Mike Battaglia FAQ
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Andrew_Heathwaite and made on 2012-04-19 22:23:34 UTC.
- The original revision id was 322950214.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
For those of you who don't know, [[Mike Battaglia]] is a total badass when it comes to explaining complicated mathy temperament stuff in ways that make good intuitive sense and answer the question of "How could this possibly be useful?". This page is a repository for some of the explanations he's offered in the Xenharmonic Alliance Facebook Group. **Q (Spectra Ce):<span class="commentBody"> My first point of confusion...the idea of a using a generator to create a temperament IE what makes a temperament generator unique and/or given the temperament, how can I find the generators and how many generators there are (IE 1 for EDO, 2 for 2D, 3 for 3D...)? --- I "know", for example, quarter comma meantone has 2 generators...but one is so damn close to the other (IE the octave as the second generator) I wonder what the point is.</span>** **<span class="commentBody">Another...say a temperament is accurate with primes 2,3,5, and 11. Does this mean it has good 5:9:11, 9:10:11...chords and how do you deduce which of the chords are available and from what roots from the "subgroup" information (or what else is needed to deduce this)?</span>** <span class="commentBody">A (Mike Battaglia): "My first point of confusion...the idea of a using a generator to create a temperament IE what makes a temperament generator unique and/or given the temperament, how can I find the generators and how many generators there are (IE 1 for EDO, 2 for 2D, 3 for 3D...)?"</span> <span class="commentBody"> Right, that's exactly right. That's what the dimensionality means, actually. An EDO is 1D, and 1D means by definition that it has 1 generator. If you already know what the dimensionality or the "rank" of a temperament is, that tells you how many generators it has.</span> <span class="commentBody"> Calculating the actual generators is a pain in the ass to do by hand. You shouldn't have to do it. You should just use the tools that we already have to do it for you.</span> <span class="commentBody"> In order for me to tell you the easiest way to find the generators, you have to tell me what you're starting with. Is it a comma, like 250/243? Is it a page on the wiki? What does a "temperament" mean to you?</span> <span class="commentBody"> "I "know", for example, quarter comma meantone has 2 generators...but one is so damn close to the other (IE the octave as the second generator) I wonder what the point is."</span> <span class="commentBody"> What do you mean it's close, exactly? One generator is like 696 cents, and the other is about 1200 cents, so they're decently far apart...</span> <span class="commentBody">"Another...say a temperament is accurate with primes 2,3,5, and 11. Does this mean it has good 5:9:11, 9:10:11...chords"</span> <span class="commentBody"> That's a tricky question, since there are lots of ways to screw up prime error measurements. But, all things considered, it's a decently good rule of thumb that if a temperament tunes the primes accurately, then in general, simple chords that use those primes are tuned decently accurately as well.</span> <span class="commentBody"> First off, I should note that lots of these mathematical questions involve computations that ARE NOT EASY for you to do by hand. That's why there are tools like Graham's temperament finder to do those things for you. It wasn't until last year that I learned the actual math behind most of this stuff, and I still use Graham's finder for most of the stuff I want to do - learning to do it yourself is only necessary if you actually want to learn the math or if you find it interesting or whatever.</span> <span class="commentBody"> Things like "overall tuning damage" measures are in this category. They're a pain in the ass to do by hand, and lots of the naive ways of doing them lead to a few snags. For instance, if you just look at unweighted, pure-octave, average prime error, you might end up in a situation where 5 is tuned flat and 3 is tuned sharp (like porcupine) and so 6/5 is tuned WAY sharp, which makes 10:12:15 much worse than you'd expect.</span> <span class="commentBody"> But if you go to Graham's finder and look at porcupine, it automatically works out one of the right ways to do this calculation and tells you that chords in porcupine are a bit more accurate, on average, than diminished temperament and a bit less accurate on average than augmented temperament.</span> <span class="commentBody"> But it's always true that the best way to figure out what chords sound best is in some tuning is to try some different things and see. For instance, 22-EDO doesn't have perfect 9/8's, nor does it have perfect 7/4's, but its 9/7's are pretty damn close to absolutely perfect. So if harmonic accuracy is what you care about, you might really chords with lots 9/7's in them in 22-EDO. And on the other hand, you might find that your ears don't give a damn about super-fine harmonic accuracy after all, like some people, so then who cares about any of this, right?</span> **<span class="commentBody">Q (Spectra Ce): "What do you mean it's close, exactly? One generator is like 696 cents, and the other is about 1200 cents, so they're decently far apart..." I mean...that they intersect at the period assuming octave equivalence...that you can stack 696 cent intervals and end up extremely close to a multiple at 1200 (if I have it right).</span>** **<span class="commentBody">"That's why there are tools like Graham's temperament finder" Which I still wonder how to apply. IE for simply looking up 81/80 in 7-limit I get [[@http://x31eq.com/cgi-bin/rt.cgi?ets=12_14c&limit=7]], including a list of prime errors. I see what you mean about prime errors IE the 5 is flat and 7 is sharp, so 7/5 is especially sharp and that accuracy of chords using 7/5 probably wouldn't be the best.</span>** **<span class="commentBody">However, say I want to find out where the 5:6:7 chords (or 6:7:10....) in that temperament are IE on which roots/base-notes and how many of them there are. How would I do that using Graham's temperament finder...or is it even possible? --- The thing that gets me is, say, if I just go by primes and think "how many temperaments for a scale that hit the primes of 3, 5, and 7 accurately within 9 tones or less?"....I get more options than I can deal with as a composer and end up getting impatient with the theory and falling back on my ears.</span>** **The other thing is the concept of five Blackwood fifths hitting the period/"root tone" dead on IE that you can chain them and end up back at the "relaxed" root tone for a chord progression. When someone points it out it seems obvious...but actually taking a temperament and finding such relations without, say, just trying to take powers of every single interval in the scale and seeing which ones eventually "intersect the root", seems very confusing. Is there a simplified way to approach the process?** <span class="commentBody">A: "What do you mean it's close, exactly? One generator is like 696 cents, and the other is about 1200 cents, so they're decently far apart..." I mean...that they intersect at the period assuming octave equivalence...that you can stack 696 cent intervals and end up extremely close to a multiple at 1200 (if I have it right)."</span> <span class="commentBody"> Right, but they end up "almost intersecting" at multiple places. They almost intersect at 12 generators, for instance. Then they intersect even more closely at 19 generators. And then they intersect even more closely at 31 generators. And then they intersect closely at 43 generators, and closely at 50 generators, and so on.</span> <span class="commentBody"> At any of these "close intersection points" you could, as you suggest above, just say "screw it, let's just make these things equal" - they're close enough. If you do so, then you'll end up with an equal temperament. But for an unequal temperament like meantone, there will be infinitely many close intersection points, depending on how close you care about.</span> <span class="commentBody">""That's why there are tools like Graham's temperament finder" Which I still wonder how to apply. IE for simply looking up 81/80 in 7-limit I get [[@http://x31eq.com/cgi-bin/rt.cgi?ets=12_14c&limit=7|http://x31eq.com/cgi-bin/]][[@http://x31eq.com/cgi-bin/rt.cgi?ets=12_14c&limit=7|rt.cgi?ets=12_14c&limit=7]] , including a list of prime errors. I see what you mean about prime errors IE the 5 is flat and 7 is sharp, so 7/5 is especially sharp and that accuracy of chords using 7/5 probably wouldn't be the best."</span> <span class="commentBody"> How did you get to this page? When you clicked on 81/80, did you see a list of temperaments and then click "Injera" for some reason?