This page describes ways to adapt conventional, European-derived harmony to work simply in magic temperament. It follows from a comment that Paul Erlich made about playing a ii⁷→V→I chord sequence on a magic guitar. It turns out that these three simple chords have problems in magic, and adaptations of conventional techniques can help to solve those problems. I take this idea to ridiculous extremes and end up proclaiming new laws of harmony.
Magic temperament is a rank 2 temperament with an optimal tuning close to 41 note equal temperament and 19 and 22 MOS subsets. It has a major third generator approximating 5:4 and a mapping to primes of 3: 5, 5: 1, 7: 12, 11: -8. I should explain it better somewhere, but I haven't, so if you don't understand that you might have trouble with the rest of the page.
I call the 19 note magic MOS "the pengcheng set" and the 22 note MOS "the haizhou set" for the rest of this page. Sorry if that confuses you, but I like to put friendly names on things. You can write them down if you have trouble remembering. These are sets of pitch classes rather than scales because I don't specify the starting pitch. (By some definitions of "mode", they're modes, but there are a lot of definitions of "mode".) I generally assume that an instrument will be restricted to a single pencheng or haizhou set at a time, and so try to get harmony to work within these sets.
The pengcheng set is important because it has 19 notes and meantone temperament also has a good 19 note MOS. The system based around conventional letter note names is usually implicitly defined around meantone, and so I'll use these names to tag pitches inside the pengcheng set. Where there's ambiguity (which means to pitches differ by the approximation of a syntonic comma or 81:80) I'll use numbers from tricycle notation without explaining them. I'll also stop saying "approximation of" when referring to frequency ratios. Assume that I'm talking about temperament here and everything is an approximation.
Magic isn't a meantone and makes 81:80 explicit. This means the concept of a major scale doesn't map to a single, obvious pitch set in magic temperament. One option is to require 5-limit harmony for a I→IV→V chord sequence. That leads to the following structure on the 5-limit lattice or Tonnetz, written as G major because this started as a discussion about guitars.
E-----B-----F♯ / \ / \ / \ / \ / \ / \ C-----G-----D-----A-1
This works with every chord except A minor (the ii chord). One way to add it in is to have two A pitches, making the whole key outside the pengcheng set.
A-3---E-----B-----F♯ \ / \ / \ / \ \ / \ / \ / \ C-----G-----D-----A-1
This allows an extended ii⁷ chord as a combination of A minor and C major. So that I can distinguish different spellings of the supertonic, I'm going to call this the "tonic ii⁷→V→I" progression. The reason is that the root of ii⁷ isn't a 3:2 perfect fifth above the dominant and so the ii⁷ doesn't really have the dominant function, but really has the tonic function in a sense that might become clearer later on if you pay attention. I could also call these pitches the "tonic G major set" in an even more confusing way, but I don't think I will.
The tonic ii⁷→V→I entails an audible comma shift between the A-3 of A minor and the A-1 of D major. This problems can be fixed by rewriting the scale to ensure common tones for this specific chord sequence.
B-----F♯
/ \ / \
/ \ / \
G-----D-----A-1---E-3
\ /
\ /
C-2
Because Am here is joined to G in the proper way by perfect fifths, I say this is the proper use of ii as the dominant, and call Am⁷→D→G on this scale the "subdominant ii⁷→V→I" and could even call this pitch set the "subdominant G major scale" (it really is a scale because it doesn't duplicate any meantone pitch classes but calling it a subdominant scale is mighty confusing because the subdominant chord itself is badly tuned). The distinction between tonic and subdominant ii is useful to have a term for because for the sake of this page (and sensible harmony in magic temperament) I'm going to say that the tonic ii and the dominant ii are not harmonically equivalent and so the tonic ii⁷→V→I and the dominant ii⁷→V→I are different chord sequences. When adapting conventional pieces to magic temperament, you have to choose between the tonic and subdominant ii because the original notation can't distinguish them. After making that choice, though, you have to stop treating them as equivalent.
The problem with the subdominant ii⁷→V→I is that the seventh of Am⁷ is a 16:9 rather than 9:5 and so the fifth between C-2 and G is bad. Also, which is really the same thing, C major (the IV chord or subdominant) is out of tune. To see the complexity of the scale in magic temperament, we can rewrite the lattice showing magic generator counts.
1-----6
/ \ / \
/ \ / \
0-----5----10-----15
\ /
\ /
14
15 generators means there are 4 of these in the pengcheng set or 7 of them in the haizhou set.
I've written before about tritone substitutions with reference to 7-limit harmony. I surely can't have been the first person to notice this relationship, but I still don't have a better citation. Tritone substitutions are well known (which means I don't want to find a citation for them.) The basic idea is that you can replace a chord with another whose root is a tritone away, and the harmony still works in some way that I'll leave vaguely undefined. The canonical example is that a V can be replaced by a ♭II in a perfect cadence, so V→I (root motion by 3:2) can be replaced by ♭II→I (root motion by 16:15). And it usually involves seventh chords.
