Tuning-Math Archive Section 5: 4575

Message: 4575 - Contents - Hide Contents Date: Sat, 13 Apr 2002 14:45:01 Subject: My Approach Generalized Diatonicity From: Carl Lumma Graham wrote...

>The rules Paul gave in his paper are objective. The rules Rothenberg >gave (propriety, efficiency and stability -- reading the original >papers is not the ideal way of learning about them ;) are also >objective. The rules you give in your paper are objective, and >correctly applied. > >Carl's rules, of which we might soon have a discussion, aren't quite >objective. He says things like "high efficiency" without putting a >number to how high.

My rules aren't mine at all -- they were all created by other people, chiefly Rothenberg and Erlich. If rules are objective in one place, they're objective in another (at least for your definition of objective). Rothenberg sorts all subsets of 12-et by stability. That's about as trivial and non-special a definition of "high" as you can get. Erlich accepts 6 out of 10 tetrads vs. 6 out of 7 triads. His rule is a "majority", but he hasn't shown why > 50% has any special meaning. I do the same.

>But perhaps he's putting forward objective rules, but allowing us to >make subjective judgements about their relative importance, which >sounds right to me. * [with cont.] (Wayb.)

The relative importance of the rules are given by their order in the list, per thing I generalized. I generalized three things, having to do with the encoding of information in: () series of absolute pitches. () series of relative pitches, in particular, the ability to change the absolute pitches but keep the relationships the same. () the simultaneity of two or more series that do one or both of the above. Let's look at each: () series of absolute pitches. This is equivalent to the first item in the Legend, "Pitches of the scale are trackable". To work with a group of things, we use short- term, or working memory. According to George Miller, human working memory can hold about 5 to 9 things. Properties 1 and 2 work together so that the pitch set can fit in working memory. I'll mention now that implicit in all of this is the assumption that we want the scale to function as an autonomous thing. The diatonic scale has interesting subsets, such as the pentatonic scale. But for our purposes, if we were interested in the pent. scale, we would put it through the list separately. 5-9 things says Miller, but what is a thing? In keeping with the parallel approach of the brain, a thing can probably be almost anything. We've said we want them to be pitches, but we allow pitches an octave apart to be "chunked" into a single thing. Paul's idea is that the 3:2 provides a weaker version of this, and because I have found tetrachordal scales to be very singable, I allow it. It sounds to me like there may be some chunking going on in arpeggiated triads... In selecting 12-tone 7-limit scales for my piano, Paul Hahn searched all rotations of the hexany 2-plex. I found the ones with chord coverage to be more singable than ones without. But in general no more than 9, 'cause you'll start dropping stuff, and no fewer than 5, because that would too simple. You want to have to work a little bit. Mind-expanding, dude. Pitch tracking is the simplest form of melody. Rothenberg says any scale can do it. Improper scales require a drone to measure things from, and work best when they are not very efficient. Proper scales can work too. I love pitch-tracking -- Persian chamber music is all about it. It's clear there's more going on in diatonic music than just pitch-tracking, though. () series of relative pitches, in particular, the ability to change the absolute pitches but keep the relationships the same. This is what I call modal transposition. You can play happy birthday in the relative minor, and it's still happy birthday. This is what propriety is all about -- when is it no longer happy birthday? Rothenberg says it becomes not happy birthday rather suddenly with respect to smooth, random changes in the pitches of the scale. Top on my list of Experiments That Should Be Funded are the ones that R. designed to test this. Modal transposition is made up of items 2 and 3 in the Legend. Item 2, "Modes of the scale function independently". Efficiency is how many notes of a scale you need to hear before you know what mode you're in. When it's low, it's bad for mode independence, because the composer has to work harder to get you to accept a New Scale Degree #1, because you still know where the old #1 is. You want to be a little lost in it. Similarly, humans can recognize consonant intervals. If every mode contains a 5:4 as a type of 3rd, then when you hear a 5:4, you know the :4 could be a New Scale Degree #1. Item 3 in the Legend is propriety, which I won't spill more ink about now. Notice that I don't generalize things about tonal music, try to pick which modes are tonal, etc., as Erlich has done. It's all fine and good, but to insist on stuff like characteristic dissonances is to put too much stock in the historical evolution of Western music, in my opinion. () the simultaneity of two or more series that do one or both of the above. This is the big one, which I call the Diatonic property, because it weeds the large number of proper, efficient, 5-10 tone scales down to very few indeed. Pay close attention. Why are parallel fifths forbidden in common-practice harmony? "They're not!" you may say, or certainly not by the time Jazz comes around. But you don't often get parallel fifths in the melody, say a vocal duet in a song, at least not for very long. My answer is: because it sounds like a timbre, which isn't harmony at all. If you harmonize in 6ths, however, you get consonance, but not synthesis. I call it "series-breaking" or "timbre-breaking" harmony. It's the last item in the Legend. Could we sing one voice in the scale, and create a timbre-breaking harmony for it _outside_ the scale? Yes, we could. And I'd love us to. But this probably would interfere with following series of absolute and relative pitches. -Carl

