..and later claimed that it had 30 consonances. Wrong. It has 31. In the original article, I also claimed that the following scale has the same number of consonances as the above scale...

                 5/3--------5/4     [Scale Y]
                / |\       / | 
               /  | \     /  |
              /  7/6--------7/4
             / / / \  \ / / /  
            4/3-/---\-1/1  /
           / |\/     \/|  /
          /  |/\     /\| /
         / 28/15------7/5
        / / / \  \ / / /
      16/15/---\-8/5  /
       |  /     \ |  /
       | /       \| /
     112/75-----28/25

Wrong again. This scale only has 30 consonances. The final score seems to be...

30 consonances
---------------
12-out-of Stellated Hexany
(6 tetrads, 1 hexany, not suitable for conventional piano)

Scale Y, above
(4 tetrads, 1 hexany, suitable for conventional piano)

31 consonances
--------------
Paul Hahn's Favorite Scale
(4 tetrads, 2 hexanies, suitable for conventional piano)

Scale Z, above
(is like Paul Hahn's (a) scale in that it is more symmetrical than his favorite, and is completed by its four tetrads unlike his favorite. But he is correct in stating that it contains an interval smaller than the 25:24, so his (a) scale is preferred if these properties are desired.)

It seems, by looking at these structures, that 31 consonances is the most we're allowed with 12 tones. A proof, based on the symmetries of the lattice or anything else, is unknown to me.

Carl

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

Topic No. 3

Date: Tue, 1 Dec 1998 14:01:59 -0500
From: "Paul H. Erlich"
To: Tuning Digest
Subject: RE: All, then best, 7-limit scales with 31 consonances
Message-ID: <85B74BA01678D211ACDE00805FBE3C050B6554@MARS>

I get the list in digest form, so I did not see Paul Hahn's very studly post when I made mine. I also quoted Bram and misattributed the quote to Paul Hahn, my mistake. Bram was thinking along the lines of Euler-Fokker/Tenney lattices (rectangular) but then understood the triangular lattice. Can we all agree that the triangular lattice is more useful? There is nothing to prevent us from defining a city-block dissonance metric like the Tenney Harmonic Distance function on the triangular lattice, where 6:5 would be represented by a single segment and thus have a smaller Harmonic Distance than 15:8, which is represented by two segments (Tenney's HD is defined on a rectangular matrix and thus gives these two intervals equal dissonance). I've suggested giving each interval a length proportional to the log of its odd limit. I came up with this before I heard of Tenney's idea, where each prime interval is given a length proportional to the log of the corresponding prime number, and before I saw any 3-d triangular lattices in music theory. I've used this metric to create diagrams of scales using multidimensional scaling. The idea is that you calculate the distance (according to this non-Euclidean metric) between all pairs of intervals and try to best approximate this distance matrix with a configuration of points in Euclidean space of low dimension. Statistical software packages like SAS should have this multidimensional scaling procedure built in. For example, a diatonic or extended meantone scale, when scaled to three dimensions, comes out as a helix, with the chain of fifths winding around the helix so that one full turn corresponds to 3-4 fifths, putting all the notes of each consonant triad near to one another. The result is the simplest and most informative diagram of the diatonic scale or extended meantone tuning that I can imagine.

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

Topic No. 4

Date: Tue, 1 Dec 1998 16:12:01 -0600 (CST)
From: Paul Hahn
To: Tuning Digest
Subject: RE: Paul Hahn
Message-ID:

On Sat, 28 Nov 1998, Paul H. Erlich wrote:

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?

Forty-eight. (Simple way to figure it: the Voronoi cell is a rhombic dodecahedron, which has 12 faces each of which is fourfold symmetric.) The original genus could only have twelve distinct orientations because it itself was fourfold symmetrical. However, if you use an assymmetrical shape, like the "Jetsons" chord 1:1 5:4 7:5 3:2, it has 48 distinct rotations and reflections. I was going to ASCII-draw them all, but then I thought better of it. 8-)>

Does that tell us the number of distinct solutions to Carl's challenge?

Y'know, I just figured out that the 396 31-interval 12-note scales I posted before are not all. There is at least one other class of them, although not nearly as large: scales consisting of a 1:1 and any 11 of the 12 pitches that form a 7-limit consonance with it. There are 12 of those (12 possible pitches to omit). But those don't even come close to approximating 12TET.

Okay, so that's 408 31-interval 12-note 7-limit scales so far. I'll try to find more. And a proof that 31 intervals is the maximum. But don't hold your breath . . .

--pH  http://library.wustl.edu/~manynote
    O
   /\        "'Jever take'n try to give an ironclad leave to
  -\-\-- o    yourself from a three-rail billiard shot?"