The 7-limit interpretation is that the ♭II⁷ is a 4:5:6:7 chord and the root motion of ♭II⁷→I is by 15:14. (The seventh degree of ♭II⁷ is really a subminor seventh or augmented sixth, but I'll keep writing it as a standard seventh because that's how tritone substitutions are usually written.) Transforming V⁷ into ♭II⁷ involves turning the 5:4 of V⁷ into the 7:4 of ♭II⁷. That entails lowering the root by 7:5 (or raising it by 10:7). So we can define 7-limit tritone substitutions in a neo-Riemannian way as lowering the root by 7:5 and changing the seventh somehow.
In magic temperament, 7:5 is approximated by 11 generators, so the tritone substitution means subtracting 11 generators from the root of the chord. 7:4 is 12 generators, so a 4:5:6:7 chord is 12 generators wide. A tritone substitution of V⁷ to ♭II⁷ turns the root from +5 to -6 generators and the seventh is +6 generators where the fifth of V was +5 generators. This suggests a tritone substitution of V means we can get away with a 7-limit chord that we couldn't afford before, if we're also using pitches with a negative generator count relative to the root. a tonic ii⁷→V⁷→I substituted as ii⁷→♭II⁷→I now looks like this:
A-3---E-----B-----F♯ \ / \ / \ / \ / \ / \ / C-----G-----D / \ / / \ / A♭----E♭
0-----5----10----15 \ / \ / \ / \ / \ / \ / 4-----9----14 / \ / / \ / 3-----8
This has a cost of 15 generators, so it has the same magic complexity as the major key I originally showed, but it allows ii⁷ an extended 5-limit tuning and the ♭II⁷ has a 7-limit tuning. There are four of them in the pengcheng set.
A syntonic comma has a complexity of 19 generators in magic temperament, co-incident with the 19 pitch classes in the pengcheng set. The tonic ii at 0 generators on this diagram would therefore be +19 generators with its subdominant spelling. A tritone substitution will take this to +8 generators, which is the E♭ already on the scale. Make this substitution, and the scale gets a bit simpler in magic temperament.
E-----B-----F♯
/ \ / \ /
/ \ / \ /
C-----G-----D
/ \ / \ /
/ \ / \ /
A♭----E♭----B♭
\ / \
\ / \
G♭----D♭
2-----7----12
/ \ / \ /
/ \ / \ /
1-----6----11
/ \ / \ /
/ \ / \ /
0-----5-----10
\ / \
\ / \
9-----14
This gives pure I and IV chords (plus others you can work out from the lattice) plus the 7-limit ♭II⁷ and an extended 5-limit ♭vi⁷ as a substitute of the subdominant ii⁷. It only needs 14 magic generator, so it's simpler than the simple major keys we started with. There are 5 of them in the pengcheng set, which is not quite enough to transpose the whole scale by a 3:2 perfect fifth. The haizhou set has three pairs of these scales separated by a 3:2 perfect fifth.
"A bit simpler in magic temperament" in this case means 11 distinct pitch classes. That looks like one pitch short of a 12 note chromatic scale, but in fact F♯ and G♭ map to the same pitch. It's well known (meaning I can't find a citation again) that a diatonic scale plus its tritone transposition includes every pitch of 12 note equal temperament. With magic temperament, there are more pitches available, so it's possible for a major key plus syntonic equivalents and tritone substitutions to use more than 12 pitch classes, so 11 is no problem.
There's a simpler tritone-substituted variation of the subdominant ii⁷→V→I chord sequence. It involves substituting the ii⁷ and leaving the V alone.
B-----F♯
/ \ / \
/ \ / \
G-----D-----A-1
/ \ /
/ \ /
E♭----B♭
\ / \
\ / \
G♭----D♭
2-----7
/ \ / \
/ \ / \
1-----6----11
/ \ /
/ \ /
0-----5
\ / \
\ / \
4-----9
That doesn't sound much like a major key, but the rule of tritone substitutions says it contains equivalent harmony in some sense or other. It covers 11 magic generators, which means there are 8 of them in each pengcheng set. This is enough that it can be transposed by a perfect fifth and remain within pengcheng. Here's an example, both C and G major with and added "F" so that all pitches of a version of the basic scales are covered:
E-----B-----F♯
/ \ / \ / \
/ \ / \ / \
C-----G-----D-----A-1
/ \ / \ / \ /
/ \ / \ / \ /
A♭3---E♭----B♭----F-3
\ / \ / \ /
\ / \ / \ /
C♭----G♭----D♭-2
2-----7----12
/ \ / \ / \
/ \ / \ / \
1-----6----11----16
/ \ / \ / \ /
/ \ / \ / \ /
0-----5----10----15
\ / \ / \ /
\ / \ / \ /
4-----9----14
The pengcheng set includes three pairs of keys like this, separated by major thirds.
