, not absolute frequencies.

Besides, a single Harmonic Major Scale starting at a certain frequency is NOT the same as the Key of an Harmonic Major starting on the same frequency. Do you object to this?

Cordially,
Ozan
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 05 Ekim 2005 Çarşamba 3:33
Subject: [tuning] Re: A 17-note scale in an alternative Western History

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> My point was that you implied no other information could be
>contained in the key,

Dear Ozan,

I implied no such thing. I simply asserted that in my experience, one
would not say "Key of Harmonic Minor" -- one would either say "Mode of
Harmonic Minor" if one didn't care about the absolute pitch level, or
one would say "Key of G# Harmonic Minor" or whatever keynote was in
place if one did care about that.

Best,
Paul

🔗Graham Breed <gbreed@gmail.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗microtonalist <mark@equiton.waitrose.com>

I'll give a few examples

19EDO, 11 note scale

22122212221
The Generator is 7/19 (seven steps of 19EDO)
The Interval is

2 2 1 2
0 2 4 5 7
1 2 3 4 5
^ - bisection point (I know it's not exactly equal, but its
two 'thirds'

This results in a harmonic grid - those grids in Balzano, as you all
know -

2
14 18
7 11
0 4
12 16
5 9
2

--------------------------
Now let's try 9 from 14EDO

212121221

The generator here is 11/14 (11 steps of 14EDO),

2 1 2 1 2 1 2
0 2 3 5 6 8 9 11
1 2 3 4 5 6 7 8

This interval can't be bisected, as this lies between notes a 4th and
a 5th up from the 'tonic'.

Nor can it be trisected (i.e. into three 'equal' parts).

I regard that the generator interval is the most 'consonant' interval
for the given scale, and should be treated as such in composition.
Where an generator can be bisected as above, then the resulting
harmonic grid components can also be regarded as consonances too.

What I find interesting about this concept of 'consonance and
dissonance' is that it is independent of JI. In fact, I would make a
case for stating that generator intervals correspond to different
models of the acoustic phenomena of modes, which as well as strings
and pipes also includes those found in metal plates and bells. Such
sonorities have long been used by people in music. Bill Sethares'
work with adapted spectra is very illuminating here. Its also the
reason why I personally don't consider JI to be the be all and end
all of music.

none of this is 'ultimate truth', but just IMHO. I'd like opinions
though.

Mark

I do have an example of a trisect harmony, but not not from where I
am typing this.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> wallyesterpaulrus wrote:
> > --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> >
> >
> >>If, for a given scale, the generator can be bisected or trisected
> >>then 'harmony' results, otherwise, the generator is the only
stable
> >>harmony of the scale.
> >
> >
> > Anyone agree with this statement? Disagreee?
>
> I don't know that I understand it. There are a lot of specialist
terms
> in there. But -- if the generator is the same as we call the
generator,
> and "bisected" means in terms of diatonic scale steps then it's a
rather
> idiosyncratic concept of "harmony". Perhaps great music will show
it to
> be significant, but I don't see it.
>
> You can always bisect a generator in this fashion in an MOS with an
odd
> number of notes to the octave, where the octave is the period. So
it's
> not a very strong condition.
>
> Modulation by a semitone is so commonplace in pop music as to be a
> cliche. But the fifth is still the strongest consonance. So the
> interval you modulate by clearly doesn't have to be the fifth --
and I
> suspect fifths will be fifths whatever the tuning system.
>
>
> Graham

🔗microtonalist <mark@equiton.waitrose.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> I'll give a few examples
>
> 19EDO, 11 note scale
>
> 22122212221
> The Generator is 7/19 (seven steps of 19EDO)
> The Interval is
>
> 2 2 1 2
> 0 2 4 5 7
> 1 2 3 4 5
> ^ - bisection point (I know it's not exactly equal, but its
> two 'thirds'

I don't know what you are saying here, but I have a related question:
would you regard this scale as essentially the same as the following
version of it in 46edo?

55255525552

If so, we have something related to small intergers ratios; we are
apparently in the realm of a temperament which splits the sixth in
half to get the generator, which is, I presume, bisection. Anyway,
since 14 steps out of 19 is so close to 5/3, it's pretty obvious to
want to make 7 out of 19 half of that, and so take this into the realm
of "sensi" (semisixths) tempering. To some extent your ears are likely
to stick you with it.

> I regard that the generator interval is the most 'consonant' interval
> for the given scale, and should be treated as such in composition.

Yes, but often it just plain isn't, and treating it as such therefore
makes no sense.

> What I find interesting about this concept of 'consonance and
> dissonance' is that it is independent of JI.

It's also independent of hearing. You can't very well take the secor,
which is a generator of 116.6 cents, and simply claim it is the most
consonant interval in your resulting scales, because your ears will
tell you otherwise.

🔗microtonalist <mark@equiton.waitrose.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> > I'll give a few examples
> >
> > 19EDO, 11 note scale
> >
> > 22122212221
> > The Generator is 7/19 (seven steps of 19EDO)
> > The Interval is
> >
> > 2 2 1 2
> > 0 2 4 5 7
> > 1 2 3 4 5
> > ^ - bisection point (I know it's not exactly equal, but its
> > two 'thirds'
>
> I don't know what you are saying here, but I have a related
question:
> would you regard this scale as essentially the same as the following
> version of it in 46edo?
>
> 55255525552
I would regard them as being essentially the same scale.

>
> If so, we have something related to small intergers ratios; we are
> apparently in the realm of a temperament which splits the sixth in
> half to get the generator, which is, I presume, bisection. Anyway,

No.

The generator is used in the set theoretic sense. In 46EDO, the
generator would be 5525 = 17 steps wide. I am not bisecting another
interval. Maybe the correct expression is to cut the generating
interval in half.

> since 14 steps out of 19 is so close to 5/3, it's pretty obvious to
> want to make 7 out of 19 half of that, and so take this into the
No, you start with 7 from 19 being the generator.

14 is also a generator of 19, but 19 is prime so all intervals will
be generators. 3 is a generator of 19, but it doesn't form a scale
with the rules I've imposed. 16/19 forms a scale, though.

>realm
> of "sensi" (semisixths) tempering. To some extent your ears are
likely
> to stick you with it.
>
> > I regard that the generator interval is the most 'consonant'
interval
> > for the given scale, and should be treated as such in
composition.
>
> Yes, but often it just plain isn't, and treating it as such
therefore
> makes no sense.
I am at this point considering the analogy of LLsLLLs scale, as the
primary focus. It chimes with JI. I am looking at patterns.
>
> > What I find interesting about this concept of 'consonance and
> > dissonance' is that it is independent of JI.
>
> It's also independent of hearing. You can't very well take the
secor,
> which is a generator of 116.6 cents, and simply claim it is the most
> consonant interval in your resulting scales, because your ears will
> tell you otherwise.
A Secor is a generator of which EDOS? The first I can see would be
31EDO, and then it would be three steps wide. I checked and it
doesn't form a scale according to the rules. 31-3 = 28 forms a scale
though.

Mark

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> The generator is used in the set theoretic sense. In 46EDO, the
> generator would be 5525 = 17 steps wide. I am not bisecting another
> interval. Maybe the correct expression is to cut the generating
> interval in half.

Well, two generators give you an approximate 5/3, four an approximate
7/5, and six an approximate 7/6, leaving a lot of room for citting
things in half. If you cut 17 steps of 46 roughly in half, as 8 and 9
steps, then you have intervals not in your scale. Splitting 7 steps of
19 into 3 and 4 doesn't work too well either; 4 steps is in the scale,
but 3 steps isn't.