</span> <span class="commentBody"> The key thing here to note is something like the "fundamental theorem of tempering": if you start with any temperament, and you eliminate one comma, the temperament you get with is one dimension LESS than the one you started with. For instance, if you start with 7-limit JI, that's a 4D system, because the four generating intervals are 2/1, 3/1, 5/1, and 7/1 (or 2/1, 3/2, 5/4, 7/4, etc). If you temper out one comma, like 81/80, you get to a 3D, or "rank 3" temperament.</span> <span class="commentBody"> Here's the search for 81/80 in the 7-limit:</span> <span class="commentBody"> [[@http://x31eq.com/cgi-bin/uv.cgi?uvs=81%2F80&limit=7|http://x31eq.com/cgi-bin/]][[@http://x31eq.com/cgi-bin/uv.cgi?uvs=81%2F80&limit=7|uv.cgi?uvs=81%2F80&limit=7]]</span> <span class="commentBody"> Note at the bottom, under rank-3, there's only one. This is the only 7-limit, rank-3 temperament that exists that tempers out 81/80.</span> <span class="commentBody"> See all of the "rank 2 temperaments?" Those can be thought of as CHILDREN of the above rank-3 temperament, each of which tempers out one additional comma. So if you start at rank-3, and you temper out one more comma, you get to rank-2.</span> <span class="commentBody"> Injera, which you linked to above, doesn't just temper out 81/80, but 50/49 as well. That's the extra comma you have to add to get to Injera.</span> **<span class="commentBody">Q (Spectra Ce):</span> ^ Right, and so here we get what sounds like an explanation of why 31EDO often is used for quarter comma meantone and, in fact, also land on 12, 19, and several other good accuracy EDOs for meantone that seem to represent a fair deal of the kind of EDOs many first-time Xenharmonic musicians start with and find very accurate for quarter-comma-meantone-lik<span class="commentBody">e composition (even 12 seems good for surprisingly many, as we know).</span>** **<span class="commentBody">So it isn't a stretch to say "temperament is what happens when you simplify JI and have at least one set of two primes act as one (and the different combinations/sets represent the different "children")?"</span>** <span class="commentBody">A: there's a lot of questions going on at once here. I'm going to condense them into one post:</span> <span class="commentBody"> "However, say I want to find out where the 5:6:7 chords (or 6:7:10....) in that temperament are IE on which roots/base-notes and how many of them there are."</span> <span class="commentBody"> Common misconception: a temperament is an -infinite- set of pitches, so there are an infinite amount of any type of chord in it. I assume you're only concerned about how many chords there are in a certain scale -in- that temperament, right?</span> <span class="commentBody"> --</span> <span class="commentBody"> "The other thing is the concept of five Blackwood fifths hitting the period/"root tone" dead on IE that you can chain them and end up back at the "relaxed" root tone for a chord progression. When someone points it out it seems obvious...but actually taking a temperament and finding such relations without, say, just trying to take powers of every single interval in the scale and seeing which ones eventually "intersect the root", seems very confusing. Is there a simplified way to approach the process?"</span> <span class="commentBody"> Are you saying that, given the comma 256/243, is there an easy way to see what things are equated?</span> <span class="commentBody"> -- </span> <span class="commentBody"> "So it isn't a stretch to say "temperament is what happens when you simplify JI and have at least one set of two primes act as one (and the different combinations/sets represent the different "children")?""</span> <span class="commentBody"> In a very roundabout sense, you could say it means that "two primes act as one." You could also take it to mean that two composite ratios act as one. For instance, in meantone, 9/8 and 10/9 are equal and hence "act as one."</span> **<span class="commentBody">Q (Spectra Ce): "Are you saying that, given the comma 256/243, is there an easy way to see what things are equated?" More like...to see which kind of chord progressions are possible given a comma.</span>** **<span class="commentBody">"I assume you're only concerned about how many chords there are in a certain scale -in- that temperament, right?" Right. And I figure I've made that mistake many times IE if there's a single 7 or 9 tone scale under a temperament, I tend to use that as a general measuring stick for how "good" a temperament is far as my compositional use (including larger scales under the temperament, for example).</span>** <span class="commentBody">A: "Are you saying that, given the comma 256/243, is there an easy way to see what things are equated?" More like...to see which kind of chord progressions are possible given a comma."</span> <span class="commentBody"> Well, any chord progression that would move you around by the comma will end up getting you back to the unison. You may be interested in chord progressions that move around by simple things like 3/2, 5/4, and 6/5.</span> <span class="commentBody"> Monzos can make it easy to see what sorts of basic chord movements go into a comma. For instance, 256/243 is</span> <span class="commentBody"> |8 -5 0></span> <span class="commentBody"> And 3/2 is</span> <span class="commentBody"> |-1 1 0></span> <span class="commentBody"> But let's say we don't care about octave displacement at all, right? Then it doesn't matter if you move by 3/1 or 3/2. So we can just throw the 2-coefficient completely away, and replace it with a * to show that we don't care about it</span> <span class="commentBody"> |* -5 0> = 256/243</span> <span class="commentBody"> |* 1 0> = 3/2</span> <span class="commentBody"> And now it's pretty easy to see that going down by 5 3/2's moves you 256/243 away from where you started, counting octave equivalence. So if 256/243 is being tempered out, then it becomes equal to 1/1, right? Therefore, you can see above that going down by 5 3/2's brings you back to the tonic in Blackwood temperament.</span> <span class="commentBody"> Also, since 256/243 is tempered out, you know that 243/256 is also tempered out. So this comma also vanishes</span> <span class="commentBody"> |* 5 0> = 243/256</span> <span class="commentBody"> This also means that going up by 5 3/2's also gets you back to the tonic in Blackwood.</span> <span class="commentBody"> Of course, some chord progressions sound better than others. Which combination of 3/2's and 5/4's and 6/5's and other stuff will sound the best? I have no idea - there's no formula for that that I know of. Just figure out what you like and use it. Feel free to ask questions about different temperaments too - many of us have accumulated a database of cool things to do in them.</span> <span class="commentBody">"I assume you're only concerned about how many chords there are in a certain scale -in- that temperament, right?" Right. And I figure I've made that mistake many times IE if there's a single 7 or 9 tone scale under a temperament, I tend to use that as a general measuring stick for how "good" a temperament is far as my compositional use (including larger scales under the temperament, for example)."</span> <span class="commentBody"> So what's your question then? For an arbitrary scale, the best way is to just poke around and see. If it's something like a rank-2 temperament and you're using an MOS, you can automate some of it. But it's typically always easiest to just manually poke around and start practicing the scale.</span> <span class="commentBody">(at least for me, anyway.)</span> <span class="commentBody">If you really do only care about MOS and rank-2, then you can find the "Graham complexity" of the chord you care about, and subtract that from the number of generators you're using and add 1. But before dealing with that - is that what you're actually talking about here? The scales you tend to use are often not MOS.</span> **<span class="commentBody">Q:</span> "You may be interested in chord progressions that move around by simple things like 3/2, 5/4, and 6/5."..."And now it's pretty easy to see that going down by 5 3/2's moves you 256/243 away" That's exactly what I was talking about...I get it now, thank you. :-) Igs also brought this up to me ages ago...that having the root of the chord move by estimates of simple ratios is a very important compositional feature of a scale. --- I believe one really good way to explain this would be (if one doesn't already exist) a program to break a comma into a Monzo and then identify all possible fractions that satisfy primes used in the Monzo within a given limit. This way, if I have it right, someone could throw in a comma and get a list of root tones for possible "basic" chord progressions.