The rule that a tritone substitution replaces a pitch with another a 7:5 lower defines these relationships:
| Original pitch | G-3 | B-1 | D♯-2 | B♭-1 | D-2 | F♯-3 | A♯-1 | F-3 | A-1 | C♯-2 | E♯-3 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Substituted pitch | D♭-1 | F-2 | A-3 | F♭-2 | A♭-3 | C-1 | E-2 | C♭-1 | E♭-2 | G-3 | B-1 |
If this is a mathematical equivalence relationship, it should be possible to apply it twice and still get and equivalence. Apply 7:5 twice and apply octave equivalence (which is generally assumed in matters of harmony) and you get 50:49, which is quite a small interval. In meantone terms, you get an enharmonic diesis, which means this amounts to enharmonic equivalence. In jazz harmony this is sort of assumed but not stated, although in common practice harmony it certainly doesn't apply: differently spelled but enharmonically equivalent intervals have different dissonance properties and resolve differently. So there you go: tritone substitutions lead to enharmonic equivalence of harmony.
In magic terms, a double tritone substitution maps to a shift of -22 generators, and so is the equivalence interval of the haizhou set. This makes sense, because 7:5 is an exact half octave in 22 note equal temperament and 50:49 is tempered out. Note that the pengcheng set is also bounded by a harmonic equivalence. It's equivalence interval is the syntonic comma (81:80) and this is implicitly assumed by the notation system because there's no way to write it. So, if a chord lies entirely outside the haizhou or pengcheng set, you have a transformation that can bring it inside with harmonic equivalence.
Here's a table of enharmonic equivalences derived from double application of tritone substitutions.
| Original pitch | A♯-3 | B-1 | B♯-1 | C♯-1 | D♯-2 | E-2 | E♯-3 | F♯-3 | G♯-3 |
|---|---|---|---|---|---|---|---|---|---|
| Equivalent pitch | B♭-2 | C♭-3 | C-3 | D♭-3 | E♭-1 | F♭-1 | F-2 | G♭-2 | A♭-2 |
There's a general rule here: the diatonic pitch class goes up by one, the pitch rises 1 step on a 19 note scale, and the wheel number goes down by one to balance the other rises. The result is no movement on a 22 note scale and a 1 step rise on a 41 note scale.
Here's another way of looking at the double application of tritone substitutions. A substituted V in a V→I cadence leads to descending root progression by an interval of 15:14 or, given "marvel" 225:224 equivalence, 16:15. This interval can also be called a diatonic semitone, a secor, or a toe. The rule, then, is that rising by a 3:2 fifth and applying a tritone substitution is equivalent to rising by 15:14. It follows from this that going up by two fifths and applying two tritone substitutions is equivalent to rising by a 16:15 and a 15:14 which is the same as a rising 8:7. (This assumes 225:224 equivalence, which I do assume all over this page. Magic and meantone temperaments all temper out 225:224.)
The arithmetic adds up here: a fifth is 5 magic generators, two fifths are 10 generators and the double tritone substitution subtracts 22 generators. 10 - 22 = -12 and 12 is the number of generators to the prime 7, so -12 generators means a falling 7:4 or a rising 8:7. Alternatively, two fifths above the root takes you to the second degree of the major scale or 9:8. In E-2 major, this pitch is F♯-3. Apply the double tritone substitution to F♯-3 and you get G♭-2. This is the correct 8:7 above E-2.
In terms of ratios, 9/8 * 50/49 * 224/225 = 8/7. You could also say that a double tritone substitution means movement by 50/49 * 224/225 or 64/63. And now we're throwing around 225:224 equivalences, we can note that the 7:5 tritone is equivalent to 7/5 * 225/224 or 45:32. That, in turn, can be derived as 9/8 * 10/9 * 9/8, or three tones, hence "tritone".
Magic temperament is generated by 5:4 major thirds. The simplest harmony, then, will be generated by chains of chords separated by major thirds. Traditional European-derived harmony makes 3:2 perfect fifths the simplest interval and gives progression by fifths the central importance. We can use tritone substitutions to derive movement by third from movement by fifth. The recipe is: move up by two fifths to get a 9:8 (octave equivalently) then apply the 45:32 version of the tritone substitution to get 4:5 or a descending 5:4 major third. Alternatively, consider the tritone as 9/8 * 10/9 * 9/8 which is also 5/4 * 9/8 so a tritone is made up of a 9:8 and a 5:4.
This is also expected from the examples of tritone substitutions of subdominant ii chords. As ii substitutes to ♭vi, and an ascending sixth is octave equivalent to a descending third, it all adds up. If you follow the idea that harmony should progress by descending fifths, the rule follows that major thirds should ascend. Of course, you aren't obliged to follow that rule, and plenty of music allows roots to ascend by fifths. At least, it brings legitimacy to root progressions by major thirds.