Message: 4576 - Contents - Hide Contents Date: Sat, 13 Apr 2002 22:08:45 Subject: Re: Scales again and Crystals From: genewardsmith --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

>I am currently seeking ways to > realize the scales I have discovered thus far. There are many, many more, > and my study will (hopefully) be an unbiased survey of a good number.

If you like, I think people would be interested in seeing what you've done so far. The feedback you get will almost certainly be good for your paper, if you want to subject yourself to it.

> "These [ratio] lattices and periodicity blocks, do they resemble those > vector descriptions of crystal lattices?"

You bet. Of course as an unreconstructed mathematician, I prefer to use the word "lattice" only when the vectors form a group--that is, when the some of any two vectors is always another vector in the given set.

> At the time we were wondering if that, as a periodicity block approached as > sphere or circle in shape, and intersected it with a lattice, the radius and > centre could locate scale formations in FCC or HCP combinations of > spheres/circles.

I don't know what a FCC or HCP combination is, but I posted something about spheres and 7-limit lattices on the new tuning group, and was planning on following it up today with something on the 5-limit.

> At that point the conversation wandered onto considering if scales could be > constructed on a line between 1/1 and 2/1 as Cantor sets like Cantor Dust.

Hmmm. Have you ever seen Farey circles? If you take the rational numbers between 1 and 2, and for each p/q make a circle of radius 1/2q^2 and center [p/q, 1/2q^2], you get a fractal collection of circles which never intersect, but where the circles for adjacent Farey fractions always touch. Maybe that could inspire something.

Message: 4577 - Contents - Hide Contents Date: Sat, 13 Apr 2002 15:13:36 Subject: ! From: Carl Lumma

>Hmmm. Have you ever seen Farey circles? If you take the rational numbers >between 1 and 2, and for each p/q make a circle of radius 1/2q^2 and center >[p/q, 1/2q^2], you get a fractal collection of circles which never >intersect, but where the circles for adjacent Farey fractions always touch. >Maybe that could inspire something. ! -Carl

Message: 4578 - Contents - Hide Contents Date: Sat, 13 Apr 2002 01:07:14 Subject: Re: Scales, Analysis, Accidentals and Performance From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> ok, so maybe there were other issues some of us had with other > aspects of your paper.

Where is this paper? Is there an on-line version?

Message: 4579 - Contents - Hide Contents Date: Sat, 13 Apr 2002 01:16:58 Subject: Re: Scales, Analysis, Accidentals and Performance From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

>> ok, so maybe there were other issues some of us had with other >> aspects of your paper.

> Where is this paper? Is there an on-line version?

I think I found it: Volume 38, Number 2 (Summer 2000) of Perspectives of New Music.

Message: 4580 - Contents - Hide Contents Date: Sun, 14 Apr 2002 08:51:45 Subject: Re: A common notation for JI and ETs From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>> Here are the corresponding 5 and 7 et mappings of the non-prime

> collection of commas: >>

>> [5, 8, 12, 14, 17, 18, 21, 21, 23, 24, 25, 26, 27] >> [7, 11, 16, 20, 24, 26, 28, 30, 31, 34, 34, 37, 37] >

> Thanks for looking at this, but I don't understand. Why should we care > about their 5 and 7-ET mappings, and what do the numbers above mean? I > don't understand why any of them are greater than 1.

