------------------------------

Topic No. 4

Date: Fri, 27 Nov 1998 19:00:11 -0800 (PST)
From: "M. Schulter"
To: Tuning Digest
Subject: Re: Tuning "innovations" and rediscoveries
Message-ID:

One curious feature of the world of alternate tunings is that almost any "new" proposal is likely to have interesting precedents. Here I would like to comment both on a tuning recently proposed by Robin Perry, and on an interesting interval from Gary Morrison's 88-cent equal temperament (88-cet) as analysed by Heinz Bohlen in his entertaining article "The Musical Animal's Acoustic Adventures," Xenharmikon 17 (Spring 1998), pp. 41-56 at 49-50.

First, however, I should recall my own recent post of a system for "quasi-12-tet" tuning using two just intonation intervals to define each "quasi-700-cent" fifth: a Pythagorean schisma major third built from eight pure 3:2 fifths down or 4:3 fourths up at 8192:6561 or ~384.36 cents; and a pure 6:5 minor third at ~315.64 cents. As I soon discovered, Kirnberger and a number of other theorists in the later 18th and early 19th centuries had proposed a tuning based on an identical fifth of 16384:10935, or fourth of 10935:8192.

The only possible distinction was that these theorists generally seemed to prefer building a "quasi-500-cent" fourth from a Pythagorean apotome of seven pure fifths (2187:2048, ~113.69 cents) plus a pure 5:4 major third (~386.31 cents). In fact, this approach is a bit more economical than mine in requiring only eight pure intervals rather than nine to generate each fifth or fourth of the quasi-12-tet cycle!

From a more historico-poetic perspective, of course, my use of the schisma M3 may have reflected my fascination with this interval in an early 15th-century setting. More generally, however, the exercise may have demonstrated mainly that "new" tunings are often mostly a matter of reinventing the wheel -- or the spiral of fifths.

Nevertheless, such proposals can place traditional tunings in a new light, and suggest that attractive "reformulations" may have merits other than simply following the parameters of a traditional tuning system.

-----------------------------------
1. Robin Perry's 114/90-cent system
-----------------------------------

As my first example, I cite the case of Robin Perry, who only this month described an approximation of 7-limit just intonation based on the use of two semitones of 114 cents (114/1200 octave) and 90 cents (90/1200 octave).

Paul Hahn, in reply, suggested that such a proposal seemed just about identical to extended Pythagorean tuning with its schisma thirds and septimal schisma.

From a purely acoustical standpoint, in fact, there may seem little difference between Robin's rounded interval of 90 cents and the Pythagorean diatonic semitone or limma at ~90.224 cents, or between a rounded 114 cents and the chromatic semitone or apotome at ~113.69 cents.

However, as a Pythagorean, I found it fascinating that someone should propose such a system for its elegance and convenience quite apart from the connection with Pythagorean tradition.

Of course, the focus of Robin's proposal also permits us to appreciate some of this stylistic motivations which might apply to medieval Pythagorean tuning more than to some flavors of 20th-century just intonation music. For example, the regular Pythagorean M3 and m3 at about 407.82 cents and 294.13 cents play an integral role in 13th-14th century polyphony, but have less significance in styles where smooth rather than active thirds are the ideal.

As I recall, Robin's proposal focuses instead on approximations of 5-based and 7-based intervals, an area where a 90/114-cent tuning can have attractions quite apart from the possibility of the kinds of intervals which intrigue me as a medievalist.

In short, Robin's post includes for me a valuable insight: "If there were not already a tradition of Pythagorean tuning, a tuning with semitones of 114 cents and 90 cents might be well worth inventing."

---------------------------------------------
2. Gary Morrison, Heinz Bohlen, and 792 cents
---------------------------------------------

In his article on "The Musical Animal's Acoustic Adventures," at pp. 49-50 and Fig. 9, Heinz Bohlen analyses the interval of 792 cents produced by nine steps of Gary Morrison's 88-cet.

Bohlen, ibid. at 50, interestingly remarks that the Musical Animal (MA)

"chose something that it regarded as totally unusual: Gary Morrison's 88CET. And in order to make it a real hard test it tuned two of its sine-wave generators to an interval of 792 cent, or nine times 88 cent."

In choosing this interval of 792 cents, our MA is drawn by the consideration that it's

"far enough from the perfect minor sixth's 814 cents" -- that is, from a ratio of 8:5 in 5-based just intonation.