The more obvious way of deriving thirds from fifths is to take the major third as two whole tones, or four octave-equivalent perfect fifths. This works in meantone, and associates ascending fifths with ascending major thirds. It's a weaker rule to apply to magic temperament, though, because the relationship of four fifths is more distant and it requires syntonic comma equivalence, and it's nice to avoid that in temperaments that make the syntonic comma explicit.
In conventional music based on 12 note equal temperament, modulation or root progression by a major third is used to symmetrically divide the octave and create harmonic ambiguity. Movement by three major thirds gets you back where you started. This is usually assumed in analysis with no stipulation that it requires a circulating set of 12 nominally equal pitch classes. In magic temperament, things are very different, because there is no 12 note MOS to invoke for equivalences and the interval between three 5:4 major thirds and an octave is the same as the interval between a 6:5 minor and 5:4 major third, so it can't be "magicked" away. You can get an equivalence of 3 magic generators by combining the syntonic-comma equivalence of the pengcheng scale with the double tritone-substitution derived equivalence of the haizhou scale. On this page, though, I'm not considering pengcheng equivalences as harmonic equivalences in whatever sense tritone substitutions are equivalent. I only use equivalence of pengcheng sets to convert from conventional notation into magic temperament, because conventional notation assumes the 81:80 syntonic comma is tempered out.
In fact, everything breaks down if you consider both 3 and 19 generators as equivalences. If pitches 3 generators apart are equivalent, so are pitches 18 generators apart. And if pitches 18 and 19 generators apart are both equivalent, it must mean that pitches one generator apart are also equivalent, which leads to every pitch being equivalent to every other pitch. This is incredibly liberating for a theory of harmony, and means you can do whatever you like and label it with whatever erudite labels you like. It doesn't to much as a meaningful form of analysis, though, so I'll stick to pitches a syntonic comma apart being distinct.
Before we discard "three generator equivalence", though, I'm going to remark that it looks a lot like harmonic function. Every pitch falls on one of three wheels (as in tricycle notation) with the root, fourth, and fifth of a diatonic scale on distinct wheels. So, we can call the three wheels the tonic, subdominant, and dominant. This doesn't reproduce functional harmony, though, for the very reason that syntonic commas are not equivalence intervals. The second degree of a diatonic scale has the tonic function if its root is 10/9 but the subdominant function if its root is 9/8 whereas with functional harmony it always has the subdominant function. Funnily enough, this corresponds to whether I call it the "tonic ii" or the "subdominant ii". So, let's cast aside this crazy notion that three major thirds would in any sense bring you back to where you started from and describe how they work in real music (with magic temperament).
To make use of major thirds, let's take the pitch set above containing harmonically equivalent resources to two major keys separated by a 3:2 perfect fifth and transpose it by a 5:4 major third.
G♯----D♯----A♯1
/ \ / \ / \
/ \ / \ / \
E-----B-----F♯----C♯2
/ \ / \ / \ /
/ \ / \ / \ /
C-1---G-----D-----A-1
\ / \ / \ /
\ / \ / \ /
E♭----B♭----F-3
It's promising that this still contains the pitches of C and G major, which shows you that transposing by a single generator doesn't get you very far in magic temperament. Now, but the two pitch sets together to give a superscale containing four harmonically related major keys.
G♯----D♯----A♯1
/ \ / \ / \
/ \ / \ / \
E-----B-----F♯----C♯2
/ \ / \ / \ /
/ \ / \ / \ /
C-1---G-----D-----A-1
/ \ / \ / \ /
/ \ / \ / \ /
A♭3---E♭----B♭----F-3
\ / \ / \ /
\ / \ / \ /
C♭1---G♭----D♭2
Conventional and not-at-all-strange modulations involving tritone substitutions take us from E to B to C-1 to G. There are lots of chords here that would look normal in music based on keys as pitch centers rather than sets of allowable pitches and none of them require double sharps or flats or assume respellings based on enharmonic equivalences. The interesting thing comes when we look at this pitch set in terms of magic generators.
3-----8----13
/ \ / \ / \
/ \ / \ / \
2-----7----12----17
/ \ / \ / \ /
/ \ / \ / \ /
1-----6----11----16
/ \ / \ / \ /
/ \ / \ / \ /
0-----5----10----15
\ / \ / \ /
\ / \ / \ /
4-----9----14
This is a consecutive set of magic generators! All pitches are generated by major thirds, even when the spelling and latticing obscures the relationship. There are two of these sets in the pengcheng set, meaning we can get the pencheng set by adding either F♭-2 or E♯-3. I'll choose E♯-3 out of laziness because it makes it easier to modify the lattice diagram. Also, it means G to E♯-3 is a 7:4.