They are mappings to primes, from 2 to 41. The use of them in part is that they tell you how to go about transforming your notation to and from 41-limit JI.

>> These can be modifed to give the 41-limit interval the notation,

> considered as JI notation, would be notating. >>

>> Here are the "standard" mappings, by way of comparison: >> >> [5, 8, 12, 14, 17, 19, 20, 21, 23, 24, 25, 26, 27] >> [7, 11, 16, 20, 24, 26, 29, 30, 32, 34, 35, 36, 38] >

> Again, I don't understand what this means, but I'd like to, including > why the standard is a standard.

It's not actually an accepted standard, certainly not by Paul, but it seems to be the default meaning lurking in some discussion--that the mapping to primes is simply gotten by rounding off to the nearest integer.

> Also, I wonder if there are any other ETs between say 1000 and 1600-ET > that are 41-limit unique, or 37 limit unique, or even 31-limit unique.

I'll run a search for "standard" ones, at any rate, and report the results.

Message: 4582 - Contents - Hide Contents Date: Sun, 14 Apr 2002 10:02:02 Subject: Re: Fairy Cirlces From: genewardsmith I see Math World has an entry on it: Ford Circle -- from MathWorld * [with cont.]

Message: 4584 - Contents - Hide Contents Date: Sun, 14 Apr 2002 18:19 +0 Subject: Re: My Approach Generalized Diatonicity From: graham@xxxxxxxxxx.xx.xx Carl Lumma wrote:

> My rules aren't mine at all -- they were all created by other people, > chiefly Rothenberg and Erlich. If rules are objective in one place, > they're objective in another (at least for your definition of > objective).

Rothenberg stability is an objective measure, but "highly Rothenberg-stable" isn't an objective criterion. That's all I meant.

> * [with cont.] (Wayb.) > () series of absolute pitches. > > This is equivalent to the first item in the Legend, "Pitches of the > scale are trackable". To work with a group of things, we use short- > term, or working memory. According to George Miller, human working > memory can hold about 5 to 9 things. Properties 1 and 2 work together > so that the pitch set can fit in working memory. > > I'll mention now that implicit in all of this is the assumption that > we want the scale to function as an autonomous thing. The diatonic > scale has interesting subsets, such as the pentatonic scale. But for > our purposes, if we were interested in the pent. scale, we would put > it through the list separately.

Does it have to be a fixed set of notes, or are we allowed statistical definitions?

> 5-9 things says Miller, but what is a thing? In keeping with the > parallel approach of the brain, a thing can probably be almost > anything. We've said we want them to be pitches, but we allow pitches > an octave apart to be "chunked" into a single thing. Paul's idea > is that the 3:2 provides a weaker version of this, and because I have > found tetrachordal scales to be very singable, I allow it. It sounds > to me like there may be some chunking going on in arpeggiated triads... > In selecting 12-tone 7-limit scales for my piano, Paul Hahn searched > all rotations of the hexany 2-plex. I found the ones with chord > coverage to be more singable than ones without.

That all looks reasonable. I don't know if it's for pitch trackability or not, but you certainly want chord coverage in a diatonic scale.

> But in general no more than 9, 'cause you'll start dropping stuff, and > no fewer than 5, because that would too simple. You want to have to > work a little bit. Mind-expanding, dude.

Except you're also allowing 10?

> Pitch tracking is the simplest form of melody. Rothenberg says any > scale can do it. Improper scales require a drone to measure things > from, and work best when they are not very efficient. Proper scales > can work too. I love pitch-tracking -- Persian chamber music is all > about it. It's clear there's more going on in diatonic music than > just pitch-tracking, though.

You could also say that the flexibility in tuning of a pitch should be smaller than the difference between pitches. So far, you're assuming the tuning isn't flexible.