** **"But before dealing with that - is that what you're actually talking about here? The scales you tend to use are often not MOS." I usually don't. However, for the sake of ease of teaching this to people new to the art...let's assume they are MOS or rank 2. I'm really just trying to tackle the problem of "given x scale...how many relatively resolved chords (IE of similar consonance to a minor chord) are there and where can they be found** **I figure given A) A list of chords good enough to be used as resolutions in a scale (and hopefully also just a full list of chords) B) A list of possible root tones for chord progression many musicians will simply be able to jump in and start composing...even if such compositions seem rather formulaic and "pop-like". It would at least pop their minds open to think "hey, this isn't random alien music after all...I can do this!".** **<span class="commentBody">And given the number of "resolved" chords in the scale and the number of root-tone progressions based on commas possible...composers can hopefully get a quick idea how much flexibility a scale has far as composition to eliminate some of the usual "too many options" issues of xenharmonic scales.</span>** <span class="commentBody">A: "But before dealing with that - is that what you're actually talking about here? The scales you tend to use are often not MOS." I usually don't. However, for the sake of ease of teaching this to people new to the art...let's assume they are MOS or rank 2. I'm really just trying to tackle the problem of "given x scale...how many relatively resolved chords (IE of similar consonance to a minor chord) are there and where can they be found."</span> <span class="commentBody"> Work out how many stacked generators you need to express the chord, which is also known as the "Graham complexity" of the chord. Then subtract that from the number of stacked generators needed to express the whole scale, and then add 1. The remainder is the number of times the chord appears in the scale.</span> <span class="commentBody"> For instance, 4:5:6 in meantone has a Graham complexity of 4, because you need to stack four generators (fifths) to get to it: 1) C-G</span> <span class="commentBody"> 2) G-D</span> <span class="commentBody"> 3) D-A</span> <span class="commentBody"> 4) A-E.</span> <span class="commentBody"> Now, say you want to find out how many 4:5:6's are in meantone[7]. Meantone[7] is actually generated by 6 stacked fifths</span> <span class="commentBody"> 1) C-G</span> <span class="commentBody"> 2) G-D</span> <span class="commentBody"> 3) D-A</span> <span class="commentBody"> 4) A-E</span> <span class="commentBody"> 5) E-B</span> <span class="commentBody"> 6) B-F#</span> <span class="commentBody"> To find out how many 4:5:6's are in meantone[7], just do 6-4+1 = 3.</span> <span class="commentBody"> The above is measuring things in units of "# of stacked generators." You can also use units of "# of notes", not the number of generators. For example, meantone[7] has 7 notes</span> <span class="commentBody"> C-G-D-A-E-B-F#</span> <span class="commentBody"> And you need a 5-note chain to express a major chord</span> <span class="commentBody"> C-G-D-A-E</span> <span class="commentBody"> So you can see above that there are going to be three different instances of this 5-note chain in meantone[7], which will end up being</span> <span class="commentBody"> C-G-D-A-E</span> <span class="commentBody"> ....G-D-A-E-B</span> <span class="commentBody"> ........D-A-E-B-F#</span> <span class="commentBody"> consequently, there are 3 major chords, which are here:</span> <span class="commentBody"> C-G-xx-xx-E</span> <span class="commentBody"> ....G-D-xx-xx-B</span> <span class="commentBody"> ........D-A-xx-xx-F#</span> <span class="commentBody"> If you subtract notes from notes, and add 1, you get the same thing, which is 7-5+1 = 3 major chords in meantone[7].</span> <span class="commentBody"> The above is just so you can conceptually see a bit of how it works (and if you're confused, just ask me to clarify more). Assuming you get the gist, you can also just use a shortcut and subtract the Graham complexity (in stacked generators) from the MOS size (in notes). Just don't confuse the two units or you'll screw yourself over.</span> <span class="commentBody">"I figure given A) A list of chords good enough to be used as resolutions in a scale (and hopefully also just a full list of chords)"</span> <span class="commentBody"> To answer question A: I'm not sure I've heard any adequate explanation of how something like a "resolution" works. It's pretty clear to me that, as you say here, we need to figure it out. Dustin's been throwing around a lot of interesting ideas about this, especially in BP, which seems like it's a step in the right direction as far as pinning down a theory of melodic resolution is concerned.</span> <span class="commentBody"> However, there is one useful thing I and others have noticed in terms of chords that do, for whatever reason, tend to sound "resolved." I note that an important class of chords which tend to sound resolved in some meaningful sense are ROOTED chords, which are chords of the form a:b:c:d:... that have the property that a is a power of 2.</span> <span class="commentBody"> OK, wtf does that mean? It's actually simple - here are some chords with that property</span> <span class="commentBody"> 4:5:6</span> <span class="commentBody"> 8:9:10:12</span> <span class="commentBody"> 4:7:9:11</span> <span class="commentBody"> 2:3:5:7:9:11:13</span> <span class="commentBody"> OK, you get the idea. What's the point? Psychoacoustic theories that explain what's going on here are a dime a dozen, but all I can say is to compare these chords and decide for yourself</span> <span class="commentBody"> 7:9:11</span> <span class="commentBody"> 3.5:7:9:11</span> <span class="commentBody"> 1.75:3.5:7:9:11</span> <span class="commentBody"> 4:7:9:11</span> <span class="commentBody"> and see which one seems most consonant and stable to you. To my ears, it's the latter, with the former sounding more dissonant and "augmented" or whatever.</span> <span class="commentBody"> One possible explanation is that chords like these are set up so that you can double the lowest note in the chord down some number of octaves in the bass, and your bass note will end up aligning with the virtual fundamental of the chord. Another is that chords like these allow you to take advantage of some kind of "chroma"-matching VF integration strategy like that one paper that [[https://www.facebook.com/keenanpepper|Keenan Pepper]] found. A third is that magic elves do it. Which one you believe is up to you.</span> <span class="commentBody"> Also, another important question is: what about minor chords like 6:7:9 and 10:12:15, which aren't rooted but still sound resolved? And what about 8:11:13, which is rooted and can sound unresolved? And what about 1/1-9/7-5/3, tempered by 245/243, which sounds mysteriously consonant for no reason?</span> <span class="commentBody"> Yep, you're right. I have no idea. This rooted thing is just a useful rule of thumb. All I know is that a chord like 7:9:11:13 can suddenly become way more resolved if you turn it into 4:7:9:11:13. And 14:18:21, which is supposedly the most awful dissonant thing ever, suddenly becomes a benign and harmless Earth, Wind and Fire type chord if you turn it into 8:14:18;21. So use it for what you can - otherwise take it with a grain of salt and use it for what it's worth.</span> <span class="commentBody"> Now, is this possible with powers of 3 if you hear tritave equivalence? I dunno, why not? You BP people should be telling me!</span> <span class="commentBody"> --</span> <span class="commentBody"> "B) A list of possible root tones for chord progression many musicians will simply be able to jump in and start composing...even if such compositions seem rather formulaic and "pop-like". It would at least pop their minds open to think "hey, this isn't random alien music after all...I can do this!"."</span> <span class="commentBody"> When you say "possible root tones," do you mean a list of trippy characteristic chord progressions for any temperament that sound immediately awesome? (e.g. comma pumps?) I think [[https://www.facebook.com/Harmon3D|Petr Pařízek]] is the guy you want, but yeah, we need to make an auto-comma pump generator ASAP. I totally agree with everything you just said, but it's a matter of time and resources.</span> <span class="commentBody"> So until then, feel free to just ask on XH if anyone knows of cool stuff to do with __________ temperament and I'm sure you'll get an answer. (And, TBH, any auto-comma pump generator is going to sound like crap compared to an actual composer who knows how to bring out the emotion and nuance involved by thinking outside the box a bit.)