G♯----D♯----A♯1---E♯3
/ \ / \ / \ /
/ \ / \ / \ /
E-----B-----F♯----C♯2
/ \ / \ / \ /
/ \ / \ / \ /
C-1---G-----D-----A-1
/ \ / \ / \ /
/ \ / \ / \ /
A♭3---E♭----B♭----F-3
\ / \ / \ /
\ / \ / \ /
C♭1---G♭----D♭2
3-----8----13----18
/ \ / \ / \ /
/ \ / \ / \ /
2-----7----12----17
/ \ / \ / \ /
/ \ / \ / \ /
1-----6----11----16
/ \ / \ / \ /
/ \ / \ / \ /
0-----5----10----15
\ / \ / \ /
\ / \ / \ /
4-----9----14
As well as C, E, G, and B, this gives us the keys of G♯ and D♯ major. It isn't obvious how, though, because they run off the top of the lattice diagram. I'll make it clearer by copying rows and doing a bit of renaming. (Because we're using the magic equivalences now, there isn't a unique choice of pitch name for every lattice point.)
B♯1---G♭----D♭2
/ \ / \ / \
/ \ / \ / \
G♯----D♯----A♯1---E♯3
/ \ / \ / \ /
/ \ / \ / \ /
E-----B-----F♯----C♯2
/ \ / \ / \ /
/ \ / \ / \ /
C-1---G-----D-----A-1
/ \ / \ / \ /
/ \ / \ / \ /
A♭3---E♭----B♭----F-3
\ / \ / \ /
\ / \ / \ /
C♭1---G♭----D♭2
/ \ / \ / \
/ \ / \ / \
G♯----D♯----A♯1---F♭3
This, then, is an instance of the pengcheng set showing equivalences so that magic temperament does some work for us. It includes the tritone-substituted harmonic resources of at least 6 major keys, but is still incapable of all but the simplest diatonic harmony with correct tuning. This isn't the only way of spelling it, and I'm not even proposing it as a standard, but if you made it clear by, say, writing the lattice on a score, you could write music using this set with conventional pitch names and leave out the wheel numbers (or other notation of syntonic commas) because they're all obvious given the context. You might even find existing pieces of conventional music that use tritone substitutions in such a way that they naturally work with such a set, but I don't know any.
Here's an example of adding three more pitches to get a haizhou set. This gives two tunings of A so that all diatonic chords of G major can have 5-limit tunings.
E♯2---B♯1---G♭----D♭2---A♭1
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
C♯1---G♯----D♯----A♯1---E♯3
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
A-3---E-----B-----F♯----C♯2
\ / \ / \ / \ /
\ / \ / \ / \ /
C-1---G-----D-----A-1
/ \ / \ / \ /
/ \ / \ / \ /
A♭3---E♭----B♭----F-3
/ \ / \ / \ / \
/ \ / \ / \ / \
F♭2--C♭1---G♭----D♭2---A♭1
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
C♯1---G♯----D♯----A♯1---F♭3
Previously, I've shown minor triads having the same intervals as major triads, and so approximating 10:13:15. An alternative is to use a 7-limit "subminor" third to give a 9-limit 6:7:9 chord. If you write a ii chord like this, it is a subset of the 7-limit V⁹ chord, and so the substituted 6:7:9 ♭vi is always there when you have a substituted ♭II⁹ (the substitution takes care of the 7-limit tuning). If you add a 7:4 to a 6:7:9 chord, you get 12:14:18:21. (Caveat: once you add the seventh it stops being a 9-limit chord.) In magic temperament, this covers 12 generators, so it isn't very simple but it might be interesting. Combining it with the 7-limit chord a fifth lower costs 17 generators, so there are only two ways of doing it in the pengcheng set. Here are the pitches to do a fancy version of it in G major.
E♯3
/
/
B-----F♯----C♯2
/ \ / \ /
/ \ / \ /
C-1---G-----D-----A-1
/ \ / \ /
/ \ / \ /
A♭3---E♭----B♭
I is G-B-D (or I⁹ is G-B-D-E♯-A with 9-limit tuning), ♭II⁹ is A♭-C-E♭-F♯-B♭ and ♭vi⁷ is E♭-F♯-B♭-C♯. The other key with a I→♭II⁹→♭vi⁷ of this kind in the pengcheng set above is B.
D♯----A♯1---E♯3
/ \ /
/ \ /
E-----B-----F♯
/ \ / \ /
/ \ / \ /
C-1---G-----D
The chords here are I as B-D♯-F♯ (no septimal seventh here), ♭II⁹ as C-E-G-A♯-D and ♭vi⁷ as G-A♯-D-E♯.
The magic generator counts for this pattern:
6----12----17
/ \ /
/ \ /
1-----5----11
/ \ / \ /
/ \ / \ /
0-----5----10
Like with the subdominant ii⁷→V→I, it's simpler to leave the V alone and do a tritone substitution of the ii⁷.
