Regular Mappings and Marvel Temperaments

Graham Breed

This is the annotated version of the slides for the presentation I gave at Beyond the Semitone in Aberdeen, October 2013. The annotations cover things I said, or should have said, or might have said if I’d had more time.

0.1 Motivation

Goals of Regular Temperament

Regular temperaments are systems that approximate just intonation using fewer notes. This assumes there’s something special about just intonation that you want to preserve, and if you don’t feel this, temperament won’t be useful to you. The ”regular” part means each just ratio is tuned in a consistent way, and more than one just ratio will typically approximate to the same tempered interval. The list below is roughly prioritized, and the tuning is the least important thing. There’s a trap in microtonal literature that you focus on the tuning, and people think it’s all about tuning, although the tuning is only a tool in the service of exploring new harmonies, or making just intonation scales more manageable.

• New harmonic systems

• Frugal scales

• Notation

• Generalized keyboards

• Tuning

Beyond Temperament

These ideas come from the study of temperaments, but can still be useful if you don’t use temperament. This is why I talk about a ”regular mapping paradigm” rather than ”regular temperament theory”.

1 I added some citations for the academic audience, and also to avoid giving the impression that everything I was talking about was original to me. Christian Kaernbach, who gave the talk after mine, discussed the idea that musicians might not play the (simplistic) theoretically correct pitches, and that doesn’t mean they’re wrong. That fortuitously agrees with my first point here. • Support flexible pitch instruments.

• Leave the tuning as just intonation. (Fokker 1969)

• Write just intonation as tempered notation. (Secor 1975) The mapping from just to tempered pitches remains.

What Took So Long?

This is the defeatist slide that excuses why everybody isn’t using these temper- aments and related techniques already. It furthers my agenda regarding the regular mapping paradigm in that part of Kuhn’s description of paradigm shifts is about the resistance that a new paradigm encounters. It’s also worth pointing out that these ideas won’t be equally valuable to everybody. • New pitches are disruptive.

• Meantone is good.

• Not all music needs pure harmony.

• Good mappings are easy to recognize but hard to find.

• Computers make searches and tuning easier.

0.2 Theory

The Marvel Mapping

Tempering out 225:224 (7.7 cents) relates the 7th harmonic partial to two fifths and two thirds. These lattice diagrams show how the prime number 7 ties into the 5-limit lattice. With proper spelling, the 7 should be written as an augmented sixth rather than a minor seventh. (You may choose to add a diesis¨ accidental so that the 7 can be a kind of seventh.)

2 Because the 7.7 cents get shared out among four 5-limit intervals, each 7-limit interval can be tuned to within 2 cents, which will sound as good as just intonation in most contexts. I copied the diagrams from my Tripod Notation document. The thirds are shown shorter than the fifths so that subminor thirds giving magic approxima- tions don’t end up as huge triangles. Without showing magic approximations, the triangles should be roughly equilateral, but it doesn’t really matter.

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5

4 6 9

A]

E (F])

C G D

Interval Arithmetic

I like equations, so I had to bring some equations into my music theory talk. Each shows a musical relationship, and in each case the equations fail to be exact by the 225:224 that’s being tempered out. The first is an augmented triad showing two 5:4 major thirds and a 9:7 su- permajor third adding up to an octave. (Intervals add where fractions multiply.) The Wikipedia page on augmented triads mentions the 225:224, so it must be standard theory. The second equation ties the new septimal interval to familiar extended 5-limit intervals including the larger whole tone of 9:8. The third equation is the same but with the 9:8 split into two 3:2 fifths, and covers another octave. This is the equation that describes the lattices on the previous slide. Finally, we can see that two diatonic semitones approximate to being the same. This means that you can take any marvel temperament, pile up two semitones, and call the result an 8:7. In fact, you can do more than see. You can hear audio examples of these equations in ”fractions.pdf” in the accompanying bundle. For each equation, notes of an ascending scale overlap to give each dyad on the left hand side and then the first note comes back to give the right hand dyad.