> () series of relative pitches, in particular, > the ability to change the absolute pitches > but keep the relationships the same. > > This is what I call modal transposition. You can play happy birthday > in the relative minor, and it's still happy birthday. This is what > propriety is all about -- when is it no longer happy birthday? > Rothenberg says it becomes not happy birthday rather suddenly with > respect to smooth, random changes in the pitches of the scale. Top > on my list of Experiments That Should Be Funded are the ones that R. > designed to test this.

And both stability definitions imply propriety. Is that right?

> Modal transposition is made up of items 2 and 3 in the Legend. Item > 2, "Modes of the scale function independently". Efficiency is how > many notes of a scale you need to hear before you know what mode > you're in. When it's low, it's bad for mode independence, because > the composer has to work harder to get you to accept a New Scale > Degree #1, because you still know where the old #1 is. You want to > be a little lost in it. Similarly, humans can recognize consonant > intervals. If every mode contains a 5:4 as a type of 3rd, then when > you hear a 5:4, you know the :4 could be a New Scale Degree #1. Item > 3 in the Legend is propriety, which I won't spill more ink about now.

Do you have code for calculating efficiency? I never got the hang of it. I'd rather use some measure involving the minimal sufficient sets. A full efficiency calculation should include the probability of a note being used in any given context. The minimal sufficient sets let you work backwards to see what notes should be most common to establish the key. I'd like to see some measure based on inspecting them. The classic diatonic has lots of 3 note MSSs (any 3 notes including a tritone) which means it's easy to establish the key when you want to. But it also has two 6 note insufficient sets. That is, take out either note not involving the tritone and you don't know what key you're in. That means you can also leave the key ambiguous when you want to, which is good for modulation. Any octave-based MOS will have such an n-1 not maximal insufficient subset. The "in a majority of a scale's modes" means it belongs to the majority of intervals of that diatonic class, does it?

> Notice that I don't generalize things about tonal music, try to pick > which modes are tonal, etc., as Erlich has done. It's all fine and > good, but to insist on stuff like characteristic dissonances is to > put too much stock in the historical evolution of Western music, in > my opinion.

So you're really after modality rather than tonality?

> () the simultaneity of two or more series > that do one or both of the above. > > This is the big one, which I call the Diatonic property, because it > weeds the large number of proper, efficient, 5-10 tone scales down > to very few indeed. Pay close attention.

I still don't understand what that means, in the light of the explanation.

> Why are parallel fifths forbidden in common-practice harmony? > "They're not!" you may say, or certainly not by the time Jazz comes > around. But you don't often get parallel fifths in the melody, say > a vocal duet in a song, at least not for very long. My answer is: > because it sounds like a timbre, which isn't harmony at all. > > If you harmonize in 6ths, however, you get consonance, but not > synthesis. I call it "series-breaking" or "timbre-breaking" harmony. > It's the last item in the Legend.

This is one thing Mark Gould's process leads to, when the alternating intervals are consonant. But it's also stricter because it means all the intervals have to be consonant, and they have to alternate. And it doesn't have the criterion about the consonances not being ambiguous. So his 11 from 41 note scale may be worth a look. It can be based on a 7:8:9 chord. It's outside the Miller limit, but I couldn't find anything better in the "diatonics". There may be something in the "pentatonics". The consonances of this class don't have to be the only ones adding up to the criterion (5) consonance, do they? So the generalized fifth doesn't have to be an odd number of diatonic scale steps.

> Could we sing one voice in the scale, and create a timbre-breaking > harmony for it _outside_ the scale? Yes, we could. And I'd love > us to. But this probably would interfere with following series > of absolute and relative pitches.

But how about using contrapuntal harmonisation? This one looks like it isn't so useful in that context. Graham

Message: 4585 - Contents - Hide Contents Date: Sun, 14 Apr 2002 20:11 +0 Subject: Re: Scales, Analysis, Accidentals and Performance From: graham@xxxxxxxxxx.xx.xx Me:

>> Johnny's also been very dismissive of the kind of experiments you >> propose. Although he welcomes a precision of 1 cent, he doesn't >> claim to reproduce arbitrary scores with that level of accuracy. Jon Szanto:

> Excuse me?! JR has not only claimed such, but purported to have a > personal accuracy to tolerances *smaller* than 1 cent. And claimed to > reproduce said accuracy, in pieces of music, on a regular basis.