>
> > since 14 steps out of 19 is so close to 5/3, it's pretty obvious to
> > want to make 7 out of 19 half of that, and so take this into the

> No, you start with 7 from 19 being the generator.

My point is that if you take two such generators, you get something
which is a seventh of a cent flat from a pure 5/3. This acoustical
fact cannot but affect the way this scale is heard. It has major
sixths and minor thirds in it whether you want them or not.

> > It's also independent of hearing. You can't very well take the
> secor,
> > which is a generator of 116.6 cents, and simply claim it is the most
> > consonant interval in your resulting scales, because your ears will
> > tell you otherwise.

> A Secor is a generator of which EDOS? The first I can see would be
> 31EDO, and then it would be three steps wide. I checked and it
> doesn't form a scale according to the rules. 31-3 = 28 forms a scale
> though.

Eh? Which rules are these? A good generator to use is 7/72; in that
case "Blackjack" is the scale with steps 252525252525252525252. It's
been used successfully to compose music with a number of times. In any
case, 3/31 and 28/31 should form the same scale, should they not?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗microtonalist <mark@equiton.waitrose.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> > What I find interesting about this concept of 'consonance and
> > dissonance' is that it is independent of JI. In fact, I would
make
> a
> > case for stating that generator intervals correspond to different
> > models of the acoustic phenomena of modes, which as well as
strings
> > and pipes also includes those found in metal plates and bells.
>
> I have a different suggestion -- find systems which contain the
> intervals you want, "consonant" by whatever definition you want,
> temper the system to be rank 2 (which hopefully introduces even
more
Rank 2 - lost me there.
> of these "consonant" intervals), and *then* (if you like) find an
> appropriate generator/period pair for them -- we've worked out the
> methods to do so on tuning-math. I believe that there's no reason
> that the generator (or period) *itself* needs to be consonant; the
> very fact that there *is* a generator/period pair is a mathematical
> truism; it's certainly useful to be able to find them, but I don't
> think one needs to read or inject more musical significance into
them.
I think we'll have to agree to differ here. I just don't buy the
concept of the generator being a dissonance. sorry. I make it a
premise of my thinking.
>
> I've talked to Bill Sethares about this, and he said he'd be
> interested in something along these lines for the next edition of
> this book.
Maybe it would be better to take a spectrum, take two adjacent
components, then repeat it until you form a scale with some f-fsharp
type property. Then you can temper this interval to form different.
Dunno - just a thought.

Mark

🔗microtonalist <mark@equiton.waitrose.com>

I've looked at 'blackjack', and I observe that the basic generator
7/72 is a 'third' of this scale in this form. bisecting results in a
cluster of 3 adjacent pitches, which could be construed as
a 'harmony', but I'd prefer to exclude this possibility if there was
a reasonable way of describing it consistently. I prefer to have at
least one note between the notes of the 'harmony', i.e. the generator
describes at least a fifth. (for no better reason than that is the
case for the diatonic scale) Of more concern to me is the relation
between L and s, which is 5/2 = 2.5. This is not as bad as 3 to 1 of
the diatonic in 17EDO. However, I took a look at the generator of 7,
and noticed that if the segment was extended to 31 elements, a new
scale is formed:

3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 2

I think you have a name for it. Canasta? - apologies if not

For this scale, the generator covers a 'fourth' of the scale, so the
generator cannot be 'bisected' to form a harmony. My biggest problem
with this (and blackjack) is that its notes are a semitone apart, so
i suspect that presentations of the scale would be heard as chromatic.

There is some perceptual theory that asserts that we poor human
beings can't appreciate the complexities of more than a few
simultaneous things, so maybe musical scales are kept to small number
(like 5 or 7 or 9). Of course, that doesn't mean the total gamut be
restricted to this number. Can't say whether I agree with the theory
or not, but I tried blackjack, I noted that diatonic subsets could be
formed (after a fashion). How far does 31 from 72 deviate from
31EDO?, I wonder - without bothering to look atm. Scales within
scales within scales....

As regards the scale formation in 31EDO for example and the
difference between 3 and 28, it is best to consider the f to fsharp
rule in scale transposition. Transposing the scale from 31EDO up by 3
results in the flattening of a pitch. Therefore transposing by the
complement, 28 results in the raising of a pitch. I consider that the
generator interval is the upward transposition of the scale that
results in an f to Fsharp type motion. Again by analogy with LLsLLLs
diatonic.

I have to admit that scales where L > 2 x s concern me, if only
because in the JI diatonic the chromatic interval is very definitely
smaller than the smallest interval in a scale. 25/24 as opposed to
16/15. If only I could put my finger on why this is important...

Mark

🔗Ozan Yarman <ozanyarman@superonline.com>

Paul,
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 07 Ekim 2005 Cuma 0:13
Subject: [tuning] Re: A 17-note scale in an alternative Western History

>You overlook the fact that 1/1 or 0 cents IS the tonic.

But that is always so, regardless of which key you're in, right?

Exactly.

>You want
>absolute frequencies denoted by pitch-letters in order to be able to
>define a key. I say unto you that absolute frequencies are redundant
>when defining a key. What you need instead are chromatic degrees
>within a 12-tone framework. The letter names are thusly important.
>They denote degrees, not absolute frequencies.

Then I say what you're defining is probably a mode, not a key.

How can you say that when you approve my words above for relative frequency tonics? The key determines not only the tonic, but also other significant degrees in hierarchy, harmonic textures and even style. It signifies the fact that one can modulate to other keys within a tonal framework. In simplified terms:

"Key is the diatonical scale around which a piece of music is written."

Since maqam scales can be expressed in diatonical form one way or the other, this statement holds for the Maqamat. But for the sake of preserving pitch-level which also happens to be a fundamental element of key, let me say:

"Key of Rast in Sipurde"

which translates to:

"Harmonic Major revolving around the tone-center of C"

or in other words:

"Key of C Major".

Conversely, "A Prelude in D Major" would mean "A Prelude in Rast in Bolahenk".

> Besides, a single Harmonic Major Scale starting at a certain
>frequency is NOT the same as the Key of an Harmonic Major starting
>on the same frequency. Do you object to this?

I guess the difference is that a scale is just a set of notes played
in order, while a key tells you something about an actual musical
composition, even if that composition happens to contain extra notes
that the scale doesn't.

Precisely. That is why Maqams are not to be classified as modes as if the music done with them did not evolve beyond modality.

Cordially,
Ozan

🔗Graham Breed <gbreed@gmail.com>

Mark, my last email to you bounced. I can re-send if you can suggest an address that's likely to work.

microtonalist wrote:

> Rank 2 - lost me there.

Rank 2 means there are two distinct notes used to construct the tuning. Like an octave and generator, or a large and small step.

> I think we'll have to agree to differ here. I just don't buy the > concept of the generator being a dissonance. sorry. I make it a > premise of my thinking.

You can very easily end up with meantone by that logic. Mavila is another scale with a fifth generator. Has that been mentioned? It's the opposite of the meantone diatonic:

s s L s s s L

A tempered spectrum might help, but people still like it with normal timbres.

Here's a curiosity:

2 1 1 2 2 1 1

It's the equivalent of Arabic Rast in 10-equal. It isn't an MOS, but it does only change by 1 note when you transpose it by a fifth (6 steps).