The Musical Animal concludes that

"[w]hat it heard was still close to a minor sixth, perhaps a little rougher.... The MA thought: It doesn't sound odd, just interesting, and it doesn't look bad either."

Both when first reading this and now, my immediate reaction is to point out that the minor sixth of 128:81, or ~792.18 cents, is a regular Pythagorean interval which plays a vital role in 12th-13th century polyphony, where its expansion by conjunct contrary motion to the octave is one of the most characteristic cadences of the era.

From another point of view, this interval might be described as an octave minus two 9:8 whole-tones; each voice contributes a whole-tone to the expansion into the octave:

	  f'-- +204 -- g'
	(792)        (1200)
	  a -- -204 -- g
	

Also, a very evocative resolution during the same period moves to the fifth by oblique motion, with an expressive diatonic semitone in one of the melodic lines:

	  c'-- -90 -- b
	(792)       (702)
	  e           -
	

While often used in practice, this minor sixth was interestingly often considered as a strong dissonance in the theory of the period, being frequently classed together with m2, M7, and A4 or d5. Of course, these intervals also often enjoy a bold role in practice, and Johannes de Garlandia (c. 1240?) is realistic as well as flexible when he asserts that any discord can be pleasing if it is aptly resolved to a stable concord..[1]

At any rate, it's fascinating that virtually the same interval -- here a minor sixth at or very near 792 cents -- can be either something "totally unusual" or a familiar friend depending on one's background. The difference between the rounded 88-cet interval and an 8:5 minor sixth, noted by Bohlen, is very close to the syntonic comma of 81:80 (~21.51 cents) distinguishing the Pythagorean 128:81 from 8:5.

Again, while some of us may enjoy citing historical precedent, the fresh perspective on a 792-cent interval offered by Bohlen is interesting in itself, as is the fact that such a traditional interval (or a close approximation) should be included in a new system such as 88-cet.[2]

In conclusion, it seems to me that "new" tuning proposals of this kind are indeed new points of view. While discovering connections between these "innovations" and older traditions is a delight, appreciating the new motivations and contexts which prompt their "rediscovery" can be an education on all sides.

Since we can hardly make many nontrivial tuning proposals without having a good chance of retracing precedent, it may be fortunate that at least we can enjoy and learn from this continual process of retracing and reinvention.

------------
Notes
------------

1. Garlandia ranks m6 as somewhat more tense than m7 or M6, on par with M2, and milder than m2, M7, or a tritone. Other theorists such as Franco of Cologne (c. 1260?) take a somewhat less lenient view, ranking M2, m7, and M6 together as relatively tense but to some degree "compatible," and m6 together with the strongly discordant m2, M7, and A4 or d5.

2. There are other Pythagorean affinities in 88-cet: one step at 88 cents is only about 2 cents from the Pythagorean ~90-cent limma at 256:243; 8 steps or 704 cents is only about 2 cents wide of the pure 3:2 fifth at ~702 cents; 16 steps or 1408 cents is only about 4 cents wide of the pure Pythagorean M9 at 9:4 or ~1404 cents; 17 steps or 1496 cents is again only about 4 cents from the Pythagorean m10 at 64:27 (~1494 cents); and 18 steps or 1584 cents is almost identical to a schisma M10 at 16384:6561 or ~1584.36 cents. Since 88-cet produces its closest approximation to an octave at 14 steps or 1232 cents, it is in effect almost an open tuning, so that these are only a sampling of the possibilities.

Most respectfully,

Margo Schulter
mschulter@value.net

------------------------------

Topic No. 5

Date: Fri, 27 Nov 1998 23:26:31 -0800
From: curley@ucla.edu (Doren Garcia)
To: Tuning Digest
Subject: Thanking the Wolf
Message-ID:

Tuning Buddies,

I want to take this opportunity to thank you for your opinions about how Mozart might have approached tuning. I suppose I should have been surprised at the fullness of the controversy my little question provoked, but I'm no longer a tuning virgin.

However, I have one more question (I really should know this since my buddy Ivor explained it to my tiny brain once):

I've forgotten what a 'wolf' is. I think I know, but could someone give me a brief explanation?

Thanks again, you guys are great, never condescending always helpful.

