This page is about tuning the standard dominant seventh chord mostly to be a bit better in tune than possible in meantone temperament. Quite often that means 225:224 and 100:99 being tempered out, which gives apollo planar temperament. This is the context I'm currently most interested in for getting extensions of standard harmony, what I've called evolutionary harmony, working outside of meantone temperaments. The particular rank 2 temperament classes I work with are magic temperament and andromeda temperament (which is an 11-limit extension of schismatic or helmholtz or garibaldi).
Apollo has a one to one mapping with 5-limit just intonation, meaning that you can work out chords using 5-limit logic and then temper them to get higher-limit intervals. It also means that you can write all your chords on a 5-limit lattice. Not all rank 3 temperaments work like this and it is a useful feature when you're starting with traditional chordal harmony, like I am here. That isn't the only way to work with microtonal harmony and it isn't even a very popular one, which is why I don't have good citations for a lot of what I cover here. It is an approach I think is well worth looking at.
This material should be useful for musicians working within harmonic traditions tracing back to Europe who want to move outside of 12-equal or meantone. But I also show the arithmetic for why things work and talk about where ideas are first recorded, and generally say a lot of impractical things. It turns out that there's a lot to say about dominant seventh chords once you go outside meantone. So practically minded musicians will have to be able to skip over the bits that don't interest them. I also explain standard concepts in a way that people who don't already know them probably won't understand. The idea is to make explicit what the rules are and why they came about so that you can decide whether you're likely to find them useful.
The best practical way to experiment with an apollo tuning with standard hardware is the cassava keyboard.
Where I quote books, you can find the details of what the book is in the bibliography.
After all that, the first approximation I'll look at is inconsistent with apollo temperament: simply treating the chord as a harmonic series fragment 4:5:6:7. The best way to tie this to a 5-limit context is to tune the seventh as 16/9 and temper out 64:63. This is consistent with the twelve equal divisions of the octave as featured on a piano near you, and a host of manufactured instruments. Whether the minor seventh is really heard as a 7:4 with such a crude tuning is one of those questions that I don't think is clearly defined.
There are better tunings that involve sharing the mistuning of 64:63 between the 7:4 and the 16:9, making them both 13 or 14 cents out of tune and the perfect fifth around 7 cents out of tune from 3:2. One notable rank 2 class is what we now call pajara with an optimal tuning close to 22-equal (divisions of the octave) and featured in Paul Erlich's Xenharmonikôn 17 paper, 1998. I think this is good enough to evoke the 7-limit harmony, but not close enough count as smooth. It's similar to the mistuning of the 5-limit in 12-equal.
Identification of the dominant seventh as approximating 4:5:6:7 goes at least back to Helmholtz's "On the Sensation of Tone…". On p.343 (Chapter XVII) he says "The Sevenths of these chords all come pretty near to the natural Seventh 7/4 [cents 969], and are all smaller than the Sevenths in the chords of the Seventh formed from two consonant triads [cents 1088 and 1018]." You might also note that on this page Helmholtz lowers the seventh by another comma and notes that the result is "very nearly equal to 7/4" with the discrepancy being 5120:5103. You might also have a look at the footnote on p.344 where the translator (Ellis) discusses the 4:5:6:7 tuning.
Helmholtz has more to say on p.347 and I'll reproduce as much as I can of the original typography whether you understand it or not.
The chord of the dominant Seventh, g + b₁ – d | f contains three tones belonging to the compound tone of G, namely g, b₁, and d, and the Seventh f is the dissonant tone. But we must observe that the minor Seventh g…f [or 16/9 = 64/63 × 7/4, or cents 996 = 969 + 27] approaches so near to the ratio 7/4 [cents 969] which would be almost exactly represented by g…f₁ [cents 974], that f may in any case pass as the seventh partial tone of the compound G. Singers probably often exchange the f of the chord of the dominant Seventh for f₁ partly because it usually passes into e₁, partly because they thus diminish the harshness of the dissonance.