3 5 5 9 2 × × ≈ 4 4 7 1

5 5 9 7 × × ≈ 4 4 8 4

5 5 3 3 7 × × × ≈ 4 4 2 2 2

16 15 ≈ 15 14

Question

This was an attempt to catch the audience out, but they were too canny and asked what kind of semitone I was asking about. How may semitones are there to an octave? The point about focusing on semitones was partly to subvert the ”beyond” in the conference title, but also to show that we’re expecting generalzed diatonics of around 10 notes. I could have put this at the introduction to miracle temperament, but I decided to have it to introduce the step size list instead.

Answer

I showed the logarithms so that people can see how easy the calculations are. The last one is the geometric mean of the two just intervals. It’s a rough approximation of the optimal marvel semitone.

log(2) 15
= 10.04 ... log 14

log(2) 16
= 10.74 ... log 15

log(2) 2 8
= 10.38 ... log 7

Scale Sizes

These are scale sizes rather than equal temperaments so that I don’t have to worry about ambiguous mappings. (Where there’s ambiguity, this is the

4 marvel option.) It also helps draw attention to the small numbers that represent diatonics rather than tunings. 10 and 9 both come out better than 7, so we might still expect 7 note diatonics, but also 9 and 10 note scales. For a bit more complexity, 12 is very good. There’s a notable absence of pentatonics. This is because the things that make pentatonics good in a 5-limit context put 10 and 15 into this list, but 5 itself doesn’t work so well in the 7-limit. The ranking follows cangwu badness with a dimensionless parameter of 0.001. This gives simpler scales than the default my Python module uses, and means pajara gets near the top of the rank 2 list later on. It also gives 19 as the number one division to tie in with Graham Hair’s work that was featured at the conference. I filtered out temperaments with contorsion (where each JI approximation is an even number, or something like that) because they aren’t very interesting. A lot of the information in the list ties in with that rank 2 list. If you can compare the two, you may like to do so. Note that the top three are all meantones. 19 can than pair with 41 or 22 to give magic. You can find multiple representatives of the other top rank 2 classes in the top row. The bottom row is less interesting. Because I don’t target excessively small errors, near-optimal tunings for planar marvel don’t show up. If you want one, you could try 269. The planar audio examples use 1273 ”marvel cents” to the octave. This week’s Marvel Top 40:

19 31 12 41 22 53 10 72 50 9 34 29 7 60 43 3 84 63 65 21 2 94 91 75 32 51 28 103 48 15 40 69 81 46 113 17 79 26 16 74

0.3 Harmony

Substituted Dominant Cadences

These audio examples (sdc.mp3) are to illustrate the use of tempered augmented triads. They’re written in musical notation, but they aren’t musical examples. They only show some chord progressions you could use to build music out of. What I know about augmented triads is that you can use one to connect any chord to any other chord. So, given that, I don’t know what a typical progression

5 would be to use as an example. I thought about finding one in Gesualdo, to tie in with some other things at the conference, but that would have meant work so instead I went to Wikipedia. That mentioned this ii-V-I progression with the V substituted. The accidentals are mixed 72-tone Sagittal. The sharp hooks are commas and the more rounded hooks are larger deviations. The first example has the augmented triad written as two consecutive 5-limit thirds. That’s the way Wikipedia writes it, but in marvel I find it the worst tuning. The whole chord covers an interval of 14:9 which isn’t very digestible. In the second example, the higher third is a 9:7 approximation. The chord as a whole covers 8:5. This is the best version if you like 5-limit intervals from the root. The final example has the 9:7 as the lower interval. That’s the best version if you like chords that imitate the large/small pattern of a major triad. Whether or not that’s important, it sounds fine because the span is still 8:5. 
♭

♭ 
♭


 ♯

Undiminishing Triads

This example doesn’t require marvel temperament, but I like it so I included it anyway. The first chord is a 5:6:7 diminished triad and it resolves to a 4:5:6 major triad with the middle note held constant. This follows a rule I picked up from Margo Schulter that cadences should involve stepwise contrary motion. Hear it as undiminish.mp3. If you add in the acoustic root of the 5:6:7, you get a harmonic seventh resolving to a major triad by a rising minor third. This is the second part of the example and I illustrated it with the roman numerals. VI I