B-----F♯----C♯2
/ \ / \ /
/ \ / \ /
G-----D-----A-1
/ \ /
/ \ /
E♭----B♭
2-----7----12
/ \ / \ /
/ \ / \ /
1-----6----11
/ \ /
/ \ /
0-----5
This has a complexity of 12 generators, and so there are 7 of them, and two pairs of them a 3:2 apart, in the pengcheng set. In fact, you can do this whenever you can do the simple extended 5-limit ii⁷→V→I because only the pitch requiring 12 generators (C♯-2 in the G major example) is new.
The most problematic pitch of the ii chord is the fifth degree, which is +5 generators relative to the root and +15 generators relative to the tonic. One candidate is 11:8, and interval you can generally think of as a quartertone-sharp fourth, but in evolutionary magic notation comes out as an augmented fourth. Here's G major supporting a I, V, and altered ii:
D♯
/ \
/ \
B-----F♯
/ \ / \
/ \ / \
G-----D-----A-1
\
\
C-2
2
/ \
/ \
1-----6
/ \ / \
/ \ / \
0-----5-----10
\
\
14
This gives us a kind of A diminished seventh, which is an 11-limit essentially tempered chord. Specifically, it's what the Xenharmonic wiki calls a magical seventh chord with the root chosen so that intervals relative to it are 6:5, 11:8, and 5:3. Like most of this page, it doesn't require all equivalences of magic temperement. It only needs 100:99 and 385:384 to be tempered out, which defines a supermagic temperament. It saves us one generator compared to the subdominant ii chord. Here's a grid to show that every interval is within the 11-limit.
| C | D♯ | F♯ | |
| A | 6:5 | 11:8 | 6:5 |
| C | 8:7 | 11:8 | |
| D♯ | 6:5 |
The chord can also be rotated to save another 2 generators:
B♯
\
\
D♯
/ \
/ \
B-----F♯
/ \ / \
/ \ / \
G-----D-----A-1
-2
\
\
2
/ \
/ \
1-----6
/ \ / \
/ \ / \
0-----5-----10
Intervals relative to the root are now 8:7, 11:8, 5:3.
Another way of dealing with the third is to introduce another chromaticism, and make it major.
D♯
/ \
/ \
B-----F♯----C♯
/ \ / \ /
/ \ / \ /
G-----D-----A-1
2
/ \
/ \
1-----6-----11
/ \ / \ /
/ \ / \ /
0-----5-----10
This is the most compact yet, with a complexity of 11 generators. The altered sixth chord is also an 11-limit essentially-tempered chord. Here's the interval grid to prove it:
| C♯ | D♯ | F♯ | |
| A | 5:4 | 11:8 | 6:5 |
| C♯ | 10:9 | 3:2 | |
| D♯ | 6:5 |
This only depends on 100:99 being tempered out.
So far, tritone substitutions have always been in the same direction: -11 generators. This is the direction consistent with the usual written notation and the 7-limit interpretation. Conventional theory doesn't tend to concern itself a great deal about which direction the tritones should point, though. That's partly because a lot of conventional theorists consider the octave divided into 12 nominally equal divisions, or even that dividing the octave into an even number of equal divisions is required for tritone substitutions. Anyway, even if we decide the algebra of tritone substitutions should be such that an equivalence only works in one direction, we can always cheat by saying that the tonic chord itself is being substituted in the normal direction and this will have the same effect as substituting the other chords backwards. So let's think about backwards substitutions.
Substituting by +11 generators works if we start with something on the negative side, which means the tonic ii⁷→V→I. Here it is with the ii⁷ substituted to be ♯v⁷.
C♯ D♯----A♯1 \ / \ / \ / \ / E-----B-----F♯ / \ / \ / \ / \ / \ / \ C-----G-----D-----A-1
-3 7-----12
\ / \ /
\ / \ /
1-----6-----11
/ \ / \ / \
/ \ / \ / \
0-----5----10-----15
This follows the ill defined rule of the tritone substitution transformation acting on seventh degrees to put the C♯ where it doesn't make the ♯v⁷ a composite of a 5-limit minor and major. We do keep a standard IV chord in the same pengcheng set, though, (In fact, that IV can be extended to a 9-limit IV⁹.) and there's exactly one pengcheng set that includes this scale. True parsimony means breaking the rule and allowing a better tuning of the ♯v⁷ though.
D♯----A♯1
/ \ / \
/ \ / \
B-----F♯----C♯
/ \ / \ /
/ \ / \ /
G-----D-----A-1
This happens to include a tonality diamond centered on F♯.
2-----7
/ \ / \
/ \ / \
1-----6-----11
/ \ / \ /
/ \ / \ /
0-----5-----10
This scale has a complexity of 11 generators, which matches that of the simplest conventional substitution of the subdominant ii⁷ with V→I. That means we get 8 of these in a pengcheng set including to pairs separated by a 3:2 perfect fifth. It also means that the same pengcheng set includes substituted scales for both the tonic and subdominant ii⁷→V→I. In fact, those scales have a great deal of overlap: In this tonic ii⁷→V→I equivalent, G is the pitch with 0 generators. In the subdominant ii⁷→V→I version, G is the pitch with 1 generator. So every pengcheng set includes 7 major keys that allow for both a tritone substitutions of the subdominant ii⁷ and a reverse-tritone substitution of the tonic ii⁷, both with their best extended 5-limit spelling.