Ellis slips in a footnote to say he disagrees with this. And Helmholtz brings in the crucial quote "Hence, although the chord of the dominant Seventh is dissonant, its dissonant tone so nearly corresponds to the corresponding partial tone in the compound tone of the dominant, that the whole chord may be very well regarded as a representative of that compound." That's one of the things I'll consider later as I look at the more dominant-seventh-ish tunings.
As there are plenty of references to 4:5:6:7 being treated as a tuning of a dominant seventh, it clearly belongs on this page. However, in the context of meantone, the 7:4 interval is better written as an augmented sixth than a minor seventh, so we can also think of 4:5:6:7 chords and dominant sevenths as different chords that are written in different ways. I could say more about augmented sixths, but for the rest of this rant I'm going to talk about other interpretations of the dominant seventh.
The primary tuning Helmholtz gave for the minor seventh is 16/9. That means a dominant seventh chord has the ratios 1/1 5/4 3/2 16/9. A G dominant seventh (G7 in middle tyning notation) on the lattice looks like this.
B
/ \
/ \
F----- -----G-----D
As it's the resolution to the tonic that makes this a dominant chord, I'll add the major tonic in there.
(E)----B
/ \ / \
/ \ / \
F----(C)----G-----D
This shows that the F from the dominant seventh is the perfect fourth of the major scale, and so closely related to the tonic pitch. That appears to be a good thing, and I think this is the best tuning to use in the context of a dominant seventh. There are problems though: the intervals B to F and D to F are outside the 9-limit.
To fix the tritone from B to F, we need to invoke some temperament. The ratio from B as 5:4 to F as 16:9 is 64:45 which becomes the octave complement of 7:5 when widened by 225:224. That leaves us with a 9-limit dyadic (essentially tempered) chord when you leave the fifth out of the chord.
The idea behind dyadic chords is that the concordance is determined by the roughness of the intervals in the chord. Roughness is minimized by simple ratios because then more partials of harmonic timbres line up and stop beating. Or, when tempered, the partials give nice fuzzy beating. The logic here assumes that the 7-limit tritones are good enough to be recognized as concordant intervals. I think that does work, especially if you choose timbres with prominent seventh partials. We also have to assume that the 9-limit is acceptable, because the seventh is already a 9-limit interval. The approximations here can be good enough that the concordance conferred by the ratios doesn't get spoiled. The chord as a whole can still be recognized as a coarse approximation to a 4:5:6:7 or, as the fifth is omitted, 4:5:7. I think a chord can be heard on both these levels but I think I won't go into the details now.
Recognizing the 16/9 tuning and the 7:5 approximation also has repercussions on the voicing of the chord. For the 7:5 to really be a 7:5, the third has to be voiced higher than the seventh, with the two pitches still within an octave. This conflicts with the 4:5:7 approximation, and simple root position, where you'd expect the seventh to be the highest pitch. As 16/9 has a power of 2 in the numerator, it becomes simpler the lower it's pitched. That means you could end up with an inversion where the seventh is in the bass.
Riemann doesn't like this. He says (p.58) "As the seventh in the progression D⁷—T makes the leading-tone step to the third of the tonic … and, on the other hand, the fundamental note of the tonic is required for the bass in the last, closing chord, the seventh as a bass-note for the penultimate chord is impossible" (italics in the original). This doesn't mean you can't write a chord with these intervals with the seventh in the bass, only that it it doesn't work for the specific case where it's resolving to the tonic, which is what "dominant seventh" naturally means.
If we don't leave the fifth degree out, the interval outside the 9-limit is the 32:27 pythagorean minor third between the fifth and the seventh. It is possible to make this equivalent to 13:11 by tempering out 351:352. This is a reasonable thing to do: it's consistent with 13-limit apollo and andromeda and yet more if you don't temper out 100:99. Whether this is of any significance in taming the discordance of the chord is less clear. An approximation to the interval 13:11 will bring the 11th and 13th partials close enough to give pleasant beats. That'll only be significant if the 11th and 13th partials are loud enough for their beating to be otherwise unpleasant or the ear has some other way of recognizing and accepting the ratio. Still, this is the best I can do, and you can always leave the fifth out.