             NOTE: dehyphenate node to remove spamblock.          <*>

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

Topic No. 5

Date: Tue, 1 Dec 1998 15:37:21 -0800
From: monz@juno.com
To: Tuning Digest
Subject: All, then best, 7-limit scales with 31 consonances
Message-ID: <19981201.153726.-147713.2.monz@juno.com> [Paul Hahn:]

There are twelve possible orientations for this shape in the 7-limit 3-d space, only three of which are Euler genuses (geni?).

The plural of "genus" is "genera".

- Joe Monzo
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

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

Topic No. 6

Date: Tue, 1 Dec 1998 17:47:34 -0600 (CST)
From: Paul Hahn
To: Tuning Digest
Subject: pH's sievelike memory
Message-ID:

On Tue, 1 Dec 1998 monz@juno.com wrote:

[Paul Hahn:] There are twelve possible orientations for this shape in the 7-limit 3-d space, only three of which are Euler genuses (geni?).

The plural of "genus" is "genera".

Of course it is. Even if I couldn't remember it for myself (which I couldn't at the time) I could have looked about a paragraph down in Paul E.'s message that I was replying to. Unfortunately I had already deleted that part. Silly me.

--pH  http://library.wustl.edu/~manynote
    O
   /\        "'Jever take'n try to give an ironclad leave to
  -\-\-- o    yourself from a three-rail billiard shot?"

             NOTE: dehyphenate node to remove spamblock.          <*>

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

Topic No. 7

Date: Tue, 1 Dec 1998 20:16:17 -0800
From: monz@juno.com
To: Tuning Digest
Subject: Re: TUNING digest 1598
Message-ID: <19981201.201620.-147713.4.monz@juno.com>

[Paul Erlich, TD#1598, topic 4:]

Any single interval appears just as prominently in the harmonic series as it does in the subharmonic series. It is only chords of 3 or more notes that can be classified as "otonal" or "utonal".

I thought this last point was worth reiterating. Tonality is distinctly a by-product (or a cause?) of the use of triadic harmony. In an earlier era, c. before 1500, when harmony was based on intervallic relationships and not triadic chord progressions, the tonal system was modal and not major/minor.

I've pointed out here recently that the dualistic major/minor system could only happen, as Partch observed in Genesis, [p. 88-90] because of the inherent dual relationship of a two-member ratio, and [p 109-118] the use of at least three tones to delimit a tonality.

- Joe Monzo
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

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

Topic No. 8

Date: Wed, 2 Dec 1998 02:04:41 EST
From: Ascend11@aol.com
To: Tuning Digest
Subject: Re: Margo Schulter's post: Re: What is a wolf? - piano listening trials
Message-ID: <7db319bd.3664e689@aol.com>

I'd just like to add a few brief piano test observations to Margo Schulter's full explanation of several kinds of "wolf" important in keyboard tunings and her interesting examples [TD#1597.1].

My piano is tuned with just 5/4 thirds on Eb and G, Bb and D, F and A, and C and E, eight in all, leaving the 32/25 "wolf" major thirds B to Eb, F# to Bb, C# to F, and G# to C. I tried the three inversions of the Bb-D-F# chord using:

1: bass octave Bb1-Bb2 with 8/5 minor sixth F#3-D4 on top,
2: bass octave D2-D3 with narrowed minor sixth Bb3-F#4 on top, and
3: bass octave F#2-F#3 with 8/5 minor sixth D4-Bb4 on top.

To my ears, which are neither the most acute nor the least acute, the second of these chords having the Ds octave in the bass sounded distinctly more stable and solid than did the other two chords. At the same time, all three of the chords seemed to have somewhat similar characters. Speculating a bit unknowledgeably, I could imagine Mozart using the second chord held long at some dramatic point in his music, ultimately coming to a more final kind of resolution. I believe he would treat the other inversions of this chord differently in his music. By contrast, all three inversions of the chord would not differ as much in music completely conceived in the 12-ET system. I believe that on a piano tuned to 12-ET, chords 1 and 3 would sound more stable than they do on my piano and that chord 2 would have less of a clear character, bite, and solidity than it does on my piano.

One brief additional thing - I've been measuring the actual frequencies of the partials of different individual strings on my piano to see how well it is holding its tune, and to investigate the octave stretching characteristics somewhat. The inharmonicity of the partials of the strings, particularly in the higher pitch register, is appreciable and it increases rapidly with frequency. According to Benade (see note 1), a typical rate of increase of the inharmonicity of the partials is by a factor of 2.76 per octave. He gives as typical for a string for C4 the following amounts in cents by which the 2nd through the 6th partials are sharp of what they would be if their frequencies were strictly harmonic:

Partial 2: 0.83 cents,
Partial 3: 2.25 cents,
Partial 4: 4.48 cents,
Partial 5: 6.71 cents, and
Partial 6 9.79 cents.

When I tried to analyze a C4 string on my piano, I obtained stable frequencies for partials 2 through 6 but I found that the frequency of the first partial for C4 varied by as much as 4 cents over the course of the note in a rather irregular way.