For my final trick, I'll go back to the tonic ii⁷→V⁷→I with the V⁷ substituted to get ii⁷→♭II⁷→I. Here's the scale again so you don't have to scroll up to find it.
A-3---E-----B-----F♯ \ / \ / \ / \ / \ / \ / C-----G-----D / \ / / \ / A♭----E♭
0-----5----10----15 \ / \ / \ / \ / \ / \ / 4-----9----14 / \ / / \ / 3-----8
The roots in the ii⁷→♭II⁷→I go A→A♭→G or 0→3→9 generators. This means that in magic temperament, G to A♭ is twice as large as A♭ to A. I call the interval between A♭ and A a "semitoe". (Look carefully, it isn't a spelling mistake.) A semitoe is -3 magic generators. If we add a chord at 6 generators, we can get descending root motion by equal semitoes. So where does that 6 generator (2 semitoe) pitch come from? It could be the reverse-tritone substitution of a pitch at -5 generators. That pitch would be a perfect fifth below A-3, and so D-1. So this is an enharmonic equivalent of D, the fifth degree of the scale. To check, D-2 here is 14 generators and 14 + 5 = 19, which is the complexity of the syntonic comma. D-1 and D-2 are pengcheng equivalents, which by the harmonic rules I'm imposing on this page means they're not equivalent but can both represent a D in a chord sequence of conventional music. So, the ii⁷→V⁷→I can be converted to magic temperament as A₃m⁷→D₁⁷→D₂→G₃ (wheel numbers as subscripts) and substituted as Am⁷→A♭⁷→G♯→G (with every root on wheel 3). Here's the lattice.
B♯
/ \
/ \
G♯----D♯
/ \ / \
/ \ / \
A-3---E-----B-----F♯
\ / \ / \ /
\ / \ / \ /
C-----G-----D
/ \ /
/ \ /
A♭----E♭
7
/ \
/ \
6----11
/ \ / \
/ \ / \
0-----5----10----15
\ / \ / \ /
\ / \ / \ /
4-----9----14
/ \ /
/ \ /
3-----8
We've previously seen that 8:7 (or 4 semitoes or -12 generators) can be a substitution of ii. So, a general rule is that ii and V chords get tritone substituted to have their roots on multiples of semitoes. Or, I could formulate two fundamental laws of magic harmony.
Huge swathes of Western harmony (cherry-picked from European classical music and jazz) can be generalized from these two laws. It turns out that there are 11 different chord categories (which we could call functions) corresponding to the 22 notes of the haizhou set modulo the 11 step tritone. Here's a diagram showing all the chord roots for the pengcheng set above grouped in this fashion. Roots in italics don't have 3:2 fifth above them, but might be useful for altered chords. (Note that a theorem of magic harmony states that whenever you have two pitches in the same generated scale (e.g. MOS) separated by a 3:2 fifth, you can always make triads of both the 5-limit major (4:5:6) and 5-limit minor (10:12:15) pattern.) Each root is specified by the number of generators relative to the tonic and the pitch in the key of G major. Chord sequences following the two laws of magic harmony will naturally progress down this table.
| 10 | 3 | G♭ | ||||||
|---|---|---|---|---|---|---|---|---|
| 9 | 6 | F♯ | -5 | C | ||||
| 8 | -2 | C♭ | 9 | F | ||||
| 7 | 1 | B | 12 | E♯ | ||||
| 6 | 4 | B♭ | ||||||
| 5 | -4 | E | 7 | A♯ | ||||
| 4 | -1 | E♭ | 10 | A | ||||
| 3 | 2 | D♯ | ||||||
| 2 | -6 | A♭ | 5 | D | ||||
| 1 | -3 | G♯ | 8 | D♭ | ||||
| 0 | 0 | G | 11 | C♯ | ||||
| wheel | 3 | 2 | 1 | 3 |
Any root in regular typeface except F♯-3 and A♯-1 (that is any root specified by no more than 5 generators) can have a 6:7:9 9-limit subminor triad built on it. Any root that's specified by either zero or a negative number of generators can have a full 4:5:6:7 chord built on it with pitches from the pengcheng set. Any root in normal typeface that's specified by two or more generators can have a 4:5:6:11 chord built on it.
