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Message: 9175 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:11:32 Subject: Re: More on the naming convention From: Herman Miller On Thu, 15 Jan 2004 19:36:05 -0000, "Gene Ward Smith" <gwsmith@xxxxx.xxx> wrote:>The 5-limit temperament with Ampersand's comma, 7-limit miracle, and >11-limit miracle all have identical TOP tunings of the primes they >cover. Should the Ampersand temperament simply be called miracle?I'd say "5-miracle" for Ampersand, "7-miracle" for 7-limit miracle, and "11-miracle" for 11-limit miracle. Not to be confused with "miracle-21", which is a 21-note miracle tuning. You could have both a prefix and suffix number, such as "7-meantone-19" for a 19-note scale of 7-limit meantone.>One might extend this convention to closest points among reasonable >temperament choices. For instance, the 225/224-planar temperament has >TOP tuning [1200.493660, 1901.172569, 2785.167472, 3370.211784]. The >nearest reasonable TOP point in val space to this seems to be 11-limit >marvel, with TOP tuning [1200.508704, 1901.148724, 2785.132538, >3370.019002, 4149.558115]. The projection of this point into 7-limit >val space is at a distance of 0.06867 cents from 225/224-planar TOP, >which seems to me to justify my proposal to give the 7 and 11 limit >planar temperaments the same name of "marvel". A little farther out, >at a distance of 0.08519 cents is the {225/224, 14700/14641}-planar >temperament, which is both a worse temperament and farther away.Or "7-marvel" and "11-marvel" to be specific. If the context is clear (in a list of 7-limit temperaments, for instance), just "miracle" or "marvel" would suffice. This sounds like a good plan in general. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin
Message: 9176 - Contents - Hide Contents Date: Fri, 16 Jan 2004 08:16:50 Subject: Anyone care to name a temperament? From: Gene Ward Smith There have been complaints I name far too many temperaments around here, so I hope someone will help me out. Name: ??? Seven limit projection: meantone Wedgie: [1, 4, 10, 18, 4, 13, 25, 12, 28, 16] TM comma basis: [81/80, 99/98, 126/125] Mapping: [[1, 2, 4, 7, 11], [0, -1, -4, -10, -18]] TOP: [1201.611156, 1899.198965, 2790.351234, 3371.044615, 4145.302457] Ets: 12, 31, 43, 74 The rules are that you can propose any name except "meantone".
Message: 9177 - Contents - Hide Contents
Date: Fri, 16 Jan 2004 20:18:58
Subject: 46 augmented scales
From: Gene Ward Smith
I'm getting more scales out of this than I expected; below are 46
distinct augmented scales obtained by reducing the 53 Fokker blocks to
augmented. It would be interesting to know what these look like on a
rectangular lattice; I may try Maple on that. The top generators for
augmented are 399.020 for the period and 93.145 for the "generator",
which would be one way to convert these to cents.
{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1,
0], [0, 3], [1, 1], [1, 0], [1, -3]}
{[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [0,
3], [1, 1], [1, 0], [0, 1], [1, -2]}
{[0, 0], [-1, 2], [0, -1], [-1, -1], [0, -3], [1, 2], [-1, 0], [1, 1],
[1, 0], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [0, -1], [-1, 1], [-1, -2], [-2, 1], [-1, -1], [0,
-2], [-1, 0], [-2, 0], [1, -3], [1, -1]}
{[-1, 3], [0, 0], [-2, 1], [-2, 2], [-1, -1], [-1, -4], [0, -3], [0,
-2], [-1, 0], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-2, 2], [-1, -1], [-1,
0], [1, 1], [1, 0], [1, -1], [0, 1]}
{[0, 0], [0, -1], [-1, 1], [-1, -1], [0, -2], [-1, 0], [1, 1], [1, 0],
[2, -2], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, -1], [-1, 1], [-1, -2], [-2, 1], [-1, -1], [0, -2], [-1,
0], [-2, 0], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -1], [1, 2], [-1, 0],
[1, 1], [1, 0], [1, -1], [0, 1]}
