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- Joti Rockwell October 2015
- Marcel de Velde October 2015
- Nicolas Meeùs October 2015
- Scott Murphy September 2015
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Omnibus-like progression
Dear SMT megamind, do you know any examples of, or research on, a four-voice progression that connects adjacent harmonies with two-voice semitonal contrary motion like the standard omnibus but instead employs a series of minor-minor sevenths and augmented triads in place of (or, if one wants to think geometrically, running parallel to) the more familiar series of major-minor sevenths and minor triads?
For example: ||: Cmm7 -- Baug -- Cmm42 -- Amm7 -- Caug -- Amm42 -- F#mm7 -- Faug -- F#mm42 -- Ebmm7 -- Daug -- Ebmm42 :||
It sounds jazzy to me, but my knowledge of that repertoire is rather weak.
Any thoughts private or public would be welcome.
Thanks!
-Scott
===
Scott Murphy
Professor, Music Theory
Director, Music Theory and Composition Division
Editor, SMT-V: Videocast Journal of the Society for Music Theory
University of Kansas
smurphy@ku.edu
SMT Discuss Manager: smtdiscuss@societymusictheory.org
Comments
Hello Scott,
I don't know of any examples or research relating to this progression.
But perhaps I can offer some insight based on enharmonically correct spelling.
I can interpret the augmented chords in 2 ways, though I think the first one is possibly the more logical interpretation.
||: Cmm7 -- Ebaug -- Cmm42 -- Amm7 -- Caug -- Amm42 -- F#mm7 -- Aaug -- F#mm42 -- D#mm7 -- F#aug -- D#mm42 :||
||: Cmm7 -- Cbaug -- Cmm42 -- Amm7 -- Abaug -- Amm42 -- F#mm7 -- Faug -- F#mm42 -- D#mm7 -- Daug -- D#mm42 :||
If we continue the progression we get a B#mm7 instead of a Cmm7. Showing that the progression modulates ever further.
In tuning circles we call such a progression a "comma pump", as it doesn't form a "circle" but will "drift" due to modulation by a Pythagorean comma (the difference between C and B# in pitch).
- Marcel
Sorry it was late when I replied wasn't very sharp minded, forgot to write the inversions.
||: Cmm7 -- Ebaug64 -- Cmm42 -- Amm7 -- Caug64 -- Amm42 -- F#mm7 -- Aaug64 -- F#mm42 -- D#mm7 -- F#aug64 -- D#mm42 :||
And it's possible to interpret the progression without modulation as well.
For instance in C minor:
||: Cmm7 -- Ebaug64 -- Cmm42 -- Amm7 -- Abaug -- Amm42 -- F#mm7 -- Faug -- F#mm42 -- Ebmm7 -- Gbaug64 -- Ebmm42 :||
or
||: Cmm7 -- Ebaug64 -- Cmm42 -- Amm7 -- Abaug -- Amm42 -- Gbmm7 -- Bbbaug64 -- Gbmm42 -- Ebmm7 -- Gbaug64 -- Ebmm42 :||
- Marcel
@MdeVelde, I don't think that the progression described, if played in just intonation (as I suppose is what you mean) would shift by a Pythagorean comma. The succession C7 -- A7 -- F#7 -- D#7 --B#7, four minor thirds, would lead to a B# 64 cents lower than the starting C, while a Pythagorean B# would be a Pythagorean comma (24 cents) higher than C, i.e. four syntonic commas higher than the B# resulting from four descending minor thirds. But this is pure theory, of course: the progression would have to be played in something ressembling equal temperament.
@Nicolas
Well there is reason for debate as to what constitutes "just intonation". If we interpret it to mean "justly tuned" according to how our brain quantizes the pitch space I can make a very good argument for just intonation actually being Pythagorean tuning (not limited to 12 tones per octave of course) instead of a 5-limit or higher limit harmonic overtone based tuning which has historically often been called "just intonation" (and which if following it to the logical conclusion would basically mean that our notation system does not correspond at all to how we interpret the intervals and many of the things that are taught from centuries tested classical theory should go straight in the garbage instead.. now that doesn't seem right).
But the reason this matters, even in 12 tone equal tempered tuning, is that it shows how we interpret the notes and gives a "functional" insight.
Pythagorean corresponds to how we spell the notes. A G# is not an Ab. A C-E makes a major third which is stable and an imperfect consonance, a C-Fb makes a diminished fourth and is "unstable" and dissonant. One could for instance write a final chord major triad as C-Fb-G and never be the wiser when tuning it to 12 tone equal temperament, but our brain would surely interpret that Fb as an E, yet in a different progression a dissonant C-Fb-G chord could be correct (with the Fb resolving to an Eb for instance) and could also come across as such in 12 tone equal temperament.
My point was that in order to see a progression for what it is, one should at least start analysing it with correct enharmonic spelling I think.
I appreciate the orthographic angle, Marcel -- thanks! I was offering an enharmonically expedient spelling.
@MdeVelde, I believe that we should always learn from history.
