Adventures In Just Intonation – FM over Edim7

The high brass enter with a concert F Major chord. The low brass follows with an Edim7. The question that I pose is:

Given that the F Major chord is established first and is sustained to the release,
How would you tune FM/Edim7?

Would you tune it as two separate chords or one large chord?

Below is a diagram showing the information of the FM/Edim7.

Edim7 (E=0) and FM (F=0)
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  • Simplest solution: Just Tune Edim7=0,+16,-17,+19, AND tune FM=0,-14,+2
  • Both chords would be close to the center of pitch
  • Concert E and F are bumping right up next to each other; a minor second needs to be raised +12 cents


Edim7 (E=0) and FM (FM=+12)
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Edim7 (E=0) and FM (Gliss 0 to +12)
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  • Concert F is sharp on the instrument (if you tune to tuning Bb/C)
  • Concert A ends up being closer to ZERO
  • More space between the Edim7 and the FM
  • The chord will resolve at +12 sharp


Edim7 (Edim7=-12) and FM (F=0)
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Edim7 (Gliss 0 to -12) and FM (F=0)
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  • Concert G & Db are closer to ZERO
  • Opens up the space between the Edim7 and FM
  • Concert E will be -12 flat
  • Concert Bb will be -29 flat








0 responses to “Adventures In Just Intonation – FM over Edim7”

  1. Phil Avatar

    If you analyze a polychord out of context, then there are several ways to interpret correct tuning. Wouldn’t you have to take the resolution of the chord into consideration? For example, If the polychord F Major/Edim7 resolves to Bb major, then it’s obvious that the F chord is the more important one because of the V-I motion. Therefore, the tuning of the F chord should take priority, and the Edim7 chord (which only serves to intensify the F chord) should be appropriately tuned to fit within the F tuning. Tuning the Edim7 chord first wouldn’t make sense; the V-I motion is much stronger than the enharmonic common tone diminished 7 tendency that the Edim7 chord would share with Bb major.
    However, say that the polychord resolves to Ab major. If we respell the Edim7 chord as a Gdim7 chord, then we have a very strong viidim7-I pull to resolve to Ab. F-Ab (or, Major VI-I) would not make much sense in this case, so we would tune it differently.
    Chords are static, but music has motion.
    -Phil Wingfield

    1. Brad Avatar

      Hi, Phil.

      Take a look at the examples; you will see that the chord resolves to F Major.


  2. Phil Avatar

    Also, because the chords in this case are introduced at separate times and by different voices (high brass then low brass), I think it would be better to tune them as separate chords. There are so many ways to interpret it, though.

  3. NW Avatar

    The next question is, how much does timbre affect “how” the excerpt would be heard? (vs. the pure electronic Reason tones)

    I know I’m asking more questions than I’m answering. 🙂


  4. Phil Avatar

    Then in the case of a resolution to F, I would think that the F major chord should be tuned correctly (lowered 3rd and raised 5th), and then the Edim7 chord would be tuned with regards to it’s tendency to resolve to F, since it really only serves as a huge amount of tension.
    The two biggest resolutions with the most pull are E resolving to F, and Bb resolving to A. Db resolving to C is also very strong.
    The perfect interval tuning chart really doesn’t take these tendencies into account. It’s fantastic for tuning individual triads or 7 chords, but doesn’t do much to show tuning with regards to harmonic function. Some of it is hinted at: notice the chart has a distinction between a dominant 7 and a minor 7 interval. A dominant 7 interval has a much stronger tendency to resolve downward than a minor 7 interval, hence the difference in tuning. Now, a dominant 7 and a minor 7 are superficially the same interval: 10 semitones. However, what differentiates them? Context.
    I’m not going to pretend like I know the acoustical mathematics needed to figure out the ideal tuning for the Edim7 chord in this example, but to me it seems like that’s an academic pursuit. If I were playing the Bb in that chord, I would tune it to it’s tendency to want to resolve to the A. I can tell you that I would lower it, probably by a fair amount. But as for the exact number of cents? I can’t tell you. Just like I can be sure that I would raise the E slightly, with regard to it’s tendency to want to resolve to F, just as should be done with the 7th scale degree resolving to Do in any key, but I wouldn’t be able to really tell you exactly how much.
    I suppose that in this context, it would be fair to consider the Edim7 chord as being not so much a separate chord, but as a series of added notes to the F major triad that serve to intensify a resolution.

  5. Sean Avatar

    Another possible idea, although not having heard it, I can’t say yet whether I like it or it works:

    Tune all the pitches relative to the F: Don’t worry about the chord integrity of the dim7, because it’s just there to draw attention to the eventual resolution. Treat the E as a M7 (-12?), the G as a M2 (+4), B-flat as P4 (-2), and D-flat as m6 (+14).

    Now, a consequence of this is that between E/G and between B-flat/D-flat, the fraction ratios that construct the just-intonation scale preserve the proper minor-third relationship. Getting from G to B-flat (and from D-flat to E), though, must cross over the wolf fifth somehow, breaking that pure minor third.

    +All pitches played support F harmonic series (enhanced if a low F can be sounded about two or three octaves beneath the tuba pitch via synthesizer).
    +F major triad in high brass is held constant, at normal levels of alteration, without need for further adjustment when the low brass enter.
    +Pitches sounded by low brass get adjusted by comparatively small amounts, with nothing more drastic than moving something by 14 cents.
    +As the dissonance is an effect whose purpose is to lead into the clarity of resolution a few bars later, it may not be necessary for the dim7 chord to sound perfectly in tune, or even in tune at all.

    -The dim7 chord will not be perfectly in tune: the pitches’ function is broken away from sounding as a chord in favor of reinforcing the F harmonic series.
    -In the baritones, the E (a sharp pitch) needs to be noticeably lowered, and the D-flat (in a particularly flat partial on the Yamahas) needs to be noticeably raised. The D-flat can be fingered 2-3, placing it in the F partial, which tends sharp. This might address the issue.
    -Without a reinforcement of the fundamental F beneath the low brass, it may not be clear that all pitches are being tuned in relation to that single pitch, and players (trained to listen down for intonation purposes) may have trouble finding the correct adjustments.

    Can you make up a chart and recording using this system? I’d like to hear how it compares with the other ones up already. Also, is there any way that the resolution chord can be added to the existing recordings?


  6. Nathan Pratt Avatar
    Nathan Pratt

    Echoing Phil and Sean, could we hear each example (both at 0, F raised 12, E lowered 12) being resolved to F Major? This is more of a horizontal question than a vertical question, and I think it would be easier to hear which sounds better in motion than on a static sonority.

    Also, Robert W. Smith worked with us in rehearsal on this section. He first established the high brass on FM (with a lowered 3rd and raised 5th), then established the low brass on eº7 (with a raised 3rd, lowered 5th, and raised 7th), then combined the two in dissonance, then resolved the low brass to consonance. His rehearsal technique seems to imply bitonality.

    Echoing NW, I think that the ears of the audience (including the judges) will hear two separate timbres with two separate tonalities. Since there is no harmonic or rhythmic overlap between the high brass and low brass, the audience will hear two different ideas. So I would instruct the high brass and low brass to tune separately until the tubas change to concert F. At that point, the high brass should listen down to the low brass and make sure that they are still playing their FM chord in tune.

    Perhaps attempts to resolve the tension between the eº7 and FM by adjusting one of the chords +/-12 will undermine the effect of the moment. That tension is there deliberately, so we should maximize it—with a great quality of sound. 😉

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  8. Brad Avatar

    I have posted the 3 examples plus Phelan’s Equation – 4 total. All of the examples resolve to F Major.