♯
♯



The point of this quote is to show that the cadence that I arrived at following voice leading rules in a 7-limit context gives a result that is also prominent in Tymyczko’s theory of rock harmony. The logic he followed has something I

6 don’t really understand involving Shenkerian reductions. Whether that’s valid or not, the results match observed chords. The IV-I at the end of a piece is rare, though, and I couldn’t find any examples in his presentation. The fact that my example ended up as an elaboration of a conventional progression could be good or bad. It’s good in that the 7 can be added with good voice leading in a context that listeners are likely to be familiar with. It’s bad for people who were looking for radically new harmony. (Using it as the final cadence is still a new trick, though.)

. . . we should be expecting ascending minor-third, ascending fifth, and either ascending or descending major-second progressions . . .

Dmitri Tymoczko, Doing the Time Warp from Schutz¨ to the Beatles http://dmitri.tymoczko.com/

Dominant Cadences

These examples show a standard ii-V-I progression with a dominant seventh. Each example tunes the added seventh differently. The first case is the natural extended 5-limit one where the 6:5 minor third hangs over from the minor triad before. That leads to a seventh of 9:5. It sounds fine (as all the examples do) but it means the interval of F to G is wider than the 4:3 that you would expect to tie this into a diatonic framework. The second example fixes the seventh to 16:9 (giving 9:8 to the implied octave double of the root). That means that the minor triad is Pythagorean which is unfortunate if you were looking for a 5-limit context. The last example is more radical. The dominant seventh is tuned as a marvel 4:5:6:7. (It’s still written as a seventh chord, although the spelling rules I gave before should make it an augmented sixth. Sagittal can handle this.) That makes the previous chord a subminor 6:7:9 triad. I find this 9-limit chord sounds wrong at first hearing, although in a proper musical context I can get used to it. These examples don’t require marvel temperament, and don’t show anything new. I included them to lead into the tritone substitutions.

7 Extended 5-limit 

 Pythagorish  
 9-limit 
    Tritone Substitutions

The tritone of the major triad with added Pythagorean seventh approximates a seven-limit tritone given marvel temperament. (Either a 7:5 or a 10:7 de- pending on which way up you hold it.) Because the other pitches of its tritone substitution are chromatic, we can tune them to give a 7-limit tetrad. The first example in ”tritone-substitution.mp3” is a full I-IV-ii-V-I progression with the V substituted with such a 7-limit tetrad. Normally, this progression would be a comma pump. (Jim Dalton had talked about comma pumps in the talk before mine.) Because the D gets lost in the tritone substition, the example now works fine with no pitch drift. It also happens that the root and fifth of the ii can be tuned however you like, so I also added Pythagorean and 9-limit triads to match the previous examples. There’s no need for them, however: they don’t solve any problems in the way that the tritone substitution solved the problems of pitch drift and the seventh chord not being 7-limit. (Neither of which had to be problems, of course, if you didn’t want them to be.) This example gives us another way for a seven-limit tetrad to resolve: down- ward by a diatonic semitone. The ”undiminishing triads” example already gave us a rising minor third. You could take these as standard idioms in a theory of 7-limit music. I find them more convincing than simply tuning the dominant seventh to the 7-limit. There’s a bonus audio example here: tritone-ji.mp3 which is for a ”guess the tuning” game that loses its fun when you know the filename. It happens that, because the augmented sixth isn’t tied to any pitches in the neighboring chords, there’s nothing to stop it being tuned to 7-limit just intonation. The logic behind the substitution requires marvel temperament, but that doesn’t mean you have to temper the tuning for it to work.