Is it reasonable to leave a note out because you have tuning problems? In this case, I say yes, because it's already idiomatic to leave the fifth out of a dominant seventh chord. This is explicitly stated by Riemann (pp. 57–58): "Here … the fifth of the chord of the seventh has been omitted: that is quite an ordinary proceeding (as the 5 or I respectively in a chord may in any case be occasionally omitted), and is always to be recommended when we desire to make the bass progression from fundamental note to fundamental note, and to have the subsequent tonic chord complete …" (italics in the original). The reason is that leaving the root and fifth in both chords will lead to a parallel fifth and Riemann forbids them. Whatever you think about parallel fifths, a lot of composers will leave one of the chords incomplete for this reason.
For another reference, see Levine p.27. This is in Chapter 5: Adding Notes to Three-Note Voicings so it's in the context of starting with seventh chords omitting the fifth and adding more pitches. Here are the simple rules he gives:
His II chord is a minor seventh, his V chord is a dominant seventh, and his I chord is the tonic as a major seventh chord. This doesn't mean you never sound the fifth of a dominant seventh (Levine gives this as an option on p.33) but it does mean you are allowed to leave out the fifth, and it is a good general rule with this tuning of dominant seventh.
9/5 also makes sense as the seventh of a dominant seventh chord. If you tune as 9/5, then every third is 5-limit. The chord in ratios is 1/1 5/4 3/2 9/5. It helps to see consonant intervals on a lattice. This is one of the things that scares off people who don't like theory to get too mathematical. Be afraid, in that case, because here is the extended 5-limit dominant seventh chord on a lattice
B
/ \
/ \
G-----D
\
\
F
I'm showing the chord as G because it's consistent with it being the dominant in the key of C, and I like to think in terms of C. Also, we don't need any sharps or flats. In middle tyning notation this is specifically written as "G⁷" with the 7 as a superscript referring to the seventh as 9/5. With that notation, "G7" is the 16/9 seventh, discussed below.
It's apparent from the lattice that every pitch in G⁷ is connected to every other pitch by a 5-limit line except for G and F (defined as 9/7, which is a 9-limit interval) and B and F. This means that the interval from B to F is outside the 5-limit, which is expected because it's a tritone and this chord is supposed to be a dissonance. It's also outside the 9-limit, which is the smallest odd limit this chord can work with given that we went and add a 9/5 to it. Is there a way of tempering the tritone to bring it within the 9- or 11-limit limits? Well, yes, of course there is, or I wouldn't be writing this.
The ratio representing this tritone, from 5/4 to 9/5, is 36:25 or 631 cents. Let's turn it upside-down and consider it as 25:18 instead. This interval is slightly smaller than a half-octave. The simplest interval it's likely to approximate is 7:5. The ratio between 25:18 and 7:5 is 126:125, which is a reasonable thing to temper out. It gives the starling planar temperament and includes 7-limit meantone (12 & 19) and more accurate temperament classes like myna (31 & 27) and valentine (31 & 25). I think it's well known that a dominant seventh is a 9-limit dyadic (essentially tempered) chord in meantone.
I'm specifically looking to temper out 225:224 or 100:99 today. In this case, what I'm really leading to is that 25:18 is wider than 11:8 by 100:99. That makes this tempering of this tuning of a dominant seventh an 11-limit dyadic chord.
Whether an interval approximating an 11-limit is enough to tame the discordance is one of those questions that doesn't have a clear answer. It helps that 11:8 is a simple 11-limit interval, and can be made simpler by lowering the seventh degree of the chord still further. (Even getting to 11:8 means the seventh degree has to be pitched below the third degree.) That means there's a tension between the best tuning as an approximation of 4:5:6:7 and the best tuning bringing out the internal dyads.