{[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [0,
3], [1, 1], [1, 0], [1, -3], [1, -2]}
{[-1, 4], [-1, 2], [0, -1], [-1, -2], [-2, 1], [0, -3], [0, -5], [-1,
0], [0, 3], [1, 0], [0, 1], [1, -2]}
{[-1, 4], [-1, 2], [0, -1], [-1, -2], [-2, 1], [0, -3], [0, -5], [-1,
0], [0, 3], [0, 1], [1, -2], [2, -4]}
{[0, 0], [0, 2], [-1, 2], [0, -1], [0, -3], [-2, 3], [-1, 0], [1, 1],
[1, -3], [2, -2], [1, -1], [2, -4]}
{[-1, 3], [0, 0], [0, 2], [-1, 1], [-2, 2], [-1, -1], [0, -2], [-1,
0], [1, 1], [1, 0], [1, -1], [0, 1]}
{[2, -3], [0, 0], [0, -1], [-1, 1], [0, -2], [1, 2], [-1, 0], [0, 3],
[1, 0], [2, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [-1, 2], [0, -1], [0, -3], [-2, 3], [-1, 0], [1, 1],
[1, 0], [2, -2], [1, -1], [0, 1]}
{[2, -3], [-1, 3], [-1, 1], [0, -2], [1, -4], [-2, 4], [1, 2], [0, 3],
[1, 0], [2, -1], [0, 1], [1, -2]}
{[-1, 4], [0, 2], [-1, 2], [0, -1], [0, -3], [-2, 3], [-1, 0], [1, 1],
[1, -3], [2, -2], [1, -1], [2, -4]}
{[-1, 4], [-1, 2], [0, -1], [-1, -2], [0, -3], [0, -5], [-3, 5], [-1,
0], [0, 3], [1, 0], [0, 1], [1, -2]}
{[2, -3], [-1, 3], [-1, 5], [0, 0], [0, 2], [-1, 1], [-2, 2], [-1,
-1], [0, -2], [0, -4], [1, 0], [2, -5]}
{[2, -3], [0, -1], [-1, 1], [0, -2], [1, -4], [-2, 4], [1, 2], [0, 3],
[1, 0], [2, -1], [0, 1], [1, -2]}
{[-1, 4], [0, 2], [-1, 2], [0, -1], [0, -3], [-2, 3], [-1, 0], [1, 1],
[2, -2], [1, -1], [0, 1], [2, -4]}
{[2, -3], [-1, 3], [-1, 5], [0, 0], [0, 2], [-1, 1], [-2, 2], [-1,
-1], [0, -2], [0, -4], [1, 0], [1, -1]}
{[-1, 4], [-1, 2], [0, -1], [-1, 1], [-2, 2], [-1, -1], [-1, -3], [0,
-4], [0, 3], [1, 0], [0, 1], [1, -2]}
{[-1, 4], [0, -1], [-1, 1], [-2, 2], [-1, -1], [0, -2], [-1, -3], [0,
-4], [0, 3], [1, 0], [0, 1], [1, -2]}
{[2, -3], [-1, 4], [-1, 2], [0, -1], [-1, -1], [1, 2], [0, -4], [0,
3], [1, 0], [0, 1], [-2, 5], [1, -2]}
{[2, -3], [-1, 3], [-1, 5], [0, 0], [-1, 1], [-2, 2], [-1, -1], [0,
-2], [0, -4], [1, 0], [1, -2], [2, -5]}
{[-1, 3], [0, 0], [0, 2], [-1, 1], [-2, 1], [-1, -1], [0, -2], [0,
-4], [-3, 4], [0, -6], [1, -3], [1, -1]}
{[0, 0], [0, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [0, -2], [-1,
0], [0, 3], [1, 1], [1, 0], [0, 1]}
{[0, 0], [-1, 2], [0, -1], [-1, 1], [-2, 1], [-2, 2], [-1, -1], [-1,
0], [1, 0], [1, -3], [1, -1], [1, -2]}
{[-1, 3], [0, 0], [-1, 2], [-1, -2], [-1, -1], [0, -4], [0, 4], [0,
3], [1, 1], [1, -3], [-2, 5], [1, -2]}
{[-1, 3], [0, 0], [-1, 2], [-2, 2], [-1, -1], [0, -3], [0, -4], [0,
4], [1, 1], [0, 1], [1, -2], [2, -5]}
{[0, 0], [-1, 2], [0, -1], [0, -4], [0, 4], [0, 3], [1, 1], [1, -3],
[2, -2], [2, -1], [-2, 5], [1, -2]}
{[2, -3], [-1, 5], [0, 2], [0, -1], [-1, 1], [0, -2], [-2, 3], [1,
-4], [-1, 0], [1, 0], [2, -2], [1, -1]}
{[-1, 3], [0, 0], [-1, 2], [-2, 2], [-1, -1], [0, -3], [-1, 0], [0,
4], [1, 1], [0, 1], [1, -2], [2, -5]}
{[-1, 3], [0, 0], [-1, 2], [-1, -2], [-1, -1], [0, -4], [-1, 7], [0,
4], [1, 1], [1, -3], [-2, 5], [2, -6]}
{[-1, 3], [0, 0], [-1, -1], [0, -3], [1, 2], [0, -4], [-2, 6], [0, 4],
[0, 1], [0, 5], [1, -2], [2, -5]}
{[-1, 3], [-1, 2], [-1, 1], [-1, -2], [-2, 1], [-1, -1], [1, -4], [-1,
0], [-2, 0], [1, -3], [1, -1], [1, -2]}
{[0, 0], [-1, 2], [0, -1], [-1, 1], [-2, 2], [-2, 3], [-1, 0], [1, 1],
[1, 0], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [0, -1], [-2, 2], [0, -3], [0, -2], [-2, 3], [-1, 0],
[1, 1], [1, 0], [1, -1], [0, 1]}
{[0, 0], [0, -1], [-2, 1], [-2, 2], [0, -3], [0, -2], [-2, 3], [-1,
0], [1, 0], [1, -1], [0, 1], [1, -2]}
{[-1, 4], [0, 2], [2, -7], [-1, 1], [0, -2], [-2, 3], [-1, -3], [0,
-6], [1, -5], [-4, 8], [-1, 0], [1, -1]}
{[-1, 3], [-1, -1], [0, -3], [1, 2], [-1, 7], [-2, 6], [2, -8], [-1,
0], [0, 1], [0, 5], [1, -2], [2, -4]}
{[-1, 3], [0, 0], [-1, 2], [-1, -2], [-1, -1], [0, -4], [-1, 6], [0,
4], [1, 1], [1, -3], [-2, 5], [2, -5]}
{[0, 0], [-1, 2], [0, -1], [0, -4], [-1, 6], [-3, 7], [1, -7], [3,