</span> <span class="commentBody"> --</span> <span class="commentBody"> "And given the number of "resolved" chords in the scale and the number of root-tone progressions based on commas possible...composers can hopefully get a quick idea how much flexibility a scale has far as composition to eliminate some of the usual "too many options" issues of xenharmonic scales."</span> <span class="commentBody"> That's not a terrible way to rate scales, just so long as you realize that it's a pretty crappy way to write music to just stick to the notes in a single MOS series and never modulate ever. If we did that in meantone, we'd never have augmented fifths, and minor/maj7 chords would cease to exist.</span> <span class="commentBody"> Once you allow yourself to modulate more and allow the scale to be less of a prison, you end up with "too many options" again. Except now it's like, woo hoo, so many options!</span> <span class="commentBody">There's no real formula to answer some of these questions. I mean, to a point we can answer them, but the general question of "what awesome things are there to do in a tuning" is only going to be answered by playing them more and writing guides like you said and writing actual music.</span> <span class="commentBody"> What I have is a bunch of random pieces of information in my head about random cool things you can do in different tunings, none of which fits nicely as of 2012 into any sort of magic formula that you can calculate using matrix algebra. So if you have questions about specific interesting things to do in a specific temperament, or "how to use it," why not just ask here? Igliashon Calvin Jones-Coolidge and Keenan Pepper and Ryan Avella and everyone else should feel free to field those questions too.</span> **<span class="commentBody">Q (Tutim Deft Wafil): Thanks Mike Battaglia for your wish of simplificate the overabundant information about xenharmonic things (: (: (:</span>** **My question is it: What name definitively coin to the scale comforms by a generator of aprox. 428 ~ 436 Cents (or its inverse 764 ~ 772 Cents) in which forms the scale with form LLLsLLLLsLLLLs (being 25-EDD the middle size for this scale 2-2-2-1-2-2-2-2-1-2-2-2-2-<span class="commentBody">1)?</span>** **<span class="commentBody"> Because 'Witnots', or Tetradecimal Triatonic are the names that are temporary in the Xenwiki. You can help for this small tangle of concepts. 'Supertriatonic' maybe, but ¬^'</span>** <span class="commentBody">A: so you mean around on the spectrum between 14-EDO and 11-EDO then?</span> <span class="commentBody"> Doesn't look like it has a name. Flatter of 14-EDO you get squares, sharper of 11-EDO you get sensi. Right between 14-EDO and 11-EDO is a middle ground that seems to have no name yet.</span> <span class="commentBody"> Wanna name it? What are the ratios that you see here? I assume you think the generator is about 9/7, right?</span> **<span class="commentBody">Q (Paul Erlich): I challenge anyone to try to use Mike's recipe above to correctly count the chords in Blackwood[10] (which is an MOS according to him). Or better yet, to improve the recipe so that it allows for correct counting in such cases. The familiar diminished (octatonic) scale would be another example of where many/most would have trouble applying Mike's recipe correctly . . .</span>** A:<span class="commentBody"> OK, for counting chords for fractional-octave periods, it's a bit more complex. The formula "subtract Graham complexity from number of notes in scale" still works, but the whole "Graham complexity" thing gets a bit easier to get tripped up on.</span> <span class="commentBody"> You still need to ask yourself - how many generators does it require to generate the chord in question? So consider augmented temperament, and say we want to look at 4:5:6. Here's 0 generators, leaving just a "shell" of periods</span> <span class="commentBody"> C E G# C E G# C E G# ...</span> <span class="commentBody"> now here's one generator per period, giving us augmented[6]</span> <span class="commentBody"> C Eb E G G# B C Eb E G G# B C Eb E G G# B ...</span> <span class="commentBody"> (excuse the 12-EDO meantone notation, it's just to make things simple)</span> <span class="commentBody"> You can see that one generator gets us a major chord. However, since there's 3 periods per octave, it's really that it requires three additional notes to get us the major chord, so you count it as 3.</span> <span class="commentBody"> Then if you want to know how many major chords are in augmented[9], just subtract 9 - 3 = 6.</span> <span class="commentBody"> You can do everything I said where you subtract generators from generators or notes from notes as well, just instead of adding 1, add the number of periods instead.</span> <span class="commentBody"> I don't have an easy way to draw a diagram of this, so maybe Andrew Heathwaite can come up with something elegant :) But if you understand the full-octave period case well enough, you shouldn't find it too unintuitive to jump tot his.</span> **<span class="commentBody">Q (Spectra Ce): "now here's one generator per period, giving us augmented[6]</span>** **<span class="commentBody"> C Eb E G G# B C Eb E G G# B C Eb E G G# B ..." What's the generator? It seems like it could be anything IE C to Eb or Eb to B or...</span>** <span class="commentBody">A: period - 1/3 octave, mapped as 5/4. Generator = ~300 cents, mapped as 6/5. (Or you could say it's ~300 cents, which is just 400 - 300)</span> **<span class="commentBody">Q (Spectra Ce): For commas...assume we're trying to find the root tones of all possible chords using the comma 81/80. The prime "basis" from factorization of 81 is 3,3,3,3 and the primes for 80 are 2,2,2 and 5. Given this, can we assume circles of 3/2, 9/5, 9/8 (and others?)...could be applied to chord progressions and what would the roots look like?</span>** **<span class="commentBody">Another question...how does Skee-Lo's "I Wish" use a comma other than 81/80 in its chord pattern? Yet another..what are, supposedly, 10 of the most used commas for chord progressions in modern pop music? Maybe we should start there as a building block before messing around with scale in other temperaments which temper out those same commas...</span>** <span class="commentBody">A (Keenan Pepper):The chord root goes down by a major third, up by a minor third, down by a major third, and then up by a perfect fourth. If you assume these intervals are supposed to be 6/5, 5/4, and 4/3, then it goes 1/1 8/5 48/25 192/125 128/125 and therefore returns to the same tonic only if 128/125 is tempered out.</span> <span class="commentBody">A (Mike Battaglia): "For commas...assume we're trying to find the root tones of all possible chords using the comma 81/80. The prime "basis" from factorization of 81 is 3,3,3,3 and the primes for 80 are 2,2,2 and 5. Given this, can we assume circles of 3/2, 9/5, 9/8 (and others?)...could be applied to chord progressions and what would the roots look like?"</span> <span class="commentBody"> Sure, why not? Just "factorize" 81/80 into a series of things that you multiply to get it - anything will do.</span> <span class="commentBody">...........</span>
Original HTML content:
<html><head><title>Mike Battaglia FAQ</title></head><body>For those of you who don't know, <a class="wiki_link" href="/Mike%20Battaglia">Mike Battaglia</a> is a total badass when it comes to explaining complicated mathy temperament stuff in ways that make good intuitive sense and answer the question of "How could this possibly be useful?". This page is a repository for some of the explanations he's offered in the Xenharmonic Alliance Facebook Group.<br /> <br /> <strong>Q (Spectra Ce):<span class="commentBody"> My first point of confusion...the idea of a using a generator to create a temperament IE what makes a temperament generator unique and/or given the temperament, how can I find the generators and how many generators there are (IE 1 for EDO, 2 for 2D, 3 for 3D...)? --- I "know", for example, quarter comma meantone has 2 generators...but one is so damn close to the other (IE the octave as the second generator) I wonder what the point is.</span></strong><br /> <br /> <strong><span class="commentBody">Another...say a temperament is accurate with primes 2,3,5, and 11. Does this mean it has good 5:9:11, 9:10:11...chords and how do you deduce which of the chords are available and from what roots from the "subgroup" information (or what else is needed to deduce this)?</span></strong><br /> <br /> <span class="commentBody">A (Mike Battaglia): "My first point of confusion...the idea of a using a generator to create a temperament IE what makes a temperament generator unique and/or given the temperament, how can I find the generators and how many generators there are (IE 1 for EDO, 2 for 2D, 3 for 3D...)?"