Doren
Design Student UCLA

------------------------------

Topic No. 6

Date: Sat, 28 Nov 1998 04:40:29 -0500
From: "Paul H. Erlich"
To: Tuning Digest
Subject: RE: reply to Ed Foote on Mozart Tuning
Message-ID: <85B74BA01678D211ACDE00805FBE3C050B6542@MARS>

Ed Foote wrote,

My point is that during Mozart's era, meantone was on the descent and well tempering was on the rise, and the smoothest of the former was very close to the roughest of the latter.

The smoothest meantone was very close to the roughest well-temperament? Even in music that avoids the wolf?

------------------------------

Topic No. 7

Date: Sat, 28 Nov 1998 04:51:15 -0500
From: "Paul H. Erlich"
To: Tuning Digest
Subject: reply to Bill Sethares on subharmonics
Message-ID: <85B74BA01678D211ACDE00805FBE3C050B6543@MARS>

I wrote,

Partch credits Riemann and many others in preceding his "utonality" concept. The concept has an intersting status around here, with a few giving it little to no importance (e.g., Heinz Bohlen), a few giving it near-equal status with the otonal or "overtone series" concept, such as Daniel Wolf and (implicitly) Bill Sethares, and most of us falling somewhere in between.

Bill Sethares wrote,

I'm not sure I agree with this. In TTSS, I discuss subharmonic series, and argue that *if* you have a sound whose timbre is defined by a subharmonic series, then its dissonance curve is the same as the dissonance curve for a harmonic sound. In consequence, the set of scale steps that minimize sensory dissonance for a subharmonic sound is the same (the set of JI scale tones) that minimize dissonance for a harmonic sound. However, there are many physical sources of harmonic sounds, and few sources of subharmonic tones. Hence the *if* part rarely occurs outside synthesized sounds.

No offense, but you missed my point completely. The duality concept espoused by Riemann and to some degree Partch claims that utonal and otonal chords are equally consonant, when the constituent tones are harmonic. "Subharmonic sounds" lie completely outside the realm of this discussion, and as discussed before, "subharmonics" of acoustic instruments are more accurately described as true fundamentals.

Anyway, the Plomp/Sethares theory of consonance states that the dissonance of a chord is the sum of the dissonances of the individual intervals. Now the otonal and utonal chords to be compared contain exactly the same intervals. If we ignore registral variations of the critical band, the theory therefore states that otonal and utonal chords are equally consonant, when the constituent tones have harmonic spectra.

------------------------------

Topic No. 8

Date: Sat, 28 Nov 1998 02:45:51 -0800 (PST)
From: Mark Nowitzky
To: Ortgies.Ibo@t-online.de
Cc: Tuning Digest
Subject: Trouble Posting to tuning@onelist.com
Message-ID: <2.2.16.19981128024231.3b2f8ccc@pacificnet.net>

Hi Ibo Ortgies,

You mentioned you were having trouble posting to tuning@onelist.com - the problem may be that you are emailing from ortgies.ibo@t-online.de but you are known to tuning@onelist.com as 04215289853-0001@t-online.de

The mailing list rejects posts from users it does not recognize. (I had a similar problem with my own second email address a couple of months ago.)

To fix the problem, do the following:

1) subscribe a second time to the tuning list, using the new email address ("ortgies.ibo@t-online.de")

2) to avoid getting two copies of each post, unsubscribe the old email address ("04215289853-0001@t-online.de")

Good luck!

--Mark Nowitzky

P.S.: With all that said, now I can go look at your web pages ("Weitere links f=FCr die neue mittelt=F6nige Orgel mit Subsemitonien in= Bremen-Walle", etc.)

+------------------------------------------------------+
| Mark Nowitzky                                        |
| email:  nowitzky@alum.mit.edu    AIM:  Nowitzky      |
| www:    http://www.pacificnet.net/~nowitzky          |
|         "If you haven't visited Mark Nowitzky's home |
|         page recently, you haven't missed much..."   |
+------------------------------------------------------+

------------------------------

Topic No. 9

Date: Sat, 28 Nov 1998 06:07:46 -0500
From: "Paul H. Erlich"
To: Tuning Digest
Subject: RE: Many Tones
Message-ID: <85B74BA01678D211ACDE00805FBE3C050B6546@MARS>

I wrote,

On the many-tones issue, I think the tetrachord structure helps to reduce the number of independent elements that need to be perceived...

Carl Lumma wrote,

Could you elucidate?