The particular utility of tempering out 100:99 here is that it gives an 11-limit interpretation to an extended 5-limit interval. More than that, it lets us bring the 11-limit into diatonic harmony! As a general rule, superparticular ratios (where one number is one more than the other) are the most useful to temper out because they're the smallest in size of pitch difference for a given size of numbers. The bigger the numbers, the more accurate the approximation. 100:99 has numbers a bit bigger than the syntonic comma of 81:80 used to define meantone. So, we can expect to get approximations similar to meantone in the 11-limit. Why bother to abandon meantone to get only a slight improvement? One reason is that 100:99 gives us interesting approximations, and it doesn't work well with meantone. If you temper out 81:80 and 100:99 at the same time, you also temper out 45:44 (because 81/80 × 100/99 = 45/44) and this will lead to coarser approximations. So, meantone is still good at being meantone, but there's another harmonic world with 100:99 tempered out that's also worth exploring.
There's another use to recognizing the 11-limit approximation. The 11:8 ratio points to the root of another 11-limit chord, and this root will be the seventh degree of the dominant seventh. In a V⁷ → I progression, that other chord will be the subdominant. So, we could have a IV+11 → V⁷ → I progression with common tones. One problem is that the root of the IV corresponds to 27:20 from the tonic (3:2 × 9:5 ÷ 2) when you'd normally expect it to be 4:3. That leads to a general problem with this spelling of a dominant seventh, so let's have a look at it.
For that specific context of a dominant chord resolving onto the tonic, we also need to look at full chord sequences, where we move away from the tonic and come back with more than two chords. Some general rules are that it's the ii chord that causes trouble and V7 works better with IV because they can share a pitch in an obvious way. If you want to avoid problems, stick with I → IV → vi → V⁷ → I, where the vi and V⁷ don't share any pitches (unless you extend vi to vi⁷) and you can always avoid comma shifts. Here, we're going to look at I → ii → V⁷ → I, which does have problems.
In the key of C major, we have C → Dm → G⁷ → C. To decide how to pitch the chords, I'll put the roots into tricycle notation. If C is C₁, then G⁷ is pitched on G₃. The tricycle subscripts can be correlated with degrees of a triad by definition: the root of G⁷ is the same as the fifth of C₁ and the fifth is on wheel 3 as the third note in the chord.
Note that I'm using subscripts in a way that makes this look like a chemical formula. Some people will say that I'm being pseudo-scientific here, and cloaking my work in the appearance of science without it being a genuine science. Such people will either have to hold their noses, or give me up as a lost cause. I like the subscripts. They could be confused with octave specifiers but they're smaller then the letters, which are more important.
The natural way of pitching Dm is on the fifth degree of G₃⁷. This would put it on wheel 2, so D₂m. The F, the minor third of D₂m, will then be on the next wheel, wheel 3. The F in the G₃⁷ chord is also on wheel 3 because G₃⁷ also includes D₂ and the F is also a minor third above D₂, so F₃. That means we can write the full sequence as C₁ → D₂m → G₃⁷ → C₁. It looks good: each root is on a different wheel and the roots work as pure descending fifth, which some people think is the archetype of a good chord sequence.
The C₁ → D₂m progression works because the two chords have no pitches in common. There is a problem if we add a seventh, to replace D₂m with D₂m⁷, spelled out as D₂ F₃ A₁ C₂. This C₂ is out of tune with the tonic root of C₁. Because F₃ to C₂ is a perfect fifth, the minor seventh chord will go out of tune with a different C. Some people don't like the comma shift, and this is known as a comma problem. Note that it's only a problem when we start with the tonic. If a ii⁷ → V⁷ → I comes out of nowhere, or from a different key, it still works without comma shifts between adjacent chords.
You might be thinking "ah, but can't we tune the ii⁷ as a subminor triad with added harmonic seventh, so 6:7:9 expanded to 12:14:18:21 or Dsmz in middle tyning notation". Well, yes we could do that, and how perceptive of you to find the correct middle tyning spelling. It means that C is tuned as the 21. The D of the C major scale must have ratio 8/7. The F is a 7:6 above this, which is 4/3, so we have the correct fourth degree and comma problems are resolve. But you're forgetting something, aren't you? This is in the context of G⁷ being the dominant seventh, and so F isn't tuned as the 4/3 of the major scale, so that F is still causing a comma problem.