8 7-limit      ♭ 
  ♭
Pythagorish   ♭ 
 ♭
9-limit    ♭ 
  ♭

0.4 Linear Temperaments

This Week’s Marvel Top 10

Now we’ve got to linear temperaments (or rank 2 temperaments if you want to be strict and acknowledge the cases where the octave isn’t a generator). Practical work with marvel is likely to be based on a linear temperament: there are fewer different interval classes, they have more consonances, and they fit to generalized keyboards. Any of them can play generic marvel music, and you might even get away with different marvel temperaments playing at the same time. This list is ordered by the same badness as the list of scale steps above. The second column shows pairs of scale steps that are consistent with the temperament named in the first column. Because these scale steps are counting the octaves of equal temperaments, I added a ”d” to the 34 of Wurschmidt¨ to show that this mapping is ambiguous, and we’re using the one with the less- good approximation to 7:4. In the last column, I tried to give a citation for the first description of each temperament class. I don’t think you’ll find a consistent set of criteria behind them. I spent some time tracking down Bosanquet references, though, and I’ll give you those here: page 72 of the ”Elementary Treatise” and page 388 of ”More on the Division. . . ”. The rest of this document covers the first five temperaments on the list. The next one, Pajara, deserves a mention here. It divides the octave into two equal parts, and so isn’t strictly a linear temperament, which is one excuse for excluding it. Another is that Paul Erlich gives a good description and argument for it in his 1998 Xenharmonikonˆ paper.

9 Meantone 19 & 31 Huygens 1661 (pub. 1724?) Magic 19 & 41 Secor (before 1974) Orwell 31 & 22 2001? Miracle 31 & 41 Secor 1975 Garibaldi 12 & 41 Bosanquet 1875 Pajara 12 & 22 Erlich 1998 Catakleismic 19 & 53 Secor (before 1974) Negri 19 & 10 Negri 1986 Waage 12 & 72 Waage 1985? Wurschmidt¨ 31 & 34d Wurschmidt¨ 1921?

Meantone

I hope meantone doesn’t require much introduction. Its history was featured at the conference. The approximation of 7:4 as an augmented sixth was known historically but barely used by musicians. That means we can shed a new light on it, and it is an excellent temperament that doesn’t need to be used in a retrospective context. I don’t have any more to say about it here, though.

• The 7-limit optimum is close to quarter-comma.

• A 7:4 is an augmented sixth.

• 7-limit unique.

• Two 4:5:6:7 tetrads in a 12 note scale.

• e.g. E[–G–B[–C].

Magic

I’ve been concentrating on magic (when I’ve been concentrating on music) for the last few years. It’s the next simplest marvel to meantone, and it has noticeably better 9-limit harmony. The argument for the 9-limit over the 7-limit is that you have more familiar intervals because you can use more fifths and you get two kinds of whole tone (9:8 and 10:9).

• Major third generator.

• Optimal tuning close to 41-equal.

• Seven 4:5:6:7 tetrads in a 19 note scale.

• 9-limit unique.

10 • Every 9-limit interval better than 12-equal.

• Lacking nice diatonics.

Magic Arithmetic

These equations are like the generic marvel ones above, but they show new approximations you get with magic. The first is a demonstration of the 5-limit magic comma, where you go up by five major thirds of 5:4 and then resolve by a 3:1 perfect twelfth. If you elaborate this, you can get a good disco feel. The second equation shows how a 5:3 major sixth can be divided into two equal intervals that approximate 9:7. This isn’t a very digestible 9-limit interval, but it means the result is an essentially tempered 9-limit chord. The next equation simply shows that a 36:35 quartertone approximates the same as a 25:24 chromatic semitone (smaller than a diatonic semitone). These are the two small steps by which a 5:6:7 diminished triad expands to give a 4:5:6 major triad. I’ve repeated this example in the audio for this page (magic.mp3) to demonstrate the equal melodic intervals you get from magic temperament. This also shows how a temperament can improve a chord sequence by something other than stopping small pitch shifts or overall pitch drift.

5 5 5 5 5 3 × × × × ≈ 4 4 4 4 4 1

9 9 5 × ≈ 7 7 3

36 25 ≈ 35 24

245 1 ≈ 243 1

Orwell

Orwell’s main claim to fame is the simple 11-limit extension. Even if you use it for its unique 9-limit approximations, you’re likely to run into 11-limit intervals as dissonances. The mapping is such that 10:9 and 9:8 are the most complex 9-limit intervals. So, if you want frugal harmony, you could drop the ”9” part of the 11-limit to get ”7-limit plus 11”. This also means that fifths are relatively complex, so modulations by chains of fifths are likely to cause trouble.