The first partial for my D4 was quite stable, and for the D4 I measured the following degrees of sharp inharmonicity for partials 2 through 6 (in cents):

Partial 2: 0.86,
Partial 3: 2.36,
Partial 4: 4.25,
Partial 5: 6.89, and
Partial 6: 9.83.

Thus the inharmonicity I found on my piano at that point was appreciably less (by roughly 20%) than that reported for a "typical" grand piano by Benade, but it was within the same general "ballpark".

For D5, I found the following stretches for partials 2-5:

Partial 2: 2.43,
Partial 3: 7.09,
Partial 4: 12.78, and
Partial 5: 20.89.

Computing the ratios between measured amounts of stretch for partials of D5 and of D4 we obtain:

Stretch of D5 Partial 2 divided by Stretch of D4, Partial 2 (= 2.43/.86): 2.83,
ratio for Partial 3 values: 3.00,
for Partial 4 values: 3.01, and
for Partial 5 values: 3.03.

Note: although I've given cent values to 0.01 cent, I believe they're accurate only to within about 0.1 or 0.2 cents based on repeat observations, etc.

The lesson I've learned through this exercise is that there simply is no way to tune a real piano so that there will be beatless intervals. I see that even with all the high precision equipment available to piano tuners today, a great deal depends on the ear and judgment of the piano tuner. Nevertheless I have hopes that I'll be able to improve - perhaps audibly - on the "just intonation" tuning to which my piano is presently tuned. According to Benade, on the piano fifths about 1 cent wide of just sound best, major thirds about 3.5 cents wide of just sound best, and I believe there should be a stretch of about 3 cents all over the temperament octave from F3 to F4.

I'm also finding that note frequencies change by as much as several cents from day to day, generally all moving in similar directions but with there being changes in the sizes of fairly close intervals by one, two, or possibly even more cents over the course of days or possibly even hours if the temperature changes appreciably.

Note 1: Source of Benade material: Arthur H. Benade: Fundamentals of Musical Acoustics, Dover Edition 1990, Chapter 16 sections 5 through 7 pages 313-324. (The author reports some interesting psychoacoustic results in these pages, too. The data he gives may need to be confirmed by others before being accepted as fully demonstrated)

Note 2: - an observation: although on a modern piano the differences in character between just or mean tone music and equal tempered music are striking to many people, these differences are likely to have been less marked on early pianos, particularly those in existence before about 1800, which were not strong enough to hold a bank of strings under such high tension as strings on a modern piano are kept at. At lower tension, other things being equal, the inharmonicity of the partials is greater and as a result, the stretching of the partials and consequent "tuning ambiguity" is greater for such pianos having less mechanical strength and whose strings were kept at lower tension.

Dave Hill, La Mesa, CA

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

Topic No. 9

Date: Wed, 2 Dec 1998 08:55:04 -0600 (CST)
From: Paul Hahn
To: Tuning Digest
Subject: Max-interval 7-limit JI scales, yet again
Message-ID:

On Tue, 1 Dec 1998, Paul Hahn wrote:

Y'know, I just figured out that the 396 31-interval 12-note scales I posted before are not all. There is at least one other class of them, although not nearly as large: scales consisting of a 1:1 and any 11 of the 12 pitches that form a 7-limit consonance with it. There are 12 of those (12 possible pitches to omit). But those don't even come close to approximating 12TET.

This scale has six complete tetrads (3 otonal, 3 utonal) but no complete hexanies, oddly enough.

Okay, so that's 408 31-interval 12-note 7-limit scales so far. I'll try to find more. And a proof that 31 intervals is the maximum. But don't hold your breath . . .

I have discovered another group of scales related to this one which adds another 144 scales to our total--but I also have to subtract 24, because I discovered that 24 out of the 396 I described earlier actually have 32 7-limit intervals. They are the 24 possible rotations/reflections of this:

       5:3---------5:4
       /|\         /|\
      / | \       / | \
     /  |  \     /  |  \
    /  7:6---------7:4  \
   /.-'/ \'-.\ /.-'/ \'-.\
 4:3--/---\--1:1--/---\--3:2
  |\ /     \ /|\ /     \ /|
  | /       \ | /       \ |
  |/ \     / \|/ \     / \|
28:15--------7:5--------21:20
    '-.\ /.-'   '-.\ /.-'
       8:5---------6:5

As with the other 396, it has two hexanies and four tetrads. But they are all fairly uneven, and probably not (in practice) as useful as the three I described a few messages ago (the ones Carl calls my favorite, (a), and (b)).

I think 32 really is the maximum. But I have yet to prove it.

--pH  http://library.wustl.edu/~manynote
    O
   /\        "'Jever take'n try to give an ironclad leave to
  -\-\-- o    yourself from a three-rail billiard shot?"

             NOTE: dehyphenate node to remove spamblock.          <*>

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

End of TUNING Digest 1599
*************************

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