-8], [0, 3], [1, 1], [1, -3], [2, -2]}
{[-1, 4], [0, 2], [0, -1], [-1, 1], [0, -4], [-2, 6], [1, 0], [1, -3],
[2, -2], [1, 3], [0, 5], [2, -5]}
Message: 9178 - Contents - Hide Contents Date: Fri, 16 Jan 2004 00:41:56 Subject: Re: More on the naming convention From: Carl Lumma>Not to be confused with "miracle-21", which is a 21-note miracle tuning. >You could have both a prefix and suffix number, such as "7-meantone-19" >for a 19-note scale of 7-limit meantone.The convention is square brackets for cards, "miracle[21]", etc. -Carl
Message: 9179 - Contents - Hide Contents Date: Fri, 16 Jan 2004 23:19:03 Subject: Re: Question for Dave Keenan From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> What does "yes" mean here? >> the sound holds together as a single pitch.My guess is that it will be experienced as a single pitch, but one that cannot be accurately determined. The pitch will be fuzzy or vague in a similar way to that of a harmonic note of very short duration.>>> If I take any inharmonic timbre with one loud partial and some > quiet,>>> unimportant ones (very many fall into this category), and use a >>> tuning system where >>> >>> 2:1 off by < 10.4 cents >>> 3:1 off by < 16.5 cents >>> 4:1 off by < 20.8 cents >>> 5:1 off by < 24.1 cents >>> 6:1 off by < 26.9 cents >>> >>> and play a piece with full triadic harmony, doesn't it follow > that>>> the harmony should 'hold together' the way 5-limit triads should? >>>> I don't know. What has the single loud partial got to do with it? Is >> this partial one of those mentioned above? >> No, it essentially determines the pitch of the timbre.So the waveform is essentially sinusoidal? Why not use sinusoidal waves for this thought experiment?>> We know that with quiet sine waves nothing special happens with any >> dyad except a unison, and that loud sine waves work like harmonic >> timbres presumably due to harmonics >> and combinational tones . . . Good point.>> being generated in the >> nonlinearities of the ear-brain system. >> quiet harmonic timbres don't generate combinational tones, so they > won't "work like" loud sine waves. > >> Don't we? >> That also ignores virtual pitch. A set of quiet sine waves can evoke > a single pitch which does not agree with any combinational tone . . . > at certain intervals, the pitch evoked will be least ambiguous, which > is certainly 'something special happening' . . .How many sine waves in an approximate harmonic series do you need for this to be experienced? And what arrangements work? I was only speaking of dyads.> The fact is that, when using inharmonic timbres of the sort I > described, Western music seems to retain all it meaning: certain > (dissonant) chords resolving to other (consonant) chords, etc., all > sounds quite logical. My sense (and the opinion expressed in > Parncutt's book, for example) is that *harmony* is in fact very > closely related to the virtual pitch phenomenon. We already know, > from our listening tests on the harmonic entropy list, that the > sensory dissonance of a chord isn't a function of the sensory > dissonances of its constituent dyads. Furthermore, you seem to be > defining "something special" in a local sense as a function of > interval size, but in real music you don't get to evaluate each > sonority by detuning various intervals various amounts, which > this "specialness" would seem to require for its detection. > > The question I'm asking is, with what other tonal systems, besides > the Western one, is this going to be possible in.If by "Western tonal systems", you mean any based on approximating small whole number ratios of frequency, and by "something special" you mean "consonance and dissonance between simultaneous tones when using only sine waves", then I suppose the answer is "none". What's your point?