</span><br /> <br /> <br /> <br /> <span class="commentBody"> Right, that's exactly right. That's what the dimensionality means, actually. An EDO is 1D, and 1D means by definition that it has 1 generator. If you already know what the dimensionality or the "rank" of a temperament is, that tells you how many generators it has.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Calculating the actual generators is a pain in the ass to do by hand. You shouldn't have to do it. You should just use the tools that we already have to do it for you.</span><br /> <br /> <br /> <br /> <span class="commentBody"> In order for me to tell you the easiest way to find the generators, you have to tell me what you're starting with. Is it a comma, like 250/243? Is it a page on the wiki? What does a "temperament" mean to you?</span><br /> <br /> <br /> <br /> <span class="commentBody"> "I "know", for example, quarter comma meantone has 2 generators...but one is so damn close to the other (IE the octave as the second generator) I wonder what the point is."</span><br /> <br /> <br /> <br /> <span class="commentBody"> What do you mean it's close, exactly? One generator is like 696 cents, and the other is about 1200 cents, so they're decently far apart...</span><br /> <br /> <br /> <span class="commentBody">"Another...say a temperament is accurate with primes 2,3,5, and 11. Does this mean it has good 5:9:11, 9:10:11...chords"</span><br /> <br /> <br /> <br /> <span class="commentBody"> That's a tricky question, since there are lots of ways to screw up prime error measurements. But, all things considered, it's a decently good rule of thumb that if a temperament tunes the primes accurately, then in general, simple chords that use those primes are tuned decently accurately as well.</span><br /> <br /> <br /> <br /> <span class="commentBody"> First off, I should note that lots of these mathematical questions involve computations that ARE NOT EASY for you to do by hand. That's why there are tools like Graham's temperament finder to do those things for you. It wasn't until last year that I learned the actual math behind most of this stuff, and I still use Graham's finder for most of the stuff I want to do - learning to do it yourself is only necessary if you actually want to learn the math or if you find it interesting or whatever.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Things like "overall tuning damage" measures are in this category. They're a pain in the ass to do by hand, and lots of the naive ways of doing them lead to a few snags. For instance, if you just look at unweighted, pure-octave, average prime error, you might end up in a situation where 5 is tuned flat and 3 is tuned sharp (like porcupine) and so 6/5 is tuned WAY sharp, which makes 10:12:15 much worse than you'd expect.</span><br /> <br /> <br /> <br /> <span class="commentBody"> But if you go to Graham's finder and look at porcupine, it automatically works out one of the right ways to do this calculation and tells you that chords in porcupine are a bit more accurate, on average, than diminished temperament and a bit less accurate on average than augmented temperament.</span><br /> <br /> <br /> <br /> <span class="commentBody"> But it's always true that the best way to figure out what chords sound best is in some tuning is to try some different things and see. For instance, 22-EDO doesn't have perfect 9/8's, nor does it have perfect 7/4's, but its 9/7's are pretty damn close to absolutely perfect. So if harmonic accuracy is what you care about, you might really chords with lots 9/7's in them in 22-EDO. And on the other hand, you might find that your ears don't give a damn about super-fine harmonic accuracy after all, like some people, so then who cares about any of this, right?</span><br /> <br /> <strong><span class="commentBody">Q (Spectra Ce): "What do you mean it's close, exactly? One generator is like 696 cents, and the other is about 1200 cents, so they're decently far apart..." I mean...that they intersect at the period assuming octave equivalence...that you can stack 696 cent intervals and end up extremely close to a multiple at 1200 (if I have it right).</span></strong><br /> <br /> <br /> <strong><span class="commentBody">"That's why there are tools like Graham's temperament finder" Which I still wonder how to apply. IE for simply looking up 81/80 in 7-limit I get <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=12_14c&limit=7" rel="nofollow" target="_blank">http://x31eq.com/cgi-bin/rt.cgi?ets=12_14c&limit=7</a>, including a list of prime errors. I see what you mean about prime errors IE the 5 is flat and 7 is sharp, so 7/5 is especially sharp and that accuracy of chords using 7/5 probably wouldn't be the best.</span></strong><br /> <br /> <strong><span class="commentBody">However, say I want to find out where the 5:6:7 chords (or 6:7:10....) in that temperament are IE on which roots/base-notes and how many of them there are. How would I do that using Graham's temperament finder...or is it even possible? --- The thing that gets me is, say, if I just go by primes and think "how many temperaments for a scale that hit the primes of 3, 5, and 7 accurately within 9 tones or less?"....I get more options than I can deal with as a composer and end up getting impatient with the theory and falling back on my ears.</span></strong><br /> <br /> <strong>The other thing is the concept of five Blackwood fifths hitting the period/"root tone" dead on IE that you can chain them and end up back at the "relaxed" root tone for a chord progression. When someone points it out it seems obvious...but actually taking a temperament and finding such relations without, say, just trying to take powers of every single interval in the scale and seeing which ones eventually "intersect the root", seems very confusing. Is there a simplified way to approach the process?</strong><br /> <br /> <span class="commentBody">A: "What do you mean it's close, exactly? One generator is like 696 cents, and the other is about 1200 cents, so they're decently far apart..." I mean...that they intersect at the period assuming octave equivalence...that you can stack 696 cent intervals and end up extremely close to a multiple at 1200 (if I have it right)."</span><br /> <br /> <br /> <br /> <span class="commentBody"> Right, but they end up "almost intersecting" at multiple places. They almost intersect at 12 generators, for instance. Then they intersect even more closely at 19 generators. And then they intersect even more closely at 31 generators. And then they intersect closely at 43 generators, and closely at 50 generators, and so on.</span><br /> <br /> <br /> <br /> <span class="commentBody"> At any of these "close intersection points" you could, as you suggest above, just say "screw it, let's just make these things equal" - they're close enough. If you do so, then you'll end up with an equal temperament. But for an unequal temperament like meantone, there will be infinitely many close intersection points, depending on how close you care about.</span><br /> <br /> <br /> <span class="commentBody">""That's why there are tools like Graham's temperament finder" Which I still wonder how to apply. IE for simply looking up 81/80 in 7-limit I get <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=12_14c&limit=7" rel="nofollow" target="_blank">http://x31eq.com/cgi-bin/</a><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=12_14c&limit=7" rel="nofollow" target="_blank">rt.cgi?ets=12_14c&limit=7</a> , including a list of prime errors. I see what you mean about prime errors IE the 5 is flat and 7 is sharp, so 7/5 is especially sharp and that accuracy of chords using 7/5 probably wouldn't be the best."</span><br /> <br /> <br /> <br /> <span class="commentBody"> How did you get to this page? When you clicked on 81/80, did you see a list of temperaments and then click "Injera" for some reason?</span><br /> <br /> <br /> <br /> <span class="commentBody"> The key thing here to note is something like the "fundamental theorem of tempering": if you start with any temperament, and you eliminate one comma, the temperament you get with is one dimension LESS than the one you started with. For instance, if you start with 7-limit JI, that's a 4D system, because the four generating intervals are 2/1, 3/1, 5/1, and 7/1 (or 2/1, 3/2, 5/4, 7/4, etc). If you temper out one comma, like 81/80, you get to a 3D, or "rank 3" temperament.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Here's the search for 81/80 in the 7-limit:</span><br /> <br /> <br /> <br /> <span class="commentBody"> <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/uv.cgi?uvs=81%2F80&limit=7" rel="nofollow" target="_blank">http://x31eq.com/cgi-bin/</a><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/uv.cgi?uvs=81%2F80&limit=7" rel="nofollow" target="_blank">uv.cgi?uvs=81%2F80&limit=7</a></span><br /> <br /> <br /> <br /> <span class="commentBody"> Note at the bottom, under rank-3, there's only one. This is the only 7-limit, rank-3 temperament that exists that tempers out 81/80.