Well, I think there is a kind of similarity at the perfect fifth and fourth, weaker than octave similarity but still strong enough to be the interval that many an untrained singer is off (I can't tell you how many parties I've gone to and strummed a song on guitar while listening to the dissonances made by the melody when sung off by a fifth by a drunken "performer"). So the diatonic scale, which has two identical tetrachords in every octave span, separated by either a p4 or a p5, really only has 3 or 4 fully independent pitches.

My paper describes two versions of the 22-tET decatonic scale, one of which is "maximally even" (e.g. 0 2 4 6 8 11 13 15 17 19), and the other is "tetrachordal" (e.g. 0 2 4 6 9 11 13 15 17 19) (actually, it consists of pentachords, which have the same relationship to this scale as tetrachords do to the diatonic scale). The "tetrachordal" scale sounds much better, despite having only 6 consonant tetrads to the "maximally even" scale's 8. I think this is p4/p5 similarity at work, reducing the cognitive load for one scale but not the other.

As a result of this comparison, I put the whole maximal evenness literature in the "interesting but unimportant" category.

I wrote,

compositional technique can make at least as much difference as an order of magnitude difference in the number of tones.

Carl replied,

Yes compositional context makes a huge difference. But you can't have your cake and eat it too. If you use less, in whatever way, you're using less. I tried to make the chart reflect this.

Stephen Soderberg made the point I was trying to make here. If you write a piece in 22 out of 41, you'd better run a lot of 2/41 oct. steps in a short period of time so that that can be understood as the norm. Then, the position of the three 1/41 oct. steps will stand out and the entire scale will be projected (actually, these steps might be too small, but ignore that for now). This is a different type of projection but there's no reason why highly microtonal piece can't operate on this principle (though muliply-iterated maximally-even structures are not the harmonies for which this scale was constructed! 13 out of 23 would work just as well to illustrate this point in a non-acoustical abstract world, but would not be a good scale for real harmonic music).

------------------------------

Topic No. 10

Date: Sat, 28 Nov 1998 06:54:19 -0500
From: "Paul H. Erlich"
To: Tuning Digest
Subject: RE: Paul Hahn
Message-ID: <85B74BA01678D211ACDE00805FBE3C050B6549@MARS>

Carl, I hate to break it to you, but I think you're wrong. The scale you describe has 30 7-limit consonances, but consider this 3^2 * 5 * 7 genus:

	|
	|               35:24-----35:32-----105:64
	|              / / \ \   / / \ \   / /
	|           5:3-/---\-5:4-/---\-15:8/ 
	|           /|\/     \/|\/     \/ |/
	|          / |/\     /\|/\     /\ /
	|         / 7:6-------7:4-------21:16
	|        / /   \ \ / /   \ \ / /  
	|      4:3-------1:1-------3:2        [Diagram by Carl Lumma]
	
	

By my count this has 31 7-limit consonances.

Guys, wouldn't any rotation and/or reflection of this figure in the tetrahedral/octahedral lattice work too? Here's where group theory comes in: how many elements does the symmetry group of this lattice (better known as the face-centered cubic lattice) have? Does that tell us the number of distinct solutions to Carl's challenge? Note that many of these are not Euler-Fokker genera. [see response]

I think I missed your tuning because I was looking mostly at stuff that fit roughly within the 12tET pitch classes (I didn't consider any tunings with the 64:63 comma).

It is conceivable that some of these rotations and/or reflections better fit Carl's notion of 12 pitch classes. However, I would prefer to stick to the configurations with two steps along the 3 axis, for the better melodic coherence and more intelligible chord progressions that would allow.

------------------------------

Topic No. 11

Date: Sat, 28 Nov 1998 14:13:51 +0000
From: Ortgies.Ibo@t-online.de (Ortgies Ibo)
To: Tuning Digest
Subject: Testmail
Message-ID: <3660051F.6E7@t-online.de>

Sorry for annoying anyone with this testmail. If it is on the list everything seems to work again with my e-mail

Kind regards

Ibo Ortgies

links for the new meantone-organ (with split keys)in Bremen-Walle (Germany) http://home.t-online.de/home/Ortgies.Ibo/Wallpage.htm
============================================= Organs with subsemitones: http://home.t-online.de/home/Ortgies.Ibo/Subsemi.htm
------------------------------

End of TUNING Digest 1595
*************************

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