Message: 9180 - Contents - Hide Contents Date: Fri, 16 Jan 2004 23:24:13 Subject: Re: Anyone care to name a temperament? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> There have been complaints I name far too many temperaments around > here, so I hope someone will help me out.But I also complained that far too many temperaments get named far too early (by anyone). And then later when more is learned about them other names seem more appropriate and if we change we cut ourselves off from existing material on it in the archives. Whereas if we just stuck to _describing_ it until a name was really needed, we wouldn't have this problem.
Message: 9181 - Contents - Hide Contents
Date: Fri, 16 Jan 2004 08:58:08
Subject: 11-limit Starling
From: Gene Ward Smith
This apparently should be the temperament with TM basis {126/125,
243/242}, which has the same tuning in the 7-limit as 126/125-planar.
Another closeby tuning is top for {126/125, 176/175}-planar, at a
distance of 0.114 cents. I proposed thrush some while back for this,
and so far no objections have been lodged. If we take all three
commas, we get the nonkleismic temperament, which like 7-limit
starling I've actually composed in.
Message: 9182 - Contents - Hide Contents Date: Fri, 16 Jan 2004 23:24:57 Subject: Re: 46 augmented scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I'm getting more scales out of this than I expected; below are 46 > distinct augmented scales obtained by reducing the 53 Fokker blocks to > augmented.There are so many of these I decided to do a little sorting out. By "diameter" I mean a sort of complexity measure, where the difference between the largest and smallest values for each of the generators is found and the maximum taken. Here are the scales with diameter less than eight. Diameter 3 {[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -1], [-1, 0], [1, 1],[1, 0], [0, 1], [1, -1], [-2, 2]} {[0, -2], [0, 0], [0, -1], [-1, 1], [-1, -1], [-1, 0], [1, 1], [1, 0], [0, 1], [1, -2], [1, -1], [2, -2]} {[0, -2], [-2,0], [0, 0], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [0, 1], [1, -2], [1, -1], [-2, 1]} {[1, 2], [0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -1],[-1, 0], [1, 1], [1, 0], [0, 1], [1, -1]} Diameter 5 {[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [0, 3], [1, 1], [1, 0], [0, 1], [1, -2]} {[1, 2], [0, 0], [-1, 2], [0, -1], [-1, -1], [-1, 0], [1, 1], [1, 0], [0, 1], [1, -2], [0, -3], [1, -1]} {[0, -2], [-2,0], [0, 0], [0, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [1, -3], [1, -1], [-2, 1]} {[0, -2], [0, 0], [0, 2], [-1, 1], [-1, -1], [-1, 0], [1, 1], [1, 0], [0, 1], [1, -1], [-1, 3], [-2, 2]} {[0, -2], [0, 0], [0, 2], [0, -1],[-1, 1], [-1, -2], [-1, -1], [-1, 0], [0, 3], [1, 1], [1, 0], [0, 1]} {[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -1], [-1, 0], [1, 0], [1, -3], [1, -2], [1,-1], [-2, 1], [-2, 2]} {[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, 0], [1, 1], [1, 0], [0, 1], [1, -2], [1, -1], [-2, 2], [-2, 3]} Diameter 6 {[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [0, 3], [1, 1], [1, 0], [1, -3]} {[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [0, 3], [1, 1], [1, 0], [1, -3], [1, -2]} {[0, -2], [1, 2], [0, 0], [0, -1], [-1, 1], [-1, 0], [0, 3], [1, 0], [0, 1], [1, -2], [2,-3], [2, -1]} {[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 0], [1, 1], [1, 0], [0,1], [0, -3], [1, -1], [2, -2], [-2, 3]} {[0, -2], [0, 0], [0, 2], [0, -1], [-1, 0], [1, 1], [1, 0], [0, 1], [0, -3], [1, -1], [-2, 2], [-2, 3]} {[0, -2], [0, 0], [0, -1], [-1, 0], [1, 0], [0, 1], [1, -2], [0, -3], [1, -1], [-2, 1],[-2, 2], [-2, 3]}} Diameter 7 {[0, -2], [0, 0], [-1, -1], [-1, 0], [0, 1], [1, -2], [0, -3], [1, -1],[-2, 1], [-1, 3], [-2, 2], [-1, -4]} {[0, 0], [0, 2], [-1, 2], [0, -1], [-1,0], [1, 1], [1, -3], [0, -3], [1, -1], [2, -2], [2, -4], [-2, 3]} {[-2, 0], [-1, 2], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [1, -3], [1, -2], [1, -1], [-2,1], [-1, 3], [1, -4]}