>>I've talked to Bill Sethares about this, and he said he'd be >>interested in something along these lines for the next edition of >>this book.
> > Maybe it would be better to take a spectrum, take two adjacent > components, then repeat it until you form a scale with some f-fsharp > type property. Then you can temper this interval to form different. > Dunno - just a thought. I can do searches for rank 2 temperaments for arbitrary timbres. I don't know of anybody using them in anger. If you've got an idea what scales you want, it's easier to tweak the spectrum towards the nearest, simple intervals from the scale.

Graham

🔗monz <monz@tonalsoft.com>

Hi Mark,

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> I've looked at 'blackjack', and I observe that the basic
> generator 7/72 is a 'third' of this scale in this form.
> <snip> ... I took a look at the generator of 7, and noticed
> that if the segment was extended to 31 elements, a new
> scale is formed:
>
> 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 2
>
> I think you have a name for it. Canasta? - apologies if not

Yes, the 31-tone miracle scale has been dubbed "canasta".

> ... How far does 31 from 72 deviate from 31EDO?, I wonder
> - without bothering to look atm. Scales within
> scales within scales....

I have a graphic which shows exactly that ... the 3rd one
from the top at:

http://tonalsoft.com/enc/c/canasta.aspx

Note that i used generators -15 to +15 ... using different
generator boundaries with the 72-edo tuning will result
in different correspondences with 31-edo. (For 31-edo tuning,
of course, there will be no differences because the entire
31-edo tuning is itself a version of canasta, and vice versa.)

Here's a more accurate assessment of exactly how the
72-edo and 31-edo versions of canasta differ:

........... --------------- cents --------------------
generator . 72-edo ..... 31-edo ... diff of 31 from 72

.. -15 ... 650 ........ 658.&02/31 .. + 8.064516129
.. -14 ... 766.&2/3 ... 774.&06/31 .. + 7.52688172
.. -13 ... 883.&1/3 ... 890.&10/31 .. + 6.989247312
.. -12 .. 1000 ....... 1006.&14/31 .. + 6.451612903
.. -11 .. 1116.&2/3 .. 1122.&18/31 .. + 5.913978495
.. -10 .... 33.&1/3 .... 38.&22/31 .. + 5.376344086
... -9 ... 150 ........ 154.&26/31 .. + 4.838709677
... -8 ... 266.&2/3 ... 270.&30/31 .. + 4.301075269
... -7 ... 383.&1/3 ... 387.&03/31 .. + 3.76344086
... -6 ... 500 ........ 503.&07/31 .. + 3.225806452
... -5 ... 616.&2/3 ... 619.&11/31 .. + 2.688172043
... -4 ... 733.&1/3 ... 735.&15/31 .. + 2.150537634
... -3 ... 850 ........ 851.&19/31 .. + 1.612903226
... -2 ... 966.&2/3 ... 967.&23/31 .. + 1.075268817
... -1 .. 1083.&1/3 .. 1083.&27/31 .. + 0.537634409
.... 0 ..... 0 .......... 0 ........... 0
... +1 ... 116.&2/3 ... 116.&04/31 .. - 0.537634409
... +2 ... 233.&1/3 ... 232.&08/31 .. - 1.075268817
... +3 ... 350 ........ 348.&12/31 .. - 1.612903226
... +4 ... 466.&2/3 ... 464.&16/31 .. - 2.150537634
... +5 ... 583.&1/3 ... 580.&20/31 .. - 2.688172043
... +6 ... 700 ........ 696.&24/31 .. - 3.225806452
... +7 ... 816.&2/3 ... 812.&28/31 .. - 3.76344086
... +8 ... 933.&1/3 ... 929.&01/31 .. - 4.301075269
... +9 .. 1050 ....... 1045.&05/31 .. - 4.838709677
.. +10 .. 1166.&2/3 .. 1161.&09/31 .. - 5.376344086
.. +11 .... 83.&1/3 .... 77.&13/31 .. - 5.913978495
.. +12 ... 200 ........ 193.&17/31 .. - 6.451612903
.. +13 ... 316.&2/3 ... 309.&21/31 .. - 6.989247312
.. +14 ... 433.&1/3 ... 425.&25/31 .. - 7.52688172
.. +15 ... 550 ........ 541.&29/31 .. - 8.064516129

> I have to admit that scales where L > 2 x s concern me,
> if only because in the JI diatonic the chromatic interval
> is very definitely smaller than the smallest interval in
> a scale. 25/24 as opposed to 16/15. If only I could put my
> finger on why this is important...

I haven't read the previous posts in this thread, but
the reason why this is important is primarily because
Western music, being based for the most part (especially
before 1900) either on pythagorean or meantone tunings,
makes a distinction between the chromatic-semitone and
the diatonic-semitone.

In pythagorean, the chromatic is larger than the diatonic,
and in meantone it's the other way around.

In JI it can be either way, but in general it's as you
describe, which is what meantone follows.

There's more here:

http://tonalsoft.com/enc/c/chromatic-semitone.aspx

http://tonalsoft.com/enc/d/diatonic-semitone.aspx

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

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi all,

On Fri, 7 Oct 2005, Ozan Yarman wrote:
>
> Paul,
> ----- Original Message -----
> From: wallyesterpaulrus
...[snipt]

> > Besides, a single Harmonic Major Scale starting at a certain
> >frequency is NOT the same as the Key of an Harmonic Major starting
> >on the same frequency. Do you object to this?
>
> I guess the difference is that a scale is just a set of notes played
> in order, while a key tells you something about an actual musical
> composition, even if that composition happens to contain extra notes
> that the scale doesn't.
>
> Precisely. That is why Maqams are not to be classified as modes
> as if the music done with them did not evolve beyond modality.

Ozan,
I'm not sure I believe you said that! The implication is that maqam
music is superior to modal music. But there are so many different
kinds and traditions of modal music in the world. And I don't
believe in "musical Darwinism", at least if that implies (as most
musical theorists of the late 19th to mid 20th Century CE seem to)
that the musical forms and traditions from which the latest music
has "evolved" are somehow more primitive and less worthy than that
latest. By that reasoning, that which "survives" best must be the
best music. I'm no musical snob, but I hate to think that history
will remember this as the age of Britney Spears and Chrstina
Aguilera ...

If our musical ideas do evolve, then as they do so they create a
complex, many-branching Tree of Musical Life, all of whose
branches and fruit are worthy of at least some consideration for
the parts they may play in nourishing the musical sensibilities of
past, present and future generations. Along the way, many more
and less complex musics may appear, flourish briefly, then
disappear - sometimes without trace. Yet without studying all
modal musics in depth, I think it unwarranted to dismiss them as
less than maqam music. No, all kinds of music are not equal; but
there are many different kinds of good music.