</span><br /> <br /> <br /> <br /> <span class="commentBody"> See all of the "rank 2 temperaments?" Those can be thought of as CHILDREN of the above rank-3 temperament, each of which tempers out one additional comma. So if you start at rank-3, and you temper out one more comma, you get to rank-2.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Injera, which you linked to above, doesn't just temper out 81/80, but 50/49 as well. That's the extra comma you have to add to get to Injera.</span><br /> <br /> <strong><span class="commentBody">Q (Spectra Ce):</span> ^ Right, and so here we get what sounds like an explanation of why 31EDO often is used for quarter comma meantone and, in fact, also land on 12, 19, and several other good accuracy EDOs for meantone that seem to represent a fair deal of the kind of EDOs many first-time Xenharmonic musicians start with and find very accurate for quarter-comma-meantone-lik<span class="commentBody">e composition (even 12 seems good for surprisingly many, as we know).</span></strong><br /> <br /> <strong><span class="commentBody">So it isn't a stretch to say "temperament is what happens when you simplify JI and have at least one set of two primes act as one (and the different combinations/sets represent the different "children")?"</span></strong><br /> <br /> <span class="commentBody">A: there's a lot of questions going on at once here. I'm going to condense them into one post:</span><br /> <br /> <br /> <br /> <span class="commentBody"> "However, say I want to find out where the 5:6:7 chords (or 6:7:10....) in that temperament are IE on which roots/base-notes and how many of them there are."</span><br /> <br /> <br /> <br /> <span class="commentBody"> Common misconception: a temperament is an -infinite- set of pitches, so there are an infinite amount of any type of chord in it. I assume you're only concerned about how many chords there are in a certain scale -in- that temperament, right?</span><br /> <br /> <br /> <br /> <span class="commentBody"> --</span><br /> <br /> <br /> <br /> <span class="commentBody"> "The other thing is the concept of five Blackwood fifths hitting the period/"root tone" dead on IE that you can chain them and end up back at the "relaxed" root tone for a chord progression. When someone points it out it seems obvious...but actually taking a temperament and finding such relations without, say, just trying to take powers of every single interval in the scale and seeing which ones eventually "intersect the root", seems very confusing. Is there a simplified way to approach the process?"</span><br /> <br /> <br /> <br /> <span class="commentBody"> Are you saying that, given the comma 256/243, is there an easy way to see what things are equated?</span><br /> <br /> <br /> <br /> <span class="commentBody"> -- </span><br /> <br /> <br /> <br /> <span class="commentBody"> "So it isn't a stretch to say "temperament is what happens when you simplify JI and have at least one set of two primes act as one (and the different combinations/sets represent the different "children")?""</span><br /> <br /> <br /> <br /> <span class="commentBody"> In a very roundabout sense, you could say it means that "two primes act as one." You could also take it to mean that two composite ratios act as one. For instance, in meantone, 9/8 and 10/9 are equal and hence "act as one."</span><br /> <br /> <strong><span class="commentBody">Q (Spectra Ce): "Are you saying that, given the comma 256/243, is there an easy way to see what things are equated?" More like...to see which kind of chord progressions are possible given a comma.</span></strong><br /> <br /> <strong><span class="commentBody">"I assume you're only concerned about how many chords there are in a certain scale -in- that temperament, right?" Right. And I figure I've made that mistake many times IE if there's a single 7 or 9 tone scale under a temperament, I tend to use that as a general measuring stick for how "good" a temperament is far as my compositional use (including larger scales under the temperament, for example).</span></strong><br /> <br /> <span class="commentBody">A: "Are you saying that, given the comma 256/243, is there an easy way to see what things are equated?" More like...to see which kind of chord progressions are possible given a comma."</span><br /> <br /> <br /> <br /> <span class="commentBody"> Well, any chord progression that would move you around by the comma will end up getting you back to the unison. You may be interested in chord progressions that move around by simple things like 3/2, 5/4, and 6/5.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Monzos can make it easy to see what sorts of basic chord movements go into a comma. For instance, 256/243 is</span><br /> <br /> <br /> <br /> <span class="commentBody"> |8 -5 0></span><br /> <br /> <br /> <br /> <span class="commentBody"> And 3/2 is</span><br /> <br /> <br /> <br /> <span class="commentBody"> |-1 1 0></span><br /> <br /> <br /> <br /> <span class="commentBody"> But let's say we don't care about octave displacement at all, right? Then it doesn't matter if you move by 3/1 or 3/2. So we can just throw the 2-coefficient completely away, and replace it with a * to show that we don't care about it</span><br /> <br /> <br /> <br /> <span class="commentBody"> |* -5 0> = 256/243</span><br /> <br /> <span class="commentBody"> |* 1 0> = 3/2</span><br /> <br /> <br /> <br /> <span class="commentBody"> And now it's pretty easy to see that going down by 5 3/2's moves you 256/243 away from where you started, counting octave equivalence. So if 256/243 is being tempered out, then it becomes equal to 1/1, right? Therefore, you can see above that going down by 5 3/2's brings you back to the tonic in Blackwood temperament.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Also, since 256/243 is tempered out, you know that 243/256 is also tempered out. So this comma also vanishes</span><br /> <br /> <br /> <br /> <span class="commentBody"> |* 5 0> = 243/256</span><br /> <br /> <br /> <br /> <span class="commentBody"> This also means that going up by 5 3/2's also gets you back to the tonic in Blackwood.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Of course, some chord progressions sound better than others. Which combination of 3/2's and 5/4's and 6/5's and other stuff will sound the best? I have no idea - there's no formula for that that I know of. Just figure out what you like and use it. Feel free to ask questions about different temperaments too - many of us have accumulated a database of cool things to do in them.</span><br /> <br /> <br /> <span class="commentBody">"I assume you're only concerned about how many chords there are in a certain scale -in- that temperament, right?" Right. And I figure I've made that mistake many times IE if there's a single 7 or 9 tone scale under a temperament, I tend to use that as a general measuring stick for how "good" a temperament is far as my compositional use (including larger scales under the temperament, for example)."</span><br /> <br /> <br /> <br /> <span class="commentBody"> So what's your question then? For an arbitrary scale, the best way is to just poke around and see. If it's something like a rank-2 temperament and you're using an MOS, you can automate some of it. But it's typically always easiest to just manually poke around and start practicing the scale.</span><br /> <br /> <br /> <span class="commentBody">(at least for me, anyway.)</span><br /> <br /> <span class="commentBody">If you really do only care about MOS and rank-2, then you can find the "Graham complexity" of the chord you care about, and subtract that from the number of generators you're using and add 1. But before dealing with that - is that what you're actually talking about here? The scales you tend to use are often not MOS.</span><br /> <br /> <strong><span class="commentBody">Q:</span> "You may be interested in chord progressions that move around by simple things like 3/2, 5/4, and 6/5."..."And now it's pretty easy to see that going down by 5 3/2's moves you 256/243 away" That's exactly what I was talking about...I get it now, thank you. :-) Igs also brought this up to me ages ago...that having the root of the chord move by estimates of simple ratios is a very important compositional feature of a scale. --- I believe one really good way to explain this would be (if one doesn't already exist) a program to break a comma into a Monzo and then identify all possible fractions that satisfy primes used in the Monzo within a given limit. This way, if I have it right, someone could throw in a comma and get a list of root tones for possible "basic" chord progressions.