Message: 9183 - Contents - Hide Contents Date: Fri, 16 Jan 2004 10:38:38 Subject: Re: summary -- are these right? From: Carl Lumma>> >y "unweighted" I probably mean a norm without coefficents for >> an interval's coordinates. // >> The norm on Tenney space... >> >> || |u2 u3 u5 ... up> || = >> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up| >> >> The 'coefficients on the intervals coordinates' here are >> log2(2), log2(3) etc. >> So 'unweighted', 9 has a length of 2 but 11 has a length of > 1 . . . :(Unless you use odd-limit.>>>> This ruins the correspondence with taxicab distance on >>>> the odd-limit lattice given by Paul's/Tenney's norm, >>>>>> Huh? Which odd-limit lattice and which norm? >>>> It's the same norm on a triangular lattice with a dimension >> for each odd number. >> That's not a desirable norm. >>> The taxicab distance on this lattice is log(odd-limit). >> No it isn't -- try 9:5 for example.This is what you were claiming in 1999. "" But the basic insight is that a triangular lattice, with Tenney-like lengths, a city-block metric, and odd axes or wormholes, agrees with the odd limit perfectly, and so is the best octave-invariant lattice representation (with associated metric) for anyone as Partchian as me. "">> It's also the same distance as on the Tenney lattice, >> except perhaps for the action of 2s in the latter (I >> forget the reasoning there). >> Try building up the reasoning from scratch.Here's what I was trying to remember... """ The reason omitting the 2-axis forces one to make the lattice triangular is that typically many more powers of two will be needed to bring a product of prime factors into close position than to bring a ratio of prime factors into close position. So the latter should be represented by a shorter distance than the former. Simply ignoring distances along the 2-axis and sticking with a rectangular (or Monzo) lattice is throwing away information. // ... a weight of log(axis) should be applied to all axes, and if a 2-axis is included, a rectangular lattice is OK. If a 2-axis is not included, a triangular lattice is better. // ... in an octave-specific rectangular (or parallelogram) lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In an octave-specific sense, 7:1 and 5:1 really are simpler than 7:5; the former are more consonant. 7:4 and 5:4 are each three rungs in the rectangular lattice, but they still come out a little simpler than 7:5 since the rungs along the 2-axis are so short. If you can buy that 35:1 is as simple as 7:5, then the octave- specific lattice really should be rectangular, not triangular. 35:1 is really difficult to compare with 7:5 -- it's much less rough but also much harder to tune . . . """>> I was thinking stuff like ||9|| = ||3|| = 1 >> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems >> bad though, since the 3s are pointed in the same direction. >> What lattice/metric was this about?Unweighted odd-limit taxicab. By "3+3" I meant adding two 3 vectors. The equation is 1 < 2. It violates... "" The city block distance has the very important property that if an interval arises most simply as the sum of two simpler intervals, the metric of the first interval is the sum of the metrics of the other two. "" ...where clearly you were impling log() weighting. More golden oldies at: [Paul Hahn] * [with cont.] (Wayb.) -Carl
Message: 9184 - Contents - Hide Contents Date: Fri, 16 Jan 2004 23:27:34 Subject: Re: Anyone care to name a temperament? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> There have been complaints I name far too many temperaments around >> here, so I hope someone will help me out. >> But I also complained that far too many temperaments get named far too > early (by anyone). And then later when more is learned about them > other names seem more appropriate and if we change we cut ourselves > off from existing material on it in the archives. Whereas if we just > stuck to _describing_ it until a name was really needed, we wouldn't > have this problem.Given this position, you can hardly complain about what names other people chose. You are either on the bus or off the bus.