Another problem arises if we expand the sequence to I → IV → ii → V⁷ → I or in C major C → F → Dm → G⁷ → C. For the F chord to be in tune with D₂m, it will be F₃, spelled out as F₃ A₁ C₂. That brings in the same comma shift from C₁ to C₂ that causes problems with D₂m⁷. You can resolve this problem by leaving the fifth out of the IV chord, leaving it as a bare third. Adding a major seventh also has a comma problem. Adding a minor seventh doesn't cause a comma problem, but is a chromatic pitch and that might be a problem. You can add a sixth, but then you end up with a iim⁷ chord with a different root, which doesn't help much. So, there's always some kind of problem with I → IV → ii → V⁷ → I but that doesn't mean you can't ever use it. I think the comma shift from I to IV is acceptable in this context, because it's a moving away from home and the shift emphasizes that. It is still a problem, and the V⁷ tuning with the 9/5 does cause problems in a lot of contexts, and in general V7 works more clearly as we'll move on to now.
You can add the 11-limit augmented fourth to the IV chord to get it rationalized as 8:10:11 (with the 12 left out because it disagrees with the tonic). This is a cool chord in C major because it's spelled F A B, specifically F₃ A₁ B₁, so it's the fab chord. It does have a common tone with Cmaj7 (C₁ E₂ G₃ B₁). The root motion from I to IV will be an approximation of 27:20 and, because 100:99 is tempered out, that's also an approximation of 15:11 (because 27/20 × 100/99 = 15/11). So, we have a completely diatonic chord sequence, I → IV+11 → ii → V⁷ → I, that includes root motion by an interval most simply represented with a factor of 11.
In C major, the ii⁷ → V7 → I becomes Dm⁷ → G7 → C. The G7 tuning is less problematic in context because the F is the 4/3 tuning of the fourth degree of the C major chord, and so works more naturally with subdominant chords that are theoretically tied to this pitch. With tricycle notation, the F is F₂ so the Dm⁷ is written D₁ F₂ A₃ C₁ which is the correct 9-limit tuning (10;12;15:18) with 5-limit thirds. The progression becomes C₁ → D₁m⁷ → G₃7 → C₁ and there are no comma problems provided you leave the fifth out of the V7 chord which we've already argued about and I decided was a good idea.
You can tune the ii⁷ as a 12:14:18:21 chord in this context and I think it works without comma problems, although it doesn't solve any problems either because the root isn't the D of a G₃7 chord if it had a fifth, and shouldn't really be spelled as a D at all.
It all depends on the specific context, but generally V7 with a 16/9 is the best tuning of a dominant seventh chord and it's best to leave out the fifth degree of the V7.
At this point, I've established that the V⁷ with a 9/5 and the V7 with a 16/9 have different tunings, work with different voicings, most noticeably with V7 being best with the fifth omitted, and are appropriate in different harmonic contexts. This means they're really different chords that get tempered to be the same in meantone, and so are naturally written the same way in standard notation. I don't think there's any performance tradition sufficiently precise to distinguish them, and so no standard theory with the tuning distinctions hidden within it, but this is what you find when you consider the implications of the different tunings: new harmonic theory based on two different chords that are traditionally subsumed under the "dominiant seventh" label. Or, if you include 4:5:6:7, three different chords.
Levine (p.27) suggested added ninths (major or minor) in preference to fifths as pitches for extending a dominant seventh chord. So, let's look at that. I'll talk about seconds instead of ninths to keep everything within an octave.
Firstly, we'll add a 9/8 major second to G⁷. The pitches are 1/1 9/8 5/4 3/2 9/5. On a lattice, it looks like this.
B
/ \
/ \
G-----D-----A
\ /
\ /
F
Except for the F, this is a 9-limit otonality without the seventh. That is, 8:9:10:12. The F is a major third below (or minor sixth above) the A. So, if the original dominant seventh was a dyadic chord, it will still be with the added second which means it's still an 11-limit dyadic chord with 100:99 tempered out. This chord is called G⁹ in middle tyning notation.