</strong><br /> <br /> <strong>"But before dealing with that - is that what you're actually talking about here? The scales you tend to use are often not MOS." I usually don't. However, for the sake of ease of teaching this to people new to the art...let's assume they are MOS or rank 2. I'm really just trying to tackle the problem of "given x scale...how many relatively resolved chords (IE of similar consonance to a minor chord) are there and where can they be found</strong><br /> <br /> <strong>I figure given A) A list of chords good enough to be used as resolutions in a scale (and hopefully also just a full list of chords) B) A list of possible root tones for chord progression many musicians will simply be able to jump in and start composing...even if such compositions seem rather formulaic and "pop-like". It would at least pop their minds open to think "hey, this isn't random alien music after all...I can do this!".</strong><br /> <br /> <strong><span class="commentBody">And given the number of "resolved" chords in the scale and the number of root-tone progressions based on commas possible...composers can hopefully get a quick idea how much flexibility a scale has far as composition to eliminate some of the usual "too many options" issues of xenharmonic scales.</span></strong><br /> <br /> <span class="commentBody">A: "But before dealing with that - is that what you're actually talking about here? The scales you tend to use are often not MOS." I usually don't. However, for the sake of ease of teaching this to people new to the art...let's assume they are MOS or rank 2. I'm really just trying to tackle the problem of "given x scale...how many relatively resolved chords (IE of similar consonance to a minor chord) are there and where can they be found."</span><br /> <br /> <br /> <br /> <span class="commentBody"> Work out how many stacked generators you need to express the chord, which is also known as the "Graham complexity" of the chord. Then subtract that from the number of stacked generators needed to express the whole scale, and then add 1. The remainder is the number of times the chord appears in the scale.</span><br /> <br /> <br /> <br /> <span class="commentBody"> For instance, 4:5:6 in meantone has a Graham complexity of 4, because you need to stack four generators (fifths) to get to it: 1) C-G</span><br /> <br /> <span class="commentBody"> 2) G-D</span><br /> <br /> <span class="commentBody"> 3) D-A</span><br /> <br /> <span class="commentBody"> 4) A-E.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Now, say you want to find out how many 4:5:6's are in meantone[7]. Meantone[7] is actually generated by 6 stacked fifths</span><br /> <br /> <br /> <br /> <span class="commentBody"> 1) C-G</span><br /> <br /> <span class="commentBody"> 2) G-D</span><br /> <br /> <span class="commentBody"> 3) D-A</span><br /> <br /> <span class="commentBody"> 4) A-E</span><br /> <br /> <span class="commentBody"> 5) E-B</span><br /> <br /> <span class="commentBody"> 6) B-F#</span><br /> <br /> <br /> <br /> <span class="commentBody"> To find out how many 4:5:6's are in meantone[7], just do 6-4+1 = 3.</span><br /> <br /> <br /> <br /> <span class="commentBody"> The above is measuring things in units of "# of stacked generators." You can also use units of "# of notes", not the number of generators. For example, meantone[7] has 7 notes</span><br /> <br /> <br /> <br /> <span class="commentBody"> C-G-D-A-E-B-F#</span><br /> <br /> <br /> <br /> <span class="commentBody"> And you need a 5-note chain to express a major chord</span><br /> <br /> <br /> <br /> <span class="commentBody"> C-G-D-A-E</span><br /> <br /> <br /> <br /> <span class="commentBody"> So you can see above that there are going to be three different instances of this 5-note chain in meantone[7], which will end up being</span><br /> <br /> <br /> <br /> <span class="commentBody"> C-G-D-A-E</span><br /> <br /> <span class="commentBody"> ....G-D-A-E-B</span><br /> <br /> <span class="commentBody"> ........D-A-E-B-F#</span><br /> <br /> <br /> <br /> <span class="commentBody"> consequently, there are 3 major chords, which are here:</span><br /> <br /> <br /> <br /> <span class="commentBody"> C-G-xx-xx-E</span><br /> <br /> <span class="commentBody"> ....G-D-xx-xx-B</span><br /> <br /> <span class="commentBody"> ........D-A-xx-xx-F#</span><br /> <br /> <br /> <br /> <span class="commentBody"> If you subtract notes from notes, and add 1, you get the same thing, which is 7-5+1 = 3 major chords in meantone[7].</span><br /> <br /> <br /> <br /> <span class="commentBody"> The above is just so you can conceptually see a bit of how it works (and if you're confused, just ask me to clarify more). Assuming you get the gist, you can also just use a shortcut and subtract the Graham complexity (in stacked generators) from the MOS size (in notes). Just don't confuse the two units or you'll screw yourself over.</span><br /> <br /> <br /> <span class="commentBody">"I figure given A) A list of chords good enough to be used as resolutions in a scale (and hopefully also just a full list of chords)"</span><br /> <br /> <br /> <br /> <span class="commentBody"> To answer question A: I'm not sure I've heard any adequate explanation of how something like a "resolution" works. It's pretty clear to me that, as you say here, we need to figure it out. Dustin's been throwing around a lot of interesting ideas about this, especially in BP, which seems like it's a step in the right direction as far as pinning down a theory of melodic resolution is concerned.</span><br /> <br /> <br /> <br /> <span class="commentBody"> However, there is one useful thing I and others have noticed in terms of chords that do, for whatever reason, tend to sound "resolved." I note that an important class of chords which tend to sound resolved in some meaningful sense are ROOTED chords, which are chords of the form a:b:c:d:... that have the property that a is a power of 2.</span><br /> <br /> <br /> <br /> <span class="commentBody"> OK, wtf does that mean? It's actually simple - here are some chords with that property</span><br /> <br /> <br /> <br /> <span class="commentBody"> 4:5:6</span><br /> <br /> <span class="commentBody"> 8:9:10:12</span><br /> <br /> <span class="commentBody"> 4:7:9:11</span><br /> <br /> <span class="commentBody"> 2:3:5:7:9:11:13</span><br /> <br /> <br /> <br /> <span class="commentBody"> OK, you get the idea. What's the point? Psychoacoustic theories that explain what's going on here are a dime a dozen, but all I can say is to compare these chords and decide for yourself</span><br /> <br /> <br /> <br /> <span class="commentBody"> 7:9:11</span><br /> <br /> <span class="commentBody"> 3.5:7:9:11</span><br /> <br /> <span class="commentBody"> 1.75:3.5:7:9:11</span><br /> <br /> <span class="commentBody"> 4:7:9:11</span><br /> <br /> <br /> <br /> <span class="commentBody"> and see which one seems most consonant and stable to you. To my ears, it's the latter, with the former sounding more dissonant and "augmented" or whatever.</span><br /> <br /> <br /> <br /> <span class="commentBody"> One possible explanation is that chords like these are set up so that you can double the lowest note in the chord down some number of octaves in the bass, and your bass note will end up aligning with the virtual fundamental of the chord. Another is that chords like these allow you to take advantage of some kind of "chroma"-matching VF integration strategy like that one paper that <a class="wiki_link_ext" href="https://www.facebook.com/keenanpepper" rel="nofollow">Keenan Pepper</a> found. A third is that magic elves do it. Which one you believe is up to you.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Also, another important question is: what about minor chords like 6:7:9 and 10:12:15, which aren't rooted but still sound resolved? And what about 8:11:13, which is rooted and can sound unresolved? And what about 1/1-9/7-5/3, tempered by 245/243, which sounds mysteriously consonant for no reason?</span><br /> <br /> <br /> <br /> <span class="commentBody"> Yep, you're right. I have no idea. This rooted thing is just a useful rule of thumb. All I know is that a chord like 7:9:11:13 can suddenly become way more resolved if you turn it into 4:7:9:11:13. And 14:18:21, which is supposedly the most awful dissonant thing ever, suddenly becomes a benign and harmless Earth, Wind and Fire type chord if you turn it into 8:14:18;21. So use it for what you can - otherwise take it with a grain of salt and use it for what it's worth.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Now, is this possible with powers of 3 if you hear tritave equivalence? I dunno, why not? You BP people should be telling me!</span><br /> <br /> <br /> <br /> <span class="commentBody"> --</span><br /> <br /> <br /> <br /> <span class="commentBody"> "B) A list of possible root tones for chord progression many musicians will simply be able to jump in and start composing...even if such compositions seem rather formulaic and "pop-like". It would at least pop their minds open to think "hey, this isn't random alien music after all...I can do this!"