Message: 9185 - Contents - Hide Contents Date: Fri, 16 Jan 2004 15:29:36 Subject: Re: 46 augmented scales From: Carl Lumma>> >'m getting more scales out of this than I expected; below are 46 >> distinct augmented scales obtained by reducing the 53 Fokker blocks to >> augmented. >>There are so many of these I decided to do a little sorting out. By >"diameter" I mean a sort of complexity measure, where the difference >between the largest and smallest values In cents? >for each of the generators is >found and the maximum taken. Here are the scales with diameter less >than eight.Can you think of another term? Paul Hahn has used it in a graph- theory sense... Music (and Music Theory) * [with cont.] (Wayb.) -Carl
Message: 9188 - Contents - Hide Contents Date: Fri, 16 Jan 2004 23:52:38 Subject: Re: TOP history From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Paul, could you tell us something about when and how you discovered > TOP? I'd like to add some history to my top page.It's exactly what I've been pleading to you guys to help me figure out last year and probably even earlier, except without octave- equivalence. The idea was to temper out commas uniformly over their length in the lattice, to see what error function this was optimal with respect to, and to then apply this same error function to optimize temperaments with more than one comma. The posts asking about this can be found in the archives here. When I made that 'waterfall' plot, I thought about replacing the former 'octave-equivalent' quantities, that Gene called 'heuristics', with their octave-specific versions in the Tenney lattice. Sometime shortly after that, and after I posted an example of the kind of tempering that makes the new 'heuristics' exact, I was at home and grasped that there was no 'limit' to the set of intervals satisfying the particular error function, minimax tenney-weighted error, that was being optimized here.
Message: 9189 - Contents - Hide Contents Date: Fri, 16 Jan 2004 23:54:07 Subject: Re: The Atomischisma Scale From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> This is the unique Fokker block you get by crossing a schisma with an > atom. I don't know if it is what Kirnberger got,If it's a chain of 11 schisma-flattened fifths, it is. Monzos would help.
Message: 9190 - Contents - Hide Contents Date: Fri, 16 Jan 2004 23:54:59 Subject: Re: TOP on the web From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > <paul.hjelmstad@u...> wrote: >>> What does BP stand for? Thanks >> I'm using it to mean music which excludes 2s, but originally it > referred to dividing the 12th into 13 parts.Better stick to the original definition, or anything that pertains to this labyrinthine set of webpages: The Bohlen-Pierce Site * [with cont.] (Wayb.)
Message: 9191 - Contents - Hide Contents Date: Fri, 16 Jan 2004 21:35:18 Subject: Re: 11-limit Starling From: Herman Miller On Fri, 16 Jan 2004 08:58:08 -0000, "Gene Ward Smith" <gwsmith@xxxxx.xxx> wrote:>This apparently should be the temperament with TM basis {126/125, >243/242}, which has the same tuning in the 7-limit as 126/125-planar. >Another closeby tuning is top for {126/125, 176/175}-planar, at a >distance of 0.114 cents. I proposed thrush some while back for this, >and so far no objections have been lodged. If we take all three >commas, we get the nonkleismic temperament, which like 7-limit >starling I've actually composed in.243;242 is |-1 5 0 0 -2> : it's compatible with 31, 58, 62, and 89-ET. 176;175 is |4 0 -2 -1 1> : this is one of the commas of my 11-limit extension of porcupine. Compatible starling ET's include 12, 15, 16, 31, 43, 46, 58, 73, 89. As it turns out, DNA experiments have revealed that starlings are more closely related to thrushes and mockingbirds than previously suspected. If we accept this classification, it's a good argument for keeping the name "thrush" for a temperament that has a close relationship with "starling". -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin
Message: 9192 - Contents - Hide Contents Date: Fri, 16 Jan 2004 17:56:56 Subject: Re: Question for Dave Keenan From: Carl Lumma>> >hat also ignores virtual pitch. A set of quiet sine waves can evoke >> a single pitch which does not agree with any combinational tone . . . >> at certain intervals, the pitch evoked will be least ambiguous, which >> is certainly 'something special happening' . . . >>How many sine waves in an approximate harmonic series do you need for >this to be experienced? And what arrangements work?Only 2 in many cases. -Carl