Regards,
Yahya

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🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> Paul,
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 07 Ekim 2005 Cuma 0:13
> Subject: [tuning] Re: A 17-note scale in an alternative Western History
>
> >You overlook the fact that 1/1 or 0 cents IS the tonic.
>
> But that is always so, regardless of which key you're in, right?
>
>
> Exactly.
>
>
>
> >You want
> >absolute frequencies denoted by pitch-letters in order to be able to
> >define a key. I say unto you that absolute frequencies are redundant
> >when defining a key. What you need instead are chromatic degrees
> >within a 12-tone framework. The letter names are thusly important.
> >They denote degrees, not absolute frequencies.
>
> Then I say what you're defining is probably a mode, not a key.
>
>
> How can you say that when you approve my words above for relative frequency tonics?
>The key determines not only the tonic, but also other significant degrees in hierarchy,
>harmonic textures and even style. It signifies the fact that one can modulate to other
>keys within a tonal framework. In simplified terms:
>
> "Key is the diatonical scale around which a piece of music is written."
>
> Since maqam scales can be expressed in diatonical form one way or the other, this
>statement holds for the Maqamat. But for the sake of preserving pitch-level which also
>happens to be a fundamental element of key,

That's what I was trying to say (at least as I know musicians to use the word)

>let me say:
>
> "Key of Rast in Sipurde"
>
> which translates to:
>
> "Harmonic Major revolving around the tone-center of C"
>
> or in other words:
>
> "Key of C Major".
>
> Conversely, "A Prelude in D Major" would mean "A Prelude in Rast in Bolahenk".

OK, we've come to a point of agreement then.

> > Besides, a single Harmonic Major Scale starting at a certain
> >frequency is NOT the same as the Key of an Harmonic Major starting
> >on the same frequency. Do you object to this?
>
> I guess the difference is that a scale is just a set of notes played
> in order, while a key tells you something about an actual musical
> composition, even if that composition happens to contain extra notes
> that the scale doesn't.
>
>
>
>
>
> Precisely. That is why Maqams are not to be classified as modes as if the music done
>with them did not evolve beyond modality.

"Evolve beyond" is quite a judgmental way of putting it! I don't buy into the implied
superiority this statement seems to suggest. As we know, most of the features and
meanings of "modality" have been lost over the centuries. What we call "modes" today
don't even scratch the surface of the highly developed musical practices of past millenia.
Modern tonality may one day be an equally archaic and poorly-understood concept.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

Yahya,

I'm terribly misunderstood it seems. Far be it for me to consider modal music an inferior genre of tonality, I was just insinuating that certain people might tend to think in such a chauvanistic manner and categorize the Maqamat as modes just so that they might envision Eastern music to be devoid of harmony.

As to the confusion about the rushly uttered phrase `pre-tonal`, I admit I carelessly used it as a synonym for "predecessor of Classical Tonality`. Granted, modal music has its own virtues which are just as admirable as any other culture's. Nevertheless, modality is distinct from what we conceive as `tonality`, doubtless, a misnomer for polyphonic European music composed and performed mostly after the 17th century, in that it considers fourths, fifths and octaves as restful instead of thirds.

From this perspective, the Maqamat are more tonal than modal since at least three centuries. This is the reason why I insist on the concept `key`. It has nothing to do with musical superiority for gosh sakes! This is so, even when tools of composition and harmony may be superior in design. Just because one possesses improved tools does not in anyway imply that the final product should be superior. The shortcomings of that particular way of thinking can be observed most clearly in the case of the ideological fiasco that attempted to create the Nationalist Turkish School of music. Some of us here in Turkey still insolently believe that polyphonic contemporary music of questionable Turkish identity is superior to the monophonic genres that `run amok`. I shun such opinions vehemently of course.

`Evolution`, or at least `Historical continuity` in all musics should exist, but not to the extent where posterity runs the risk of classifying past examples as less worthy. So my argument rests on these two articles:

1. Good music is not confined to any particular genre or historical epoch.
2. Maqam music is not modal as if it was still stuck in the medieval era.

Cordially,
Ozan
----- Original Message -----
From: Yahya Abdal-Aziz
To: tuning@yahoogroups.com
Sent: 12 Ekim 2005 Çarşamba 4:48
Subject: [tuning] Re: A 17-note scale in an alternative Western History

> Precisely. That is why Maqams are not to be classified as modes
> as if the music done with them did not evolve beyond modality.

Regards,
Yahya

🔗Ozan Yarman <ozanyarman@superonline.com>

Paul,
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 12 Ekim 2005 Çarşamba 23:16
Subject: [tuning] Re: A 17-note scale in an alternative Western History

OK, we've come to a point of agreement then.

Seems so my dear fellow.

>
> Precisely. That is why Maqams are not to be classified as modes as if the music done
>with them did not evolve beyond modality.

Please refer to my answer to Yahya on this issue. I have been grossly misunderstood it seems. I merely implied the rationale behind certain people's thinking.

Nevertheless, I rest my case. Maqam Music today is very much tonal.

Cordially,
Ozan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi Ozan,

On Fri, 14 Oct 2005, you wrote:
> I'm terribly misunderstood it seems. Far be it for me to consider modal
music an inferior genre of tonality, I was just insinuating that certain
people might tend to think in such a chauvanistic manner and categorize the
Maqamat as modes just so that they might envision Eastern music to be devoid
of harmony.
>
> As to the confusion about the rushly uttered phrase `pre-tonal`, I admit I
carelessly used it as a synonym for "predecessor of Classical Tonality`.
Granted, modal music has its own virtues which are just as admirable as any
other culture's. Nevertheless, modality is distinct from what we conceive as
`tonality`, doubtless, a misnomer for polyphonic European music composed and
performed mostly after the 17th century, in that it considers fourths,
fifths and octaves as restful instead of thirds.
>
> From this perspective, the Maqamat are more tonal than modal since at
least three centuries. This is the reason why I insist on the concept `key`.
It has nothing to do with musical superiority for gosh sakes! This is so,
even when tools of composition and harmony may be superior in design. Just
because one possesses improved tools does not in anyway imply that the final
product should be superior. The shortcomings of that particular way of
thinking can be observed most clearly in the case of the ideological fiasco
that attempted to create the Nationalist Turkish School of music. Some of us
here in Turkey still insolently believe that polyphonic contemporary music
of questionable Turkish identity is superior to the monophonic genres that
`run amok`. I shun such opinions vehemently of course.
>
> `Evolution`, or at least `Historical continuity` in all musics should
exist, but not to the extent where posterity runs the risk of classifying
past examples as less worthy. So my argument rests on these two articles:
>
> 1. Good music is not confined to any particular genre or historical epoch.
> 2. Maqam music is not modal as if it was still stuck in the medieval era.

I think we understand each other!

Regards,
Yahya

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🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> I did not bridge the gap between modality and tonality here as you
>presume I did.

I don't presume much about what you did. I probably just misunderstood
what you were trying to say about fifths and fourths below, and in my
usual ill manner just stated a fact that I thought might be relevant.
Could you try to clarify what you did mean?

>Since you are the expert on Western music history, why don't you
>attempt to show me if Ionian could be made synonymous with Rast?

Sorry to answer a question with a question, but how is this a
question of Western musical history? This is such a strange query
that surely we must have fallen off the tracks at some point. Forgive
my opacity and please, if you could, try to clarify what you have in
mind for me.

> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 15 Ekim 2005 Cumartesi 0:00
> Subject: [tuning] Re: A 17-note scale in an alternative Western
History
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
>
> > Nevertheless, modality is distinct from what we conceive as
> >`tonality`, doubtless, a misnomer for polyphonic European music
> >composed and performed mostly after the 17th century, in that it
> >considers fourths, fifths and octaves as restful instead of
thirds.
>
> Huh? I'm not sure what you're saying here, but thirds and sixths
were
> considered restful and stable in Western music since about 1450,
which
> quickly precipitated the development of meantone tunings;
> meanwhile 'tonality' in the Western common-practice sense didn't
come
> into prominence until about 1670, over two centuries later.
>

🔗Ozan Yarman <ozanyarman@superonline.com>

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 17 Ekim 2005 Pazartesi 21:47
Subject: [tuning] Re: A 17-note scale in an alternative Western History

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> I did not bridge the gap between modality and tonality here as you
>presume I did.