."</span><br /> <br /> <br /> <br /> <span class="commentBody"> When you say "possible root tones," do you mean a list of trippy characteristic chord progressions for any temperament that sound immediately awesome? (e.g. comma pumps?) I think <a class="wiki_link_ext" href="https://www.facebook.com/Harmon3D" rel="nofollow">Petr Pařízek</a> is the guy you want, but yeah, we need to make an auto-comma pump generator ASAP. I totally agree with everything you just said, but it's a matter of time and resources.</span><br /> <br /> <br /> <br /> <span class="commentBody"> So until then, feel free to just ask on XH if anyone knows of cool stuff to do with __ temperament and I'm sure you'll get an answer. (And, TBH, any auto-comma pump generator is going to sound like crap compared to an actual composer who knows how to bring out the emotion and nuance involved by thinking outside the box a bit.)</span><br /> <br /> <br /> <br /> <span class="commentBody"> --</span><br /> <br /> <br /> <br /> <span class="commentBody"> "And given the number of "resolved" chords in the scale and the number of root-tone progressions based on commas possible...composers can hopefully get a quick idea how much flexibility a scale has far as composition to eliminate some of the usual "too many options" issues of xenharmonic scales."</span><br /> <br /> <br /> <br /> <span class="commentBody"> That's not a terrible way to rate scales, just so long as you realize that it's a pretty crappy way to write music to just stick to the notes in a single MOS series and never modulate ever. If we did that in meantone, we'd never have augmented fifths, and minor/maj7 chords would cease to exist.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Once you allow yourself to modulate more and allow the scale to be less of a prison, you end up with "too many options" again. Except now it's like, woo hoo, so many options!</span><br /> <br /> <br /> <span class="commentBody">There's no real formula to answer some of these questions. I mean, to a point we can answer them, but the general question of "what awesome things are there to do in a tuning" is only going to be answered by playing them more and writing guides like you said and writing actual music.</span><br /> <br /> <br /> <br /> <span class="commentBody"> What I have is a bunch of random pieces of information in my head about random cool things you can do in different tunings, none of which fits nicely as of 2012 into any sort of magic formula that you can calculate using matrix algebra. So if you have questions about specific interesting things to do in a specific temperament, or "how to use it," why not just ask here? Igliashon Calvin Jones-Coolidge and Keenan Pepper and Ryan Avella and everyone else should feel free to field those questions too.</span><br /> <br /> <br /> <strong><span class="commentBody">Q (Tutim Deft Wafil): Thanks Mike Battaglia for your wish of simplificate the overabundant information about xenharmonic things (: (: (:</span></strong><br /> <strong>My question is it: What name definitively coin to the scale comforms by a generator of aprox. 428 ~ 436 Cents (or its inverse 764 ~ 772 Cents) in which forms the scale with form LLLsLLLLsLLLLs (being 25-EDD the middle size for this scale 2-2-2-1-2-2-2-2-1-2-2-2-2-<span class="commentBody">1)?</span></strong><br /> <strong><span class="commentBody"> Because 'Witnots', or Tetradecimal Triatonic are the names that are temporary in the Xenwiki. You can help for this small tangle of concepts. 'Supertriatonic' maybe, but ¬^'</span></strong><br /> <br /> <span class="commentBody">A: so you mean around on the spectrum between 14-EDO and 11-EDO then?</span><br /> <br /> <br /> <br /> <span class="commentBody"> Doesn't look like it has a name. Flatter of 14-EDO you get squares, sharper of 11-EDO you get sensi. Right between 14-EDO and 11-EDO is a middle ground that seems to have no name yet.</span><br /> <br /> <br /> <br /> <span class="commentBody"> Wanna name it? What are the ratios that you see here? I assume you think the generator is about 9/7, right?</span><br /> <br /> <strong><span class="commentBody">Q (Paul Erlich): I challenge anyone to try to use Mike's recipe above to correctly count the chords in Blackwood[10] (which is an MOS according to him). Or better yet, to improve the recipe so that it allows for correct counting in such cases. The familiar diminished (octatonic) scale would be another example of where many/most would have trouble applying Mike's recipe correctly . . .</span></strong><br /> <br /> A:<span class="commentBody"> OK, for counting chords for fractional-octave periods, it's a bit more complex. The formula "subtract Graham complexity from number of notes in scale" still works, but the whole "Graham complexity" thing gets a bit easier to get tripped up on.</span><br /> <br /> <span class="commentBody"> You still need to ask yourself - how many generators does it require to generate the chord in question? So consider augmented temperament, and say we want to look at 4:5:6. Here's 0 generators, leaving just a "shell" of periods</span><br /> <br /> <span class="commentBody"> C E G# C E G# C E G# ...</span><br /> <br /> <span class="commentBody"> now here's one generator per period, giving us augmented[6]</span><br /> <br /> <span class="commentBody"> C Eb E G G# B C Eb E G G# B C Eb E G G# B ...</span><br /> <br /> <span class="commentBody"> (excuse the 12-EDO meantone notation, it's just to make things simple)</span><br /> <br /> <span class="commentBody"> You can see that one generator gets us a major chord. However, since there's 3 periods per octave, it's really that it requires three additional notes to get us the major chord, so you count it as 3.</span><br /> <br /> <span class="commentBody"> Then if you want to know how many major chords are in augmented[9], just subtract 9 - 3 = 6.</span><br /> <br /> <span class="commentBody"> You can do everything I said where you subtract generators from generators or notes from notes as well, just instead of adding 1, add the number of periods instead.</span><br /> <br /> <span class="commentBody"> I don't have an easy way to draw a diagram of this, so maybe Andrew Heathwaite can come up with something elegant :) But if you understand the full-octave period case well enough, you shouldn't find it too unintuitive to jump tot his.</span><br /> <br /> <strong><span class="commentBody">Q (Spectra Ce): "now here's one generator per period, giving us augmented[6]</span></strong><br /> <br /> <strong><span class="commentBody"> C Eb E G G# B C Eb E G G# B C Eb E G G# B ..." What's the generator? It seems like it could be anything IE C to Eb or Eb to B or...</span></strong><br /> <br /> <span class="commentBody">A: period - 1/3 octave, mapped as 5/4. Generator = ~300 cents, mapped as 6/5. (Or you could say it's ~300 cents, which is just 400 - 300)</span><br /> <br /> <strong><span class="commentBody">Q (Spectra Ce): For commas...assume we're trying to find the root tones of all possible chords using the comma 81/80. The prime "basis" from factorization of 81 is 3,3,3,3 and the primes for 80 are 2,2,2 and 5. Given this, can we assume circles of 3/2, 9/5, 9/8 (and others?)...could be applied to chord progressions and what would the roots look like?</span></strong><br /> <br /> <strong><span class="commentBody">Another question...how does Skee-Lo's "I Wish" use a comma other than 81/80 in its chord pattern? Yet another..what are, supposedly, 10 of the most used commas for chord progressions in modern pop music? Maybe we should start there as a building block before messing around with scale in other temperaments which temper out those same commas...</span></strong><br /> <br /> <span class="commentBody">A (Keenan Pepper):The chord root goes down by a major third, up by a minor third, down by a major third, and then up by a perfect fourth. If you assume these intervals are supposed to be 6/5, 5/4, and 4/3, then it goes 1/1 8/5 48/25 192/125 128/125 and therefore returns to the same tonic only if 128/125 is tempered out.</span><br /> <br /> <span class="commentBody">A (Mike Battaglia): "For commas...assume we're trying to find the root tones of all possible chords using the comma 81/80. The prime "basis" from factorization of 81 is 3,3,3,3 and the primes for 80 are 2,2,2 and 5. Given this, can we assume circles of 3/2, 9/5, 9/8 (and others?)...could be applied to chord progressions and what would the roots look like?"</span><br /> <br /> <br /> <br /> <span class="commentBody"> Sure, why not? Just "factorize" 81/80 into a series of things that you multiply to get it - anything will do.</span><br /> <br /> <span class="commentBody">...........</span></body></html>