Message: 9193 - Contents - Hide Contents Date: Fri, 16 Jan 2004 17:54:23 Subject: Re: summary -- are these right? From: Carl Lumma>>>> >y "unweighted" I probably mean a norm without coefficents for >>>> an interval's coordinates. >> //>>>> The norm on Tenney space... >>>> >>>> || |u2 u3 u5 ... up> || = >>>> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up| >>>> >>>> The 'coefficients on the intervals coordinates' here are >>>> log2(2), log2(3) etc. >>>>>> So 'unweighted', 9 has a length of 2 but 11 has a length of >>> 1 . . . :( >>>> Unless you use odd-limit. >>Please elaborate on how that's 'unweighted' in your view.On a unit-length odd-limit lattice both 9 and 11 have length 1. I'm not claiming anything necessarily good about this, note. I am asking for comments about it however.>>>>>> This ruins the correspondence with taxicab distance on >>>>>> the odd-limit lattice given by Paul's/Tenney's norm, //>>>> It's the same norm on a triangular lattice with a dimension >>>> for each odd number. >>>>>> That's not a desirable norm. Why not?>>>> The taxicab distance on this lattice is log(odd-limit). >>>>>> No it isn't -- try 9:5 for example. >>>> This is what you were claiming in 1999. >> >> "" >> But the basic insight is that a triangular lattice, with >> Tenney-like lengths, a city-block metric, and odd axes or >> wormholes, agrees with the odd limit perfectly, and so is >> the best octave-invariant lattice representation (with >> associated metric) for anyone as Partchian as me. >> "" >>Right -- you need those odd axes, which screws up uniqueness, >and thus most of how we've been approaching temperament.But does the metric agree with log(odd-limit) or not? For 9:5, log(oddlimit) is log(9). If you run it through the "norm" you get... 2log(3) + log(5). Not the same, it seems. However if you followed the lumma.org/stuff/latice1999.txt link, apparently Paul Hahn did present a metric that agrees with log(odd-limit).>>>> It's also the same distance as on the Tenney lattice, >>>> except perhaps for the action of 2s in the latter (I >>>> forget the reasoning there). >>>>>> Try building up the reasoning from scratch. >>>> Here's what I was trying to remember... > >citation?You wrote it in 1999. I'm afraid I can't tell you anything more specific than that.>> """ >> The reason omitting the 2-axis forces one to make the lattice >> triangular is that typically many more powers of two will be >> needed to bring a product of prime factors into close position >> than to bring a ratio of prime factors into close position. So >> the latter should be represented by a shorter distance than the >> former. Simply ignoring distances along the 2-axis and sticking >> with a rectangular (or Monzo) lattice is throwing away >> information. >> // >> ... a weight of log(axis) should be applied to all axes, and >> if a 2-axis is included, a rectangular lattice is OK. If a >> 2-axis is not included, a triangular lattice is better. >> // >> ... in an octave-specific rectangular (or parallelogram) >> lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In >> an octave-specific sense, 7:1 and 5:1 really are simpler than >> 7:5; the former are more consonant. 7:4 and 5:4 are each three >> rungs in the rectangular lattice, but they still come out a little >> simpler than 7:5 since the rungs along the 2-axis are so short. >> If you can buy that 35:1 is as simple as 7:5, then the octave- >> specific lattice really should be rectangular, not triangular. >> 35:1 is really difficult to compare with 7:5 -- it's much less >> rough but also much harder to tune . . . >> """To my mind the good thing about the Tenney/rectangular approach is that it gives log(n*d).>>>> I was thinking stuff like ||9|| = ||3|| = 1 >>>> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems >>>> bad though, since the 3s are pointed in the same direction. >>>>>> What lattice/metric was this about? >>>> Unweighted odd-limit taxicab. >>In which 9 has its own axis . . . so the following: >>> By "3+3" I meant adding two 3 >> vectors. The equation is 1 < 2. >>does not apply.Sure it does. As you say, the 9 appears in two places. If the metric comes out the same either way, I don't see how this fact would "screws up uniqueness, and thus most of how we've been approaching temperament." When I said "ok" above, I meant it does not violate the triangle inequality. But it does 'screw up uniqueness'. -Carl
Message: 9195 - Contents - Hide Contents Date: Fri, 16 Jan 2004 02:08:12 Subject: The Atomischisma Scale From: Gene Ward Smith This is the unique Fokker block you get by crossing a schisma with an atom. I don't know if it is what Kirnberger got, because if it is, it won't be in the Scala archive because the numers in the quotients are too large to load (this provides and example of why monzos would sometimes be useful.) I've presented it below as a Scala file, but it isn't really. There is very little difference between this and 12- equal, proving that 12-et is really a form of 5-limit just intonation. ! atomschis.scl Atom Schisma Scale 12 ! 156348578434374084375/147573952589676412928 134217728/119574225 1307544150375/1099511627776 18014398509481984/14297995284350625 10935/8192 1709671705179880612640625/1208925819614629174706176 16384/10935 14297995284350625/9007199254740992 2199023255552/1307544150375 119574225/67108864 295147905179352825856/156348578434374084375 2
Message: 9196 - Contents - Hide Contents Date: Fri, 16 Jan 2004 02:12:38 Subject: Re: TOP on the web From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> What does BP stand for? ThanksI'm using it to mean music which excludes 2s, but originally it referred to dividing the 12th into 13 parts.