I found an exchange on the big list, from the 8th of March. This bit seems to be Carl Lumma:

> If we pretend the "holy grail" of auditory scene analysis, the > "unmixer" (which may just have been realized by these guys: > www.appliedneurosystems.com) exists, feed it a recording of say, > a string quartet, apply pitch tracking to each part, and do a > statistical analysis on the vertical relationships, I wager > we'd find that they are centered around JI, not 72 or 31, EVEN > IF THE MUSIC WAS NOTATED IN 72 OR 31, and NO MATTER HOW GOOD THE > PERFORMERS ARE. This is basically the assumption behind my > recent posts. I probably should have said this earlier. :)

And this is Johnny Reinhard: """ I really do not want to be insulting, but this is idiocy. Assumptions are a poor basis for preaching on the internet. Feeding a recording proves nothing. It is a bit shocking that Carl does not understand this. For myself, I refuse to be reduced to Carl's expectations. He is now doing real harm regarding what people feel they should believe about microtonal performance and he has practically no experience as a performing musician. Maybe he tells his doctors how to diagnose his ailments rather than accept professional medical advice. """ And this from 10th June last year, in reply to you """ This has become funny, now. There is no measurement you will be able to do based on released music. Sorry it doesn't work that way. The onus is not on me to prove anything. And with this I stop discussing what was originally a question to me about notation. With notation I use 1 cent deviation and have explained why. """ If that isn't "very dismissive" of experiments based on recordings, then sue me. He has claimed 1 cent accuracy, but not in an objective way, following a score. Graham

Message: 4586 - Contents - Hide Contents Date: Sun, 14 Apr 2002 23:01:49 Subject: Re: My Approach Generalized Diatonicity From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> * [with cont.] (Wayb.)

The rules on this page are fairly useless unless there are definitions to go with them. Maybe you and Monzo could get together and define Rothenberg efficient, and so forth?

Message: 4588 - Contents - Hide Contents Date: Sun, 14 Apr 2002 00:27:00 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#3994]: >

>> I think I lost some schismas for alternate 17 and 19 commas you > found. Can

>> you remind me of those? >

But I thought about it this afternoon and found one for the 19-as- sharp comma, 19456:19683. The schisma is 531392:531441 (2^6*19^2*23:3^12, ~0.160 cents). This occurs when the 19-as-sharp comma is approximated by adding the main 19-comma (512:513) to the 23- comma (729:736). It vanishes in both 217 and 1600-ET, but not in 311- ET. --George

Message: 4589 - Contents - Hide Contents Date: Mon, 15 Apr 2002 12:53 +0 Subject: Re: My Approach Generalized Diatonicity From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <4.2.2.20020414225815.01eae230@xxxxx.xxx> Carl Lumma wrote:

> The entire model assumes a fixed set of notes, but could probably be > successfully extended with statistical definitions. I don't see how > it would buy you anything when searching for scales, though.

One way of interpreting the 7 from 10 MOS, with propriety grid 1 2 1 2 1 2 1 3 3 3 3 3 3 2 4 5 4 5 4 4 4 6 6 6 6 5 6 5 7 8 7 7 7 7 7 9 9 8 9 8 9 8 is that the 3 could be either 5:4 or 6:5. That means it counts as two consonances although it only looks like one. You could say 7:6 is allowed as a consonance, but only in the diatonic pitch class with 3. Or that 6:5 can be consonant, but only if it isn't in the root, like fourths are treated in traditional counterpoint. I don't know how it's going to end up, but other scales might do the same kind of things.

>> You could also say that the flexibility in tuning of a pitch should

> be >smaller than the difference between pitches. > > Sounds like a good idea. Lumma stability should catch some of this > where flexibility is defined as not breaking propriety. What did > you have in mind?

In the above example, allowing 5:4 and 6:5 to inhabit the same chromatic pitch class would break that rule. But it still looks okay for propriety -- in fact, it all the examples of fixed scales I've worked out are strictly proper. So the statistical stability would increase.