11 • Optimal 19/84 generator.

• Two 4:5:6:7 tetrads in a 13 note scale.

• 9-limit unique.

• Simple 11-limit extension.

• 9 note scale with dense harmony.

The 9 note scale shows promise for improvisation because every interval of the right size to be in the 11-limit approximates something from the 11-limit. The remaining steps approximate the 15-limit. That means you can play random harmony and nothing is horribly out of tune. Some chords are still a lot better than others, though, like the rough 11-limit approximations with poor chord voicings.

Orwell Arithmetic

These audio examples (in orwell.mp3) have a certain attitude to them. That’s unintentional, because they’re mechanical performances of the equations, but still, enjoy them. The first equation shows the 7:6 generator building up to an 8:5 minor sixth. Then we have an alternative that ends on a 7:4. Then there are two examples with 11-limit approximations. They highlight the two primary ways that marvel can be extended into the 11-limit. The first implies 99:98 being tempered out, giving minerva temperament. The other tempers out the 385:384 comma that gives the mainstream 11-limit marvel. Magic and miracle (coming up next), also extend well as mainstream 11-limit marvels, but this document isn’t really about the 11-limit.

7 7 7 8 × × ≈ 6 6 6 5

6 8 9 7 × × ≈ 5 7 7 4

7 7 11 × ≈ MINERVA 6 6 8

8 12 5 × ≈ MARVEL 7 11 4

12 Miracle

Miracle gets as close as most of us will need to the optimal marvel tuning. It also has good 11-limit approximations without being horribly complex. George Secor proposed it as the basis for a keyboar and notation for Harry Partch’s famous 43 note scale from Genesis of a Music. It keeps each of those 43 notes distinct and works a little better with the earlier Exposition of Monophony scale.

• Semitone generator.

• Optimal tuning close to 72-equal.

• Eight 4:5:6:7 tetrads in a 21 note scale.

• 11-limit unique.

• 10 note near-equal scale.

Decimal Scale

Instead of equations, here’s a 10 note miracle scale to demonstrate how the intervals add up. It extends well to a very unequal 21 note scale that doesn’t quite manage a complete 11-limit chord. You keep adding the generator that could be 16:15 or 15:14.

7 10 5 7

8 11 5 7 9 4

15 15 15 15 15 15 15 15 15 12 14 14 14 14 14 14 14 14 14 11

7 6

3 4 2 3

2 1

Garibaldi (Schismatic)

There’s a lot I could say about garibaldi (or schismatic temperaments in general or Erv Wilson’s ”cassandra” extension) but in the talk itself I didn’t have time to say much at all. So, the key idea is that the optimal garibaldi tuning is close to Pythagorean. And, as the other side of this, you can interpret extended Pythagorean intonation using garibaldi approximations. From that, you can see that garibaldi will work

13 with progressions and modulations by fifths in a fashion a lot of musicians will be familiar with. Because of the historical importance of Pythagorean intonation, there are a number of historical contexts that you could associate with it. But I haven’t found any early references to show that people understood it, and I’ve given up expecting any enlightenment to come from this direction. As a modern temperament, garibaldi is certainly usable. It also has its bad side, however. Because it’s also generated by fifths, it gives similar scales to meantone of 12 and fewer notes. But as meantone itself is so convenient, garibaldi leaves you dealing with the commas that are no longer tempered out. Also, the pythagorean thirds approximate nothing better than the 5-limit intervals you’d get better with meantone, leaving garibaldi with simple intervals that don’t have an 11-limit interpretation. So, garibaldi’s worth a look, and helps you to understand the geometry of pythagorean intonation, but I’m finding the theoretically better temperaments to really be more promising.

• Near-Pythagorean tuning.

• Two 4:5:6:7 tetrads in a 17 note scale.

• 9-limit unique.

• Positive keyboard mapping. (Bosanquet 1875, Wilson 1975)

• Elusive Arab/Persian connection.

Thank You

For references, slides and examples, see:

http://x31eq.com/Aberdeen

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