I think I'll pass this once Paul. More urgent attires await my attention.

>Since you are the expert on Western music history, why don't you
>attempt to show me if Ionian could be made synonymous with Rast?

Truly, I am quite tired of discussing this matter, especially seeing as my knowledge on Western music history cannot possibly match your intellect and experience in the field. Afterall, what do I know on modes or maqams? I'm just an insignificant amateur mumbling to myself constantly.

Cordially,
Ozan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 18 Ekim 2005 Salı 0:14
Subject: [tuning] Re: A 17-note scale in an alternative Western History

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Truly, I am quite tired of discussing this matter, especially seeing
>as my knowledge on Western music history cannot possibly match your
>intellect and experience in the field.

Nonsense.

Is it?

>Afterall, what do I know on modes or maqams?

You know far more than me, at least on the latter. On the former, there
simply appears to be more than one definition in circulation.
The 'richer' definition has been brought up here many times -- it's
been months now -- but unfortunately, especially when people were
explicitly using it, you seemed to insist on interpreting their
statements using the 'poorer' definition. I'm sure you weren't doing
this intentionally -- we all tend to react all too quickly when we see
certain "buzzwords". And I'm glad to see the beginnings of apology and
reconciliation show their heads here.

I beg to differ concerning the adjectives "richer" and "poorer". I would rather use the terms "muddled" and "elegantly concise" in contrast. My purpose from the beginning is to divorce the term "mode" from all instances of cross-cultural referentialism regarding non-Western schools of music.

Come to think of it, I don't see anyone calling Persian Dastgahs modes. Why the priviledge?

> I'm just an insignificant amateur mumbling to myself constantly.

Your opinions and contributions are as valuable as those of anyone
else. Let's not get hung up on mere words when a far grander musical
universe is our subject of inquiry and passion.

The grander it is, the more insignificant we become in comparison! You vindicate my words. ;)

Cordially,
Ozan

🔗Carl Lumma <clumma@yahoo.com>

🔗microtonalist <mark@equiton.waitrose.com>

Hi,

no I didn't. My problem with this type of scale formulation is that
our concepts of generators are incompatible. If I were starting from
a specific interval and making it a generator, I would extend the
generator until I found start and end points that were sufficiently
close together to close into an EDO. I would then take a segment that
conforms to some general rules about scale structure. The generator
would then be some number of steps wide. If this interval were
dividable into two or three parts (like the 5th of the diatonic
scale) then I could form a triadic/tetradic harmony. Otherwise, just
the bare interval of the generator would be the 'harmony' of the
scale.

What your concept of a generator is, is to produce a scale using
similar technques, but that the generator takes no further part in
the harmony, certainly not as a consonance.

It seems that we must agree to differ. It is an 'axiom' of my concept
of tonality that the generator is a consonance, in fact *the*
consonance of the scale. Of course I only consider octave repeating
scales.

What may be happening here is that the resulting EDO (31 or 72 for
example) has several generators, and that the resulting scale can be
interpreted in the context of a generator that is not the generator
of that scale. For example, suppose you plotted the 21 blackjack from
31 EDO on the 31EDO circle of fifths (rather than in terms of the 3
generator. It is this which may be causing the acoustic 'confusion'.
As another example of this confusion, take the following scale

22122212221 from 19EDO. This has a generator of 7/19. But it also
contains instances of interval 11, which of course is the 19EDO
fifth. It also contains instances of 6 and 5 from 19EDO, major and
minor thirds. However, the scale functions more in terms of intervals
3 and 4, forming its generator 7, so I avoid the 6 and 5s except
where they can be used in the 3 and 4 context.

As a suggestion, have you tried using your scales in terms of the
generator rather than any of the other possible combinations of tones?

Mark

I'll look at the 163cent generator and report back.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> Hi Mark,
>
> I guess you missed this (?):
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> >
> > > There is some perceptual theory that asserts that we poor human
> > > beings can't appreciate the complexities of more than a few
> > > simultaneous things, so maybe musical scales are kept to small
> number
> > > (like 5 or 7 or 9).
> >
> > Hi Mark, I understand Blackjack may have too many notes for you.
> But
> > there are plenty of scales (that I find musically useful) with a
> > reasonable number of notes, and plenty of consonant intervals,
> where
> > the period is an octave and the generator is a dissonance. Did
you
> ever
> > receive the paper I (thought I) sent you? If not, maybe you've
> seen,
> > for example, the Porcupine 7- and 8-note scales mentioned on one
of
> > these lists. In a 5-limit sense, the generator is a dissonance
> since
> > it's around 163 cents, yet there is a good complement of nice
> > (approximate) 5-limit triads in each of these scales. And
Porcupine
> is
> > just one example of many.
>
> What say you?
> -Paul
>

🔗Carl Lumma <clumma@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> It seems that we must agree to differ. It is an 'axiom' of my concept
> of tonality that the generator is a consonance, in fact *the*
> consonance of the scale. Of course I only consider octave repeating
> scales.

One of Lincoln's corny gags was to ask "If you count a tail as a leg,
how many legs does a dog have?" To the response "Five", the comeback
is "No, four. Calling a tail a leg doesn't make it a leg." You can
treat anything you like formally as a consonance or a dissonace, but
you can't force your ears to hear it that way. What, then, is the point?

> What may be happening here is that the resulting EDO (31 or 72 for
> example) has several generators, and that the resulting scale can be
> interpreted in the context of a generator that is not the generator
> of that scale. For example, suppose you plotted the 21 blackjack from
> 31 EDO on the 31EDO circle of fifths (rather than in terms of the 3
> generator.

However, Blackjack in 72 EDO cannot be so plotted, as 72 has six
circles of fifths. It can be so plotted in 41 EDO, but not in a way
consistent with how it is in 31 EDO. Blackjack is not really definable
in terms of any generator you choose.

> It is this which may be causing the acoustic 'confusion'.

*What* acoustic confusion? What basis is there for the claim that
something is "confused"?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> Hi,
>
> no I didn't. My problem with this type of scale formulation is that
> our concepts of generators are incompatible. If I were starting
from
> a specific interval and making it a generator, I would extend the
> generator until I found start and end points that were sufficiently
> close together to close into an EDO.

That's not at all incompatible with my approach, and indeed this is
how one comes to the numbers 7, 8, 15 . . . as the number of notes
per octave one is likely to use for an octave-repeating Porcupine
scale. Why do you say 'incompatible'?

> What may be happening here is that the resulting EDO (31 or 72 for
> example) has several generators, and that the resulting scale can
be
> interpreted in the context of a generator that is not the generator
> of that scale. For example, suppose you plotted the 21 blackjack
from
> 31 EDO on the 31EDO circle of fifths (rather than in terms of the 3
> generator.

OK, then you have several connected chains of 2-3 fifths (3-4 notes),
and breaks in-between

> It is this which may be causing the acoustic 'confusion'.

What acoustic 'confusion' are you referring to?

> As a suggestion, have you tried using your scales in terms of the
> generator rather than any of the other possible combinations of
>tones?