> That is efficiency. R. claims to have trick of enumerating the sets > easily. Manuel might know more. I forget if I ever was able to get > him R.'s code, but he obviously can do it in scala.

How do you calculate it in Scala, for an ET subset?

> Yes. Once you can do modality, I suspect tonality or something > equally interesting could follow in any number of ways, and I don't > want to cut anything good out by mistake. But it is possible I > just don't understand tonality.

I certainly don't understand tonality. But there may be ways of getting it to work without all the modal criteria being fulfilled. Ambiguous intervals look important.

> I might even say that I see diatonicity as making very precise the > difference between additive synthesis with fancy envelopes and music. > We change the fundamental to get melody. That's the first break. > > If we want to play two melodies at once, we have to make sure the > virtual fundamentals don't move in parallel with one of them or the > voices will timbre-fuse. Power chords on electric guitars lead to > a melodic style, for example. One might say the diatonic scale > prevents the virtual fundamental mechanism from being used to chunk > the two voices, by scrambling the melody there in the time domain > a bit.

I'm not sure virtual fundamentals should be brought into this. Terhardt relates the traditional root to virtual pitch, so major and minor triads come out the same.

>> And it doesn't have the criterion about the consonances not being >> ambiguous. >

> In the propriety-breaking sense? I don't have Mark's paper, but > I just remembered Gene implying he found it on-line. Anyway, I > deal with ambiguities when determining the stability of the scale, > but can't think what they would hurt by occurring on the interval > class(es) responsible for the diatonic property.

Gene implied he found it, but as he can get at a university library it might have been there. I'm not familiar with Balzano, but it looks like he does the same thing building scales with alternating intervals. What Mark adds is the interpretation of those intervals as ratios. If those consonances can be ambiguous, it does open up new ways of interpreting the 7 from 10 scale, which is what I'm looking at now.

>> So his 11 from 41 note scale may be worth a look. It can be based >> on a 7:8:9 chord. It's outside the Miller limit, but I couldn't find >> anything better in the "diatonics". There may be something in the >> "pentatonics". >

> It's two big to claim it as diatonic, in my book. You're going to > hear melodies as a subset of the 41 whether you like it or not. > If some stable subsets have the diatonic property (unstable ones > wouldn't be the subsets you'd wind up hearing), then they would be > candidates. Else it goes into the category of making a diatonic > harmony with pitches outside the core melodic scale, which damages > something in my opinion, but is still interesting.

When I originally tried it out, after he mentioned it on the big list, I found myself not using notes too far from the tonic. So it might break down tetrachordally -- except it doesn't have tetrachords. The definition of 9:8 is such that there can be no 4:3. So the almost-periodicity would have to be about 9:7. Perhaps each part could stay in one such period. 4 3 4 4 4 3 4 4 4 3 4 4 3 4 4 is the "tetrachord". 4+4 4+3 is the 7:8:9. Another consonance is 4+4+3 as 6:5. I can't find anything else promising by this method. 6:7:8 works with the classic pentatonic. 7:11 is only an even number of steps in 3-equal, below 43-equal. 5:7 gives a 9 from 11 scale, but it's nearly 50 cents from 5:6:7 with the optimal tuning. The ideal defining chord is large and simple enough to be comprehensible, and has internal intervals close enough to be the same diatonic class, but also different enough to have a good approximation in the chromatic.

>> The consonances of this class don't have to be the only ones adding >> up to the criterion (5) consonance, do they? So the generalized fifth >> doesn't have to be an odd number of diatonic scale steps. > > God, no.

In that case less equal chords, like 7:8:11 might work.

>> But how about using contrapuntal harmonisation? This one looks like >> it isn't so useful in that context. >

> Either you're hearing a lot of the same relationship or you're not. > Try writing contrapuntal music in the wholetone scale and get back > to me.

As long as we have a variety of consonances to hand, it shouldn't matter if they're all the same diatonic interval class or not. Graham

Message: 4590 - Contents - Hide Contents Date: Mon, 15 Apr 2002 14:27:28 Subject: Re: My Approach Generalized Diatonicity From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>> >hat is efficiency. R. claims to have trick of enumerating the sets >> easily. Manuel might know more. I forget if I ever was able to get >> him R.'s code, but he obviously can do it in scala.