The 9/8 becomes a 9/4 when you move it up an octave to be the ninth it started as. Theory says that's likely to be a good voicing for this tuning of the ninth chord. All pitches except the seventh fit the harmonic series, and even the seventh is close enough to 7/4 to work as a coarse approximation, with the finer 11-limit approximations assisting the smoothness.
Adding a major second to the G7 chord with the 16/9 seventh makes more sense if the seventh is 10/9 rather than 9/8. It's easy to see why: 10/9 and 16/9 both have a 9 in the denominator, meaning the ratio between them is 10:16 or 5:8, a 5-limit minor sixth. Here it is on a lattice, with the fifth omitted, and the tonic triad in parentheses.
A----(E)----B / \ / \ / / \ / \ / F----(C)----G
This chord is called G9 in middle tyning notation. It's a helpful coincidence that G7 extends to G9 and G⁷ extends to G⁹. At least, it might be a coincidence, or I might have designed the notation so that it works like this. You decide.
The A is a 9-limit interval from every other pitch in the diagram, so the result is still a 9-limit dyadic chord. In fact, it looks a lot like F should be the root and it will work well with the F and A pitched low. You'll have to be careful to ensure the right root gets established.
Let's look at adding a minor second now. When you temper out 225:224, the 16:15 minor second becomes identical to 15:14. This interval on its own puts the chord into the 15-limit whichever way you rationalize it. Adding it above the root does work as a dyadic 15-limit chord when paired with the 9/5 seventh of G⁷ given apollo temperament:
The chord is still dyadic with the 16/9 seventh, without any need to invoke 13 identities (or intervals with a 13 in the ratio). Between the 16/15 second and the 16/9 seventh is a 5-limit major sixth of 5:3.
As we're looking at 11-limit harmony, let's try adding an 11/8 to the dominant seventh chords. With 100:99 tempered out, this is an augmented fourth, and in middle tyning notation it's specifically notated as "+11" (where "♯11" has a distinct meaning). Levine didn't say we could add an augmented fourth to a dominant seventh, but I'm going to do it anyway. It can be an interesting chord even if it wouldn't be accepted as a dominant seventh in any tradition.
As the dominant seventh pattern is a major triad with an added seventh, and adding a +11 to a major triad trivially gives you an 11-limit result (8:10:11:12), We can get an 11-limit chord provided the ratio between the seventh and the +11 can be brought within the 11-limit. In meantone, this interval is a diminished fourth, which approximates 9:7, so we're looking for 9:7 approximations.
With a V⁷ chord, you get the 9:7 approximation when 56:55 is tempered out. That's because the interval between the 9/5 seventh and the 11/8 augmented fourth is 72:55 (9/5 ÷ 11/8 = 72/55) and that differs from 9:7 by 56:55 (72/55 ÷ 9/7 = 56/55). This is in the less-accurate-than-meantone range, so it doesn't interest me today and I won't say more about it.
With a V7 chord, the interval between the 11/8 and 16/9 does approximate 9:7 when both 225:224 and 100:99 are tempered out. (16/9 ÷ 11/8 × 225/224 ÷ 100/99 = 9/7). That means V7+11 is an 11-limit dyadic chord in apollo temperament.
This chord can't be extended to an 8:9:10:11 chord plus the seventh because I showed above that the added second works better as 10/9 than 9/8 in this context. We can still get an 11-limit dyadic V9+11 chord, however. With 100:99 tempered out, 11:10 and 10:9 approximate to be the same (because 10/9 ÷ 11/10 = 100/99, which is easy to work out because 10/9 ÷ 11/10 = 10/9 × 10/11 = (10 × 10)/(9 × 11) = 100:99). The interval between 11/10 and 11/8 is 10:8 or 5:4 (trivially because the numerators are the same).
In C₁ major, this chord is G₃ A₃ B₁ C₁♯ F₂. If we revoice it with the F in the bass, you can see that it's a string of whole tones. F₂ 9:8 G₃ 10:9 A₃ 9:8 B₁ 10:9 C₁♯.
Here's a little diagram that shows the scale/chord as an extension of G7 and shows the different consonances it contains.