I don't know quite what this means. I put the scale on my keyboard
and play around. I try to create a nice melody and/or chord
progressions. There are a lot of concordances, so my ear is drawn to
them, and to different ways of resolving discordances to these
concordances. The 163-cent intervals sound discordant in most
contexts and timbres -- the only exception, with many harmonic
timbres, is the approximate 8:9:10:11:12 chord, in which three of the
163-cent intervals serve the functions of 9:10, 10:11, and 11:12,
respectively, and the sound is somewhat concordant. The
composition 'Glassic' only has a brief section in Porcupine-7, and
then moves on to other scales. Igliashon Jones said he has a pop song
in Porcupine-8, but he hasn't shared it yet.

> Mark
>
> I'll look at the 163cent generator and report back.

Thanks -- and I hope 'look' means 'play' and 'listen' and
perhaps 'compose', and not just to the generator, but to the scales.

🔗Carl Lumma <clumma@yahoo.com>

🔗microtonalist <mark@equiton.waitrose.com>

There are a number of non-set theoretic rules which I impose. And a
number of rules which I ignore. It's just a rag-bag of things. No
theory at all really. I personally don't buy the dissonant generator
idea at all. The only thing which could be reagrded as a rule is the
scale tree I posted to the list some while back. I suspect this is
nothing new either.

My only guiding principle is to aruge by analogy with the structure
of the heptatonic diatonic and pentatonic scales.

It's not much, but it does for me.

Mark

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> Hi Mark,
>
> How does your approach differ from that of Balzano, if at
> all?
>
> -Carl
>
> > no I didn't. My problem with this type of scale formulation is
> > that our concepts of generators are incompatible. If I were
> > starting from a specific interval and making it a generator, I
> > would extend the generator until I found start and end points
> > that were sufficiently close together to close into an EDO. I
> > would then take a segment that conforms to some general rules
> > about scale structure. The generator would then be some number
> > of steps wide. If this interval were dividable into two or three
> > parts (like the 5th of the diatonic scale) then I could form a
> > triadic/tetradic harmony. Otherwise, just the bare interval of
> > the generator would be the 'harmony' of the scale.
> ...
>

🔗microtonalist <mark@equiton.waitrose.com>

Who really cares about theory anyway. as I say elswhere - I don't buy
the idea that the generator is a dissonance. You're not right, and
neither am I.

I tried playing with blackjack and canasta, and with other 'diatonic-
like' scales with more than 11 notes. All they sounded like was
chromatic scales to my cloth ear. I don't accept blackjack as a quasi
diatonic like scale - it sounds like a temperament of some quasi
chromatic. In which case the true scales are subsets of 21 or 31 or
whatever.

Perhaps....

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> > It seems that we must agree to differ. It is an 'axiom' of my
concept
> > of tonality that the generator is a consonance, in fact *the*
> > consonance of the scale. Of course I only consider octave
repeating
> > scales.
>
> One of Lincoln's corny gags was to ask "If you count a tail as a
leg,
> how many legs does a dog have?" To the response "Five", the comeback
> is "No, four. Calling a tail a leg doesn't make it a leg." You can
> treat anything you like formally as a consonance or a dissonace, but
> you can't force your ears to hear it that way. What, then, is the
point?
>
> > What may be happening here is that the resulting EDO (31 or 72
for
> > example) has several generators, and that the resulting scale can
be
> > interpreted in the context of a generator that is not the
generator
> > of that scale. For example, suppose you plotted the 21 blackjack
from
> > 31 EDO on the 31EDO circle of fifths (rather than in terms of the
3
> > generator.
>
> However, Blackjack in 72 EDO cannot be so plotted, as 72 has six
> circles of fifths. It can be so plotted in 41 EDO, but not in a way
> consistent with how it is in 31 EDO. Blackjack is not really
definable
> in terms of any generator you choose.
>
> > It is this which may be causing the acoustic 'confusion'.
>
> *What* acoustic confusion? What basis is there for the claim that
> something is "confused"?
>

🔗Carl Lumma <clumma@yahoo.com>

> Who really cares about theory anyway.

If one is doing theory, one hopes to care about it.

> as I say elswhere - I don't buy the idea that the generator
> is a dissonance.

The tunings Gene and Paul and talking about tend to have
lots of consonances per note, and though having the generator
be consonant can help with this, it isn't necessary. So
why should we care about it?

> I tried playing with blackjack and canasta, and with
> other 'diatonic-like' scales with more than 11 notes. All
> they sounded like was chromatic scales to my cloth ear. I
> don't accept blackjack as a quasi diatonic like scale - it
> sounds like a temperament of some quasi chromatic.

Great, since that's what it's supposed to be (IIRC). It's
not meant as a generalized diatonic.

> In which case the true scales are subsets of 21 or 31 or
> whatever.

That's certainly valid. Have you tried the 10-note
Miracle MOS?

Though since not all music is diatonic-like, why should
we use the word "true" like this? Maybe somebody wants
a 21-note tonerow (or just very chromatic melodies).

> My only guiding principle is to aruge by analogy with the
> structure of the heptatonic diatonic and pentatonic scales.

I think that's a good principle. But *which* structural
properties of the diatonic and pentatonic scales do we
want to generalize? It matters.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗microtonalist <mark@equiton.waitrose.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> > My only guiding principle is to aruge by analogy with the structure
> > of the heptatonic diatonic and pentatonic scales.
>
> Are you familiar with Yasser?
Yes. But he assumes his supradiatonic is built out of 19EDO fifths, but
then he forms his harmony of two hexads, neither of which contain these
aforementioned fifths - in fact in his supradiatonic the 'perfect'
fifth is a dissonance. Just like your scales - the generator is a
dissonance. I analysed his supradiatonic and it is best seen as being
built from 8/19 not 11/19. His analogy fails to work in that upward
transposition byleads to a flattening of a note in the scale. The f -
fsharp analogy is from Balzano not Yasser. Plus I find his infra
diatonic really odd.

Also, nineteen 3/2s are very far from closing into a circle (137cents
off if my calcs are correct), so 19EDO is a strange temperament, well
to my mind. It does a good 6/5 tho'.

His book is also a very tedious read IMHO.
>
> > It's not much, but it does for me.
>
> Why not start with your ears?
>
Often you have to confront your ears with something totally new. i
start with a scale and then assess it aurally, not the other way round.
it is also important to consider Sethares' comments about timbre and
scales when assessing a scale. See his experiment with a dissonant 2/1
and a consonant 32/15. Start with your ears...???

🔗Graham Breed <gbreed@gmail.com>

I've never read Yasser, so I'll have to absent myself from the first part of this. But anyway

microtonalist wrote:

> Also, nineteen 3/2s are very far from closing into a circle (137cents > off if my calcs are correct), so 19EDO is a strange temperament, well > to my mind. It does a good 6/5 tho'.

You must care about natural consonance to care about these calculations. But if you like the 6/5, why not make it the generator? Or, at least, the 5/3 that can be divided more easily. Unfortunately, that gives an 11 note scale, so it's slightly beyond your limit. Here's the lattice

0
5 13
10 18
15 4
1 9
6 14
0

Using hanson temperament, you could also tune it to 53-equal or you know what other equal temperament. Or even not use an equal temperament. I don't know why you insist on an equal chromatic as you've done away with Balzano's silly group theoretic trick. A historical consideration of the diatonic scale wouldn't need equal temperament either.

> Often you have to confront your ears with something totally new. i > start with a scale and then assess it aurally, not the other way round. > it is also important to consider Sethares' comments about timbre and > scales when assessing a scale. See his experiment with a dissonant 2/1 > and a consonant 32/15. Start with your ears...???