>How do you calculate it in Scala, for an ET subset?

I haven't understood how Rothenberg calculated it for ET subsets. And I was only interested in the general method, which I've implemented. So for ET subsets the calculation can be done more efficiently, perhaps not in all cases, but I don't do that. The only exception is for Myhill scales, for which there's an easy analytical result. Manuel

Message: 4591 - Contents - Hide Contents Date: Mon, 15 Apr 2002 11:30:02 Subject: Re: My Approach Generalized Diatonicity From: Carl Lumma

>> >he entire model assumes a fixed set of notes, but could probably be >> successfully extended with statistical definitions. I don't see how >> it would buy you anything when searching for scales, though. >

>One way of interpreting the 7 from 10 MOS, with propriety grid > >1 2 1 2 1 2 1 >3 3 3 3 3 3 2 >4 5 4 5 4 4 4 >6 6 6 6 5 6 5 >7 8 7 7 7 7 7 >9 9 8 9 8 9 8 > >is that the 3 could be either 5:4 or 6:5.

The interval between 5:4 and 6:5 is very high in harmonic entropy. I don't think any single interval can reasonably be said to serve as a 5:4 and 6:5. If you claim 11:9 is consonant, I'd consider it.

>That means it counts as two consonances although it only looks >like one.

Something has got to change, to change the context in which the single interval is heard. Triads where the other interval changes would count. Maybe even octave registration changes could count.

>You could say 7:6 is allowed as a consonance,

And I do.

>but only in the diatonic pitch class with 3. ? >Or that 6:5 can be consonant, but only if it isn't in the root, like >fourths are treated in traditional counterpoint. I don't know how >it's going to end up, but other scales might do the same kind of >things. Phooey!

>>> You could also say that the flexibility in tuning of a pitch should >>> be smaller than the difference between pitches. >>

>> Sounds like a good idea. Lumma stability should catch some of this >> where flexibility is defined as not breaking propriety. What did >> you have in mind? >

>In the above example, allowing 5:4 and 6:5 to inhabit the same chromatic >pitch class would break that rule. But it still looks okay for propriety >-- in fact, it all the examples of fixed scales I've worked out are >strictly proper. So the statistical stability would increase.

Oh, you mean to actually re-tune it one way or the other, not just have it be re-interpreted in the Erlich sense. If you can do this without breaking propriety, fine. But then there would be no point in expressing the scale without the adjustments in the first place.

>> That is efficiency. R. claims to have trick of enumerating the sets >> easily. Manuel might know more. I forget if I ever was able to get >> him R.'s code, but he obviously can do it in scala. >

>How do you calculate it in Scala, for an ET subset? show data

>> I might even say that I see diatonicity as making very precise the >> difference between additive synthesis with fancy envelopes and music. >> We change the fundamental to get melody. That's the first break. >> >> If we want to play two melodies at once, we have to make sure the >> virtual fundamentals don't move in parallel with one of them or the >> voices will timbre-fuse. Power chords on electric guitars lead to >> a melodic style, for example. One might say the diatonic scale >> prevents the virtual fundamental mechanism from being used to chunk >> the two voices, by scrambling the melody there in the time domain >> a bit. >

>I'm not sure virtual fundamentals should be brought into this. >Terhardt relates the traditional root to virtual pitch, so major >and minor triads come out the same.

The triads come out the same, because of the strength of the fifth, and/or the 19:16 approximation of the minor third in 12-et. It definitely has to do with the vf mechanism. I'm not sure if the vf changing in the way I described is right.

>>> And it doesn't have the criterion about the consonances not being >>> ambiguous. >>

>> In the propriety-breaking sense? I don't have Mark's paper, but >> I just remembered Gene implying he found it on-line. Anyway, I >> deal with ambiguities when determining the stability of the scale, >> but can't think what they would hurt by occurring on the interval >> class(es) responsible for the diatonic property. >

>Gene implied he found it, but as he can get at a university library it >might have been there. I'm not familiar with Balzano, but it looks like >he does the same thing building scales with alternating intervals. What >Mark adds is the interpretation of those intervals as ratios.