F G A B C♯ F
9:8 10:9 9:8 10:9 9:7
|------\/------||------\/------|
5:4 5:4
|----------\/----------||------\/------|
7:5 10:7
|----------\/----------|
11:8
|------------------\/-----------------|
7:4
|--------------\/-------------|
14:9
|----------\/---------|
7:5
|------\/-----|
5:4
As well as the G7, this is an extension of A⁷ (but not with an 11:8 above the root). It also includes a French sixth: F A B D♯. That a French sixth can be essentially tempered to the 9-limit is already known, and the first reference I can find is from Paul Hahn in 1998;
I seem to recall having offered this chord (1/1 5/4 7/5 7/4) some time back as a small-integer-ratio "interpretation" of the French sixth chord. Since the 225/224 vanishes in my fave, 31TET, it works in that tuning as well--in fact, I'm beginning to realize that the 225/224 may be as significant in my septimal messing about as the 81/80 is to the diatonic scale.
So, what happens if we add another whole tone to get a hexatonic whole tone scale?
It looks like a good rule to continue the pattern of alternating 9:8 and 10:9 whole tones, which will give us two chains of 5:4 major thirds. That means the two scales to look at are F₂ 9:8 G₃ 10:9 A₃ 9:8 B₁ 10:9 C₁♯ 9:8 D₂♯ and E₂♭ 10:9 F₂ 9:8 G₃ 10:9 A₃ 9:8 B₁ 10:9 C₁♯. As I already showed that 5 notes from a string of alternating whole tones are 11-limit dyadic in apollo, the only intervals we need to consider now are the intervals between the two extremes of these runs. We can also simplify arithmetic by observing that every pair of 9:8 and 10:9 result in a 5:4 (because 9/8 × 10/9 = 5/4 if this wasn't obvious from a standard diatonic scale) and a pair of 5:4 intervals together approximate 14:9 when 225:224 is tempered out (because 5/4 × 5/4 ÷ 225/224 = 14/9).
The first case to consider is F₂ 9:8 G₃ 10:9 A₃ 9:8 B₁ 10:9 C₁♯ 9:8 D₂♯ where the interval F₂ to D₂♯ is 14/9 × 9/8 = 7/4. This is an 11-limit interval, so this chord is still 11-limit dyadic in apollo temperament. If you're familiar with tricycle notation, you don't even need the ratio arithmetic, because you can observe that D₂♯ to F₂ is two diatonic semitones on the same wheel, and this is known to be an 8:7. This chord is known and listed under Chords of magic on the Xenharmonic Wiki as the number 2 hexad (correctly marked as "apollo"). Gene Ward Smith wrote this page on 22nd December 2011.
The other whole tone scale to check is E₂♭ 10:9 F₂ 9:8 G₃ 10:9 A₃ 9:8 B₁ 10:9 C₁♯. The interval from E₂♭ to C₁♯ is 140:81 (14/9 × 10/9 = 140/81). In meantone, this is another augmented sixth, so how well does it approximate 7:4? 140/81 ÷ 7/4 = 80/81, so it undershoots by a syntonic comma. As these two scales both look the same in meantone, this isn't an interesting approximation. What we can do is compare it to 12:7, which is a slightly smaller interval. Then, 140/81 ÷ 12/7 = 245/243, and 245:243 is tempered out in magic temperament, so this is a chord of magic. Counting magic generators relative to F₂ (the lowest note in the generator chain) we get
E₂♭ 10:9 F₂ 9:8 G₃ 10:9 A₃ 9:8 B₁ 10:9 C₁♯. 9 0 10 1 11 2
That means this chord is the first hexad in Chords of magic on the Xenharmonic Wiki. The last "whole tone" is an approximation to 7:6, which is definitely getting on the large side.
I've been going through the archives of the Mills Tuning List to find discussions of dominant sevenths. I can't find any references to the essentially tempered chords (or necessarily tempered chords as they'll have been back then) I talk about above. There are various messages comparing dominant sevenths to 4:5:6:7 that I won't mention. Here are some older and interesting ones that consider the specific tunings with 9/5 or 16/9. Some of them are replies to messages where I can't find the original.