So what timbre do you use to assess the scale aurally? If you don't have a drive to use alternative timbres, it makes sense to stay close to the harmonic series. There's a lot of it around. Octaves, fourths and fifths have a degree of universality as well. 32/15 only works as a substitute for the octave because it isn't that far away. I've already mentioned mavila, which has a generator close enough to a fifth to be made a good consonance with a tweak of the timbre.

Another thing, if you're pushing ear-guided theory it might help to describe what you hear. It's a general problem with speculative theory that it seems to be abstract until you understand the musical thinking behind it.

Graham

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> >
> > --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> >
> > > My only guiding principle is to aruge by analogy with the
structure
> > > of the heptatonic diatonic and pentatonic scales.
> >
> > Are you familiar with Yasser?

> Yes. But he assumes his supradiatonic is built out of 19EDO fifths,
but
> then he forms his harmony of two hexads, neither of which contain
these
> aforementioned fifths - in fact in his supradiatonic the 'perfect'
> fifth is a dissonance.

Kind of strange when you actually listen to it, IMO.

> Just like your scales - the generator is a
> dissonance. I analysed his supradiatonic and it is best seen as
being
> built from 8/19 not 11/19.

To Yasser and most of us, these are the same thing.

> His analogy fails to work in that upward
> transposition byleads to a flattening of a note in the scale.

His analogy? Or yours?

> The f -
> fsharp analogy is from Balzano not Yasser. Plus I find his infra
> diatonic really odd.

I find his infra diatonic to be by far the best part of his whole
theory. What's odd about it?

> Also, nineteen 3/2s are very far from closing into a circle
(137cents
> off if my calcs are correct), so 19EDO is a strange temperament,

If each 3:2 approximation is OK on its own, who cares whether the
same number of *pure* 3:2s forms a circle? This seems a very abstract
and arbitrary theoretical consideration, far removed from anything
audible.

> Start with your ears...???

Yes, and I believe that applies to assessing Sethares as well as
Yasser (and everything else). I will grant you that it takes a *long*
period of unlearning and re-learning to even begin to be able to use
a new scale in "its own right" . . .

🔗microtonalist <mark@equiton.waitrose.com>

I agree that diatonic scales can be understood only in terms of the L
and s scalestep concepts. As soon as you assign them a distinct
proportion, then a 'universe set' does happen. Meantone tunings of the
19th century were most probably open - i.e. not 31edo, and I've often
thought that the other patterns of Ls scales could have open forms too,
wherever the ratio between L and s is irrational.

Talking philosophically, I suppose I belong to those who explore a
pattern for what it is - a pattern - and wondering about the 'shape' of
the pattern. Kind of meta-tonality thinking. What I don't see much
sense in is mathematics with n-tones, like some of the serial and pitch-
class set stuff that is very much in vogue.

Arguing by analogy is far from rigorous, and amongst the patterns there
are probably many 'dud' scales. You could argue that most of my work
is 'eye music'. but even Bach was accused of that... (tho' i'm not
comparing myself to JS)

It depends upon what is meant by microtonality. Xentonal, hypertonal,
spectral...

Mark

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> > My only guiding principle is to aruge by analogy with the structure
> > of the heptatonic diatonic and pentatonic scales.
>
> That's really my guiding principle too, though it has seemed to guide
> us in different directions, due to philosophical differences. For one
> thing, I don't buy the idea of the (usually 12-note) "universe set"
> that pervades music-theory academia -- I believe the diatonic scale
can
> be understood in its own right, without reference to a chromatic set
> which contains it.
>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi Mark,

Some comments & questions below.

On Fri, 21 Oct 2005, "microtonalist" <mark@equiton..> wrote,
in reply to "wallyesterpaulrus":
>
> I agree that diatonic scales can be understood only in terms of the L
> and s scalestep concepts. As soon as you assign them a distinct
> proportion, then a 'universe set' does happen. ...

In a Pythagorean scale, we can fix the size of L at 9/8 and s at
3/2 * 8/9 * 8/9 * 8/9 = 256/243, so that the ratio L:s = 32:27.
The Pythagorean diatonic scale has steps LLsLLLs.

In a JI diatonic scale, we can fix the fifth at 3/2, fourth at 4/3,
major third at 5/4 and major second at 9/8. The step from major
second to major third is 5/4 / 9/8 = 10/9 <-- a little less than a
major second; in fact a comma of size 9/8 / 10/9 = 81/80.
Clearly we don't have just one size L of Large step; rather we
have two, say L = 9/8 and M = 10/9.
The JI diatonic scale has steps LMsLLMs. Still, L~=M.

After all this preamble (sorry!) - does the phenomenon you're
talking about - a 'universe set' - still happen?

> Talking philosophically, I suppose I belong to those who explore a
> pattern for what it is - a pattern - and wondering about the 'shape' of
> the pattern. Kind of meta-tonality thinking. ...

See, the two diatonic patterns above are _very nearly_ the
same thing. I guess I'm asking you: does the difference
matter to you?

> Arguing by analogy is far from rigorous, and amongst the patterns there
> are probably many 'dud' scales.

How far away from the "idealised" diatonic scale, with its
two classes of step size, Large and small, can we go and
still retain the same essential feel? In your experience,
will the same musical techniques work when we vary the
steps sizes by, say, 10%? Or 20%?

Regards,
Yahya

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🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi all,

On Fri, 21 Oct 2005 "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > In a Pythagorean scale, we can fix the size of L at 9/8 and s at
> > 3/2 * 8/9 * 8/9 * 8/9 = 256/243, so that the ratio L:s = 32:27.
> > The Pythagorean diatonic scale has steps LLsLLLs.
>
> The relevant ratio is usually taken as the ratio of logarithms, and
> so in this case we'd look at log(9/8)/log(256/243) = 2.26.

Thanks, Gene, for this correction. In other words, how many
of the smaller interval make up the larger interval. Makes very
good sense.

> In the case
> of 12-et, it goes down to exactly 2, and keeps shrinking as the fifth
> gets flatter, to 5/3 at 31-et, and 3/2 at 19-et.

What are the highest and lowest values for the ratio log L / log s
in common (microtonal) use?

Obviously, 1.5 = 3/2 is fairly low, and if we went all the way down
to 1, we'd have 7-EDO, in which the distinction between L and s
disappears entirely - and with it, presumably, the possibility of
any leading-tone effect, which is such a characteristic of the way
we tend to use diatonic scales.

But I can readily imagine the ratio getting down to 1.333' = 4/3
or even 1.25 = 5/4, and the scale being recognisably "diatonic".

And someone has already commented on the possibility of the
ratio reaching as high as 3/1. What about 4/1? 5/1?

Regards,
Yahya

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🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

Hi Gene and Yahya,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > What are the highest and lowest values for the
> > ratio log L / log s in common (microtonal) use?
>
> The range is broad. Here are some landmarks:

Again, i would have included 53-edo with its 9/4 ratio.