The table preserves reasonable compatibility with the traditional understanding of G major. The diatonic chords G, D, Em, Bm, and C are ordered according to the spiral of fifths, which isn't too much of a surprise because ordering by fifths was one of the assumptions that went into this theory. C→Bm→Em→D→G or more generally IV→iii→vi→V→I even sounds like a plausible chord sequence for a real song to use but I don't know any real songs that use it. Am is missing from the pengcheng set, but is represented by altered chords and its tritone substitutions D♯m⁷ and E♭m⁷. You may think F♯ is wrongly placed, as it's the fifth above B and should be below C, which is on the other side of the spiral of fifths from G. The law of tritone substitutions says that F♯ (at least this F♯) must be equivalent to C, so there's no escaping this.
Note that this table doesn't show much evidence of diatonic pitch classes. You could formalize a harmonic theory based on a function for each diatonic pitch classes, with pitches related by sharps and flats treated equivalently, but it would have difficulty with tritone substitutions. Diatonic scales don't have any particular significance in magic temperament (there's no proper 7 note MOS) and so they've been abstracted out of this definition of a major key. (The choice of pitch names is also a little arbitrary. C♭ could also be B♯ because the two are pengcheng equivalents and although there is a perfect fifth C♭ to G♭ B♯ is required for this pitch when it's part of a G♯ chord.)
You may also find it amusing that we have a major key here without its ii chord but still with 14 different roots on which 5-limit (or higher) chords can be built when a gaggle of musicians are under the impression that the conventional theory from which this is derived only supports 12 distinct pitches to the octave. Tritone substitutions lead to a kind of enharmonic equivalence, but not the kind that leads to a 12 note scale.
For advanced students, here is a table categorizing all pitches of the 41 note magic MOS. "All pitches" means we don't worry about the 3:2 fifths being available.
| 10 | -19 | G | -8 | C♯ | 3 | G♭ | 14 | C | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9 | -16 | G♭ | -5 | C | 6 | F♯ | 17 | C♭ | ||||
| 8 | -13 | F♯ | -2 | C♭ | 9 | F | 20 | B | ||||
| 7 | -10 | F | 1 | B | 12 | E♯ | ||||||
| 6 | -18 | B | -7 | E♯ | 4 | B♭ | 15 | E | ||||
| 5 | -15 | B♭ | -4 | E | 7 | A♯ | 18 | E♭ | ||||
| 4 | -12 | A♯ | -1 | E♭ | 10 | A | ||||||
| 3 | -20 | E♭ | -9 | A | 2 | D♯ | 13 | A♭ | ||||
| 2 | -17 | D♯ | -6 | A♭ | 5 | D | 16 | G♯ | ||||
| 1 | -14 | D | -3 | G♯ | 8 | D♭ | 19 | G | ||||
| 0 | -11 | D♭ | 0 | G | 11 | C♯ | ||||||
| wheel | 1 | 3 | 2 | 1 | 3 | 2 |
Really, this should be left as pure theory and not sullied with practical music making. There's always somebody who takes these ideal theoretical ideas and tries to fit them to a keyboard, though. Here are Scala files for the pengcheng file as on the magic temperament home page:
In addition, here's a Scala keyboard mapping to get G in the right place and some ZynAddSubFX tunings to save trial-and-error setting of the right shift:
Here's a table to table to show what pitches end up on a 5 octave keyboard. If you don't have exactly these keyboard octaves, hopefully you can work out the pattern.
| Pitch | F | F♯ | G♭ | G | G♯ | A | A♯ | B♭ | B | C♭ | C | C♯ | D♭ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Pitch | D♭ | D | D♯ | E♭ | E | F♭ | F | F♯ | G♭ | G | G♯ | A♭ | A♯ |
| Pitch | A♯ | B♭ | B | C♭ | C | C♯ | D♭ | D | D♯ | E♭ | E | F♭ | F |
| Pitch | F | F♯ | G♭ | G | G♯ | A | A♯ | B♭ | B | C♭ | C | C♯ | D♭ |
| Pitch | D♭ | D | D♯ | E♭ | E | F♭ | F | F♯ | G♭ | G | G♯ | A♭ | A♯ |
| Keyboard key name | C | D | E | F | G | A | B | C |
There are gaps in the pengcheng scale. You don't always get both A♭ and A. This is because the keyboard mapping repeats every 18 notes, so pitches are in the same place every to octaves and it makes sense in terms of tripod notation. it isn't ideal for the evolutionary approach on this place, but it will do for now. The tuning should be fixed so that MIDI key 57 is G. If you don't know which key is MIDI key 57, and you're using ZynAddSubFX, you can find out using the "m" and "M" buttons under "Min.k" and "Max.k". Remember to reset the keyboard range after doing this.
You can change the keyboard mapping to get a key other than G major. Stickers and masking tape and the like can help to remind you what the sounding pitches are. This is convenient enough that you can get real music working and, whether or not you take the 11 harmonic functions seriously, you should be able to adapt whatever understanding you have of whatever music conventions to work with magic temperament.
If it doesn't sound good, don't worry. It must mean you're doing it wrong because the theory is correct.