That sounds good -- Balzano's alternating intervals are useless for what I want to do because they're not consonant.

>If those consonances can be ambiguous, it does open up new ways of >interpreting the 7 from 10 scale, which is what I'm looking at now.

You mean Rothenberg ambiguous, or ambiguous in the sense that they may have more than one harmonic series representation?

>> It's two big to claim it as diatonic, in my book. You're going to >> hear melodies as a subset of the 41 whether you like it or not. >> If some stable subsets have the diatonic property (unstable ones >> wouldn't be the subsets you'd wind up hearing), then they would be >> candidates. Else it goes into the category of making a diatonic >> harmony with pitches outside the core melodic scale, which damages >> something in my opinion, but is still interesting. >

>When I originally tried it out, after he mentioned it on the big list, I >found myself not using notes too far from the tonic.

Any subsets in particular?

>>> The consonances of this class don't have to be the only ones adding >>> up to the criterion (5) consonance, do they? So the generalized fifth >>> doesn't have to be an odd number of diatonic scale steps. >> >> God, no. >

>In that case less equal chords, like 7:8:11 might work. You bet!

>>> But how about using contrapuntal harmonisation? This one looks like >>> it isn't so useful in that context. >>

>> Either you're hearing a lot of the same relationship or you're not. >> Try writing contrapuntal music in the wholetone scale and get back >> to me. >

>As long as we have a variety of consonances to hand, it shouldn't matter >if they're all the same diatonic interval class or not.

Wrong! That's what diatonicity is all about -- tying scale objects to harmonic objects. There are a million ways to have lots of different consonances, but very few to have them make sense in terms of scale intervals. -Carl

Message: 4592 - Contents - Hide Contents Date: Mon, 15 Apr 2002 20:33:35 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#3994]: >

>> I think I lost some schismas for alternate 17 and 19 commas you > found. Can

>> you remind me of those? >

> You didn't lose them; I never found any. I was wondering why you > didn't bring up the fact that (513/512) * (2187/2176) != (4131/4096) > when you proposed consolidating my 17-limit-in-183 and 23-limit-in- > 217 approaches. Because of that, in notating a given ET I am > restricted to using only one (or only the other) of the symbols for a > 17-comma if the inequality doesn't vanish in that ET. > > But I thought about it this afternoon and found one for the 19-as- > sharp comma, 19456:19683. The schisma is 531392:531441 > (2^6*19^2*23:3^12, ~0.160 cents). This occurs when the 19-as-sharp > comma is approximated by adding the main 19-comma (512:513) to the 23- > comma (729:736). It vanishes in both 217 and 1600-ET, but not in 311- > ET.

Judging from your message #4008, you found this one also: symb lft-flgs rt-flgs ------------------------ 19' = 23 + 19 but I don't think you mentioned it specifically. The one thing that still bothers me is that there are two useful 17- commas, 2176:2187 and 4096:4131, and neither one is arrived at by a schisma. Would you consider adding a flag for 4096:4131? If the flags for the two 17-commas are used together, we would then have a simple way to notate tones modified by a Pythagorean comma, which might be more useful to theorists than composers, but useful nonetheless. --George

Message: 4593 - Contents - Hide Contents Date: Mon, 15 Apr 2002 21:16:43 Subject: Re: My Approach Generalized Diatonicity From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> In the propriety-breaking sense? I don't have Mark's paper, but > I just remembered Gene implying he found it on-line.

I found on-line how to find it in the library. I suspect the UC library has Perspectives on New Music, so you can find it also.

Message: 4594 - Contents - Hide Contents Date: Mon, 15 Apr 2002 14:34:10 Subject: Re: My Approach Generalized Diatonicity From: Carl Lumma

>> >n the propriety-breaking sense? I don't have Mark's paper, but >> I just remembered Gene implying he found it on-line. >

>I found on-line how to find it in the library. I suspect the UC >library has Perspectives on New Music, so you can find it also.

They do; I need to get over there. -Carl