Here's an expanded version of Gene's list, with 53-edo,
pythagorean, and some other EDOs and important meantones,
and with the ratio given in both rational (for EDOs)
and decimal format:

ratio of log(L)/log(s)

.. edo .. ratio .. decimal

.... 5 ... infinity (and all multiples of 5 with same mapping)
... 42 ... 8:1 ... 8.0
... 37 ... 7:1 ... 7.0
... 32 ... 6:1 ... 6.0
... 27 ... 5:1 ... 5.0
... 49 ... 9:2 ... 4.5
... 22 ... 4:1 ... 4.0
... 39 ... 7:2 ... 3.5
... 17 ... 3:1 ... 3.0
... 46 ... 8:3 ... 2.66...
... 29 ... 5:2 ... 2.5
... 41 ... 7:3 ... 2.33...
pythagorean ..... ~2.260016753
... 53 ... 9:4 ... 2.25
... 12 ... 2:1 ... 2.0
1/6-comma m.t. .. ~1.819203619
... 55 ... 9:5 ... 1.8
... 43 ... 7:4 ... 1.75
1/5-comma m.t. .. ~1.748010733
... 31 ... 5:3 ... 1.66...
1/4-comma m.t. .. ~1.649392797
golden m.t. ..... ~1.618033989 (= phi)
... 50 ... 8:5 ... 1.6
LucyTuning ...... ~1.558617346
... 19 ... 3:2 ... 1.5
... 45 ... 7:5 ... 1.4
... 26 ... 4:3 ... 1.33...
... 33 ... 5:4 ... 1.25
... 40 ... 6:5 ... 1.2
... 47 ... 7:6 ... 1.166...
.... 7 ... 1:1 ... 1.0 (and all multiples of 7 with same mapping)
... 23 ... 3:4 ... 0.75
... 16 ... 2:3 ... 0.66...
.... 9 ... 1:2 ... 0.5
... 11 ... 1:3 ... 0.33...

Woolhouse specifically mentioned the 8/5 ratio of log L / log s
(but not exactly with that terminology) of 50-edo.

🔗monz <monz@tonalsoft.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Rich Holmes <rsholmes@mailbox.syr.edu>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Gene,

On Sun, 23 Oct 2005 "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> > Hi all,
> >
> > On Fri, 21 Oct 2005 "Gene Ward Smith" wrote:
>
> > What are the highest and lowest values for the ratio log L / log s
> > in common (microtonal) use?
>
> The range is broad. Here are some landmarks:
>
> 5: infinity
> 37: 7
> 32: 6
> 27: 5
> 22: 4
> 17: 3
> 12: 2
> 31: 5/3
> 19: 3/2
> 26: 4/3
> 33: 5/4
> 7: 1
> 23: 3/4
> 16: 2/3
> 9: 1/2
> 11: 1/3
>
> The 9 and 11 are pretty cheesy, but anything from 2/3 to infinity
> seems to be fair game.

Thanks for this list.

> > Obviously, 1.5 = 3/2 is fairly low, and if we went all the way down
> > to 1, we'd have 7-EDO, in which the distinction between L and s
> > disappears entirely - and with it, presumably, the possibility of
> > any leading-tone effect, which is such a characteristic of the way
> > we tend to use diatonic scales.
>
> Some wild and dangerous types such as Paul and Herman are willing to
> go *below* 1.

I'm picturing The Ancient Mariner's Map of Musical Tunings - off
to the west of the Pillars of Hercules (and Pythagoras) lie The
Wild Waters, fearsomely inscribed "Here There Be Beastes" ...

Regards,
Yahya

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

On Mon, 24 Oct 2005 "monz" wrote:
>
> Hi Gene and Yahya,
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > > What are the highest and lowest values for the
> > > ratio log L / log s in common (microtonal) use?
> >
> > The range is broad. Here are some landmarks:
>
>
> Again, i would have included 53-edo with its 9/4 ratio.
>
> Here's an expanded version of Gene's list, with 53-edo,
> pythagorean, and some other EDOs and important meantones,
> and with the ratio given in both rational (for EDOs)
> and decimal format:
>
>
> ratio of log(L)/log(s)
>
> .. edo .. ratio .. decimal
>
> .... 5 ... infinity (and all multiples of 5 with same mapping)
> ... 42 ... 8:1 ... 8.0
> ... 37 ... 7:1 ... 7.0
> ... 32 ... 6:1 ... 6.0
> ... 27 ... 5:1 ... 5.0
> ... 49 ... 9:2 ... 4.5
> ... 22 ... 4:1 ... 4.0
> ... 39 ... 7:2 ... 3.5
> ... 17 ... 3:1 ... 3.0
> ... 46 ... 8:3 ... 2.66...
> ... 29 ... 5:2 ... 2.5
> ... 41 ... 7:3 ... 2.33...
> pythagorean ..... ~2.260016753
> ... 53 ... 9:4 ... 2.25
> ... 12 ... 2:1 ... 2.0
> 1/6-comma m.t. .. ~1.819203619
> ... 55 ... 9:5 ... 1.8
> ... 43 ... 7:4 ... 1.75
> 1/5-comma m.t. .. ~1.748010733
> ... 31 ... 5:3 ... 1.66...
> 1/4-comma m.t. .. ~1.649392797
> golden m.t. ..... ~1.618033989 (= phi)
> ... 50 ... 8:5 ... 1.6
> LucyTuning ...... ~1.558617346
> ... 19 ... 3:2 ... 1.5
> ... 45 ... 7:5 ... 1.4
> ... 26 ... 4:3 ... 1.33...
> ... 33 ... 5:4 ... 1.25
> ... 40 ... 6:5 ... 1.2
> ... 47 ... 7:6 ... 1.166...
> .... 7 ... 1:1 ... 1.0 (and all multiples of 7 with same mapping)
> ... 23 ... 3:4 ... 0.75
> ... 16 ... 2:3 ... 0.66...
> .... 9 ... 1:2 ... 0.5
> ... 11 ... 1:3 ... 0.33...
>
>
> Woolhouse specifically mentioned the 8/5 ratio of log L / log s
> (but not exactly with that terminology) of 50-edo.
>

Monz,

Thank you for this information. I can use this as a list
of diatonic ( and "anti-diatonic"? "sub-diatonic"? )
scales for experimentation.

Regards,
Yahya

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🔗monz <monz@tonalsoft.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> I agree that diatonic scales can be understood only in terms of the
L
> and s scalestep concepts.

Not at all what I had in mind. I was speaking of Fokker periodicity
blocks and the like -- have you read my _Forms Of Tonality_ paper?

http://www.lumma.org/tuning/erlich/

> As soon as you assign them a distinct
> proportion, then a 'universe set' does happen.

Huh? How so? What if they're in the golden ratio? Or did you mean a
distinct *rational* proportion?

> Meantone tunings of the
> 19th century were most probably open - i.e. not 31edo, and I've
often
> thought that the other patterns of Ls scales could have open forms
too,
> wherever the ratio between L and s is irrational.
>
> Talking philosophically, I suppose I belong to those who explore a
> pattern for what it is - a pattern - and wondering about
the 'shape' of
> the pattern. Kind of meta-tonality thinking. What I don't see much
> sense in is mathematics with n-tones, like some of the serial and
pitch-
> class set stuff that is very much in vogue.
>
> Arguing by analogy is far from rigorous, and amongst the patterns
there
> are probably many 'dud' scales. You could argue that most of my
work
> is 'eye music'. but even Bach was accused of that... (tho' i'm not
> comparing myself to JS)
>
> It depends upon what is meant by microtonality. Xentonal,
hypertonal,
> spectral...
>
> Mark
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> >
> > > My only guiding principle is to aruge by analogy with the
structure
> > > of the heptatonic diatonic and pentatonic scales.
> >
> > That's really my guiding principle too, though it has seemed to
guide
> > us in different directions, due to philosophical differences. For
one
> > thing, I don't buy the idea of the (usually 12-note) "universe
set"
> > that pervades music-theory academia -- I believe the diatonic
scale
> can
> > be understood in its own right, without reference to a chromatic
set
> > which contains it.
> >
>