Message: 9197 - Contents - Hide Contents Date: Fri, 16 Jan 2004 02:16:57 Subject: Re: TOP history From: Carl Lumma> Paul, could you tell us something about when and how you > discovered TOP? I'd like to add some history to my top page.Well you can see from my recent post that Dave was pretty close for fixed scales. Also of excerpts from the list in 1999 may be of interest, forthcoming in a forthcoming post of mine. -Carl
Message: 9198 - Contents - Hide Contents Date: Fri, 16 Jan 2004 03:10:36 Subject: 12-et commas From: Gene Ward Smith We can find an infinity of ever more precise 5-limit 12 equal commas by taking the ratios of the logs of two of them, such as the diaschisma and the schisma, and using the numerators and denomenators of the convergents as exponents. In this way we can get Fokker blocks as close as we like to 12-equal inside the 5-limit. If I knew what this was good for, I'd be happy. Here are the first commas from the ratio of the logs of the diaschisma and schisma: [161, -84, -12] [-20462, 10676, 1525] [102471, -53464, -7637] [-327875, 171068, 24436] [7315721, -3816960, -545229] [-7643596, 3988028, 569665] [825180493, -430535956, -61499384] [-832824089, 434523984, 62069049] [2490828671, -1299583924, -185637482] [-28231939470, 14729947148, 2104081351]
Message: 9199 - Contents - Hide Contents Date: Fri, 16 Jan 2004 07:48:22 Subject: 28 meantone scales From: Gene Ward Smith My final talley for the Fokker classification project was 53 scales, clearly a propitious number. Tempering that by marvel reduced the count not at all, but tempering my meantone brought it down to 28 scales, as follows: [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6] [-5, -4, -1, 0, 1, 2, 3, 4, 5, 6, 9, 10] [-5, -2, 0, 1, 2, 3, 4, 5, 6, 8, 9, 11] [-5, -3, -2, 0, 1, 2, 3, 4, 5, 6, 8, 11] [-5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 12] [-5, -2, -1, 1, 2, 3, 4, 5, 6, 8, 9, 12] [-5, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12] [-5, -4, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12] [-5, -4, -3, -2, 1, 2, 3, 4, 5, 6, 11, 12] [-5, -4, -3, 1, 2, 3, 4, 5, 6, 10, 11, 12] [-5, -1, 0, 1, 3, 4, 5, 6, 8, 9, 10, 14] [-5, -2, 0, 1, 3, 4, 5, 6, 8, 9, 11, 14] [-5, -4, -3, -2, 3, 4, 5, 6, 11, 12, 13, 14] [-5, -4, 1, 2, 3, 4, 9, 10, 11, 12, 17, 18] [-5, -1, 0, 3, 4, 5, 8, 9, 10, 13, 14, 18] [-5, -1, 0, 1, 4, 5, 6, 9, 10, 14, 15, 20] [-5, 0, 1, 5, 6, 9, 10, 11, 14, 15, 16, 20] [-5, 0, 1, 4, 5, 6, 9, 10, 11, 14, 15, 20] [-5, 0, 1, 5, 6, 10, 11, 14, 15, 16, 20, 21] [-5, -4, 0, 1, 2, 5, 6, 10, 11, 15, 16, 21] [-5, -4, 3, 4, 5, 6, 11, 12, 13, 14, 21, 22] [-5, -4, -3, 3, 4, 5, 6, 11, 12, 13, 14, 22] [-5, 3, 4, 5, 6, 11, 12, 13, 14, 20, 21, 22] [-5, -1, 3, 4, 8, 9, 12, 13, 14, 17, 18, 22] [-5, -1, 0, 3, 4, 5, 8, 9, 13, 14, 18, 22] [-5, -1, 0, 3, 4, 8, 9, 13, 14, 17, 18, 22] [-5, 1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 24] [-5, 0, 1, 5, 6, 10, 11, 15, 16, 20, 21, 26] ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
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