I can see no reason to consider that just intonation is Pythagorean tuning. The name "just intonation" originates in Sauveur's description of the "Systême juste" in his Méthode générale pour former les systêmes tempérés de musique, published in the Mémoires of the Académie des Sciences in Paris in 1707 (pp. 203-222). Sauveur stresses that this system is unusable in practice because it results in a pitch drift. This, he says, "marks the necessity of a tempered system" (p. 208). Sauveur's Système juste corresponds to what is nowadays described as 5-limit. No system using higher harmonics (i.e. with higher prime limit) has ever "historically" been called just intonation.
C-F♭ indeed makes a diminished fourth, as you say; it does in any tuning. However, in Pythagorean tuning specifically, the diminished fourth is almost exactly a just major third, the difference being a schisma (2 cents). This was used as early as in the 14th and 15th centuries to obtain pure thirds in Pythagorean intonation, as explained in several papers by Mark Lindley. It led to Ramos de Pareja (1482) advocating true pure thirds, and opened the path to meantone tuning.
It seems not without importance to realize that descending minor thirds in any documented temperament other than equal (e.g. any historical variety of meantone) eventually produce the enharmony of an octave that is wider than just; Pythagorean is the only tuning where the resulting octave would be narrower.
I don't quite see why you say that "Pythagorean corresponds to how we spell the notes". Correctly spelling the notes is important in any tuning other than equal, and probably more so in meantone tuning because the distance between enharmonic equivalents is wider. It remains that if the omnibus is used as an ostinato, one eventually has to accept the enharmony, as John Bull had to (in meantone tuning, probably!) in his Fantasy on the hexachord in the Fitzwilliam Virginal Book, when the music passes through the A-D♭-E triad (see my "Enharmonic keyboard" article in the New Grove).
@Nicolas
I meant our notation system and our music theory is based on a chain of fifths and octaves.
...Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#.. etc.
For instance a major third is "functionally" made up out of 4 perfect fifths. A circle progression will show such for instance. It is perfectly tuneable just in Pythagorean as this tuning corresponds to how our brain "functionally" handles the intervals. It is like you say indeed not possible to tune it in 5-limit "just intonation" in any acceptable manner (it gives either wolfs or comma drifting or comma shifts, all of which both sound unacceptable and make no sense / do not correspond to how our brain interprets such simple progressions).
If we were to notate in 5-limit for instance we would need a completely different notation system. One which displays the difference between a 5/4 and an 81/64 etc. One would need to notate the chain of octaves, chain of fifths and the chain of the 5/4. The resulting notation system would be completely non corresponding to our current music theory. Like I said, we would have to throw much of it in the trash. Most of it would make no sense whatsever anymore in such a system where a major third is no longer made up out of 4 perfect fifths etc. Yet clearly we have a music theory and notation system right now that DOES make sense to our brain and is the result of many centuries of thought by the best composers and theorists. The additional theory of 5-limit tuning is merely an additional afterthought in this regard, and again a much contested one since the beginning of tuning theory (canonici vs harmonici for instance). And one which I'll say again does not correspond at all to the rest of our music theory. All theorists I've read on 5-limit tuning, including Ramos, Zarlino, Rameau, Helmholtz etc. were hopelessly naive in this regard and none tied 5-limit to any actual musically functionally results, it was just theory and a theory that does not actually work in practice.
Aditionally I've performed for about 10 years now precise tuning comparisons between various limit "JI" systems and Pythagorean JI of actual music (if you wish I can send you some examples) and it is quite clear to even the untrained ear that with actual music Pythagorean sounds in tune to the brain where 5-limit and 1/4 comma meantone is noticeably out of tune in comparison. Yes a 1/1 5/4 3/2 major triad sound nice in isolation, the harmonics line up to give that tight "buzz". But in context of actual music that 5/4 major third is heard as too low and simply our of tune.
The name "just intonation" may have been linked to 5-limit tuning originally. But the meaning of the words "just" and "intonation" indicate that it is the "correct" tuning, "pure" tuning. Sauveur wrongly thought this was 5-limit. It is Pythagorean instead that is the "just" / correct / pure tuning that corresponds to how our brain handles the intervals of music.
If I'm interpreting you correctly you also make a case for enharmonic equality being "functional" / real? If so I must disagree. It is a case of interpretation, no matter the example we can have steps by a Pythagorean comma like a Gb going up to a F# there is no problem with that it is "functional" (unlike the Syntonic comma steps).
Yes I knew it was expedient spelling.
I just though I could say somethinig perhaps meaningful about it by laying the spelling bare. But then realised in my second message that there is an alternative interpretation. And I actually have no idea which one is more likely. Both seem plausible? The modulating one and the one with the progression "keeping touch" with for instance Cminor. Can anybody make an argument in favor of one or the other?
I mean, is there a long term memory somewhere of a tonic so that one of the modulations is interpreted not as a minor third down but as an augmented second instead so we arrive again at the C minor instead of B# minor? In some other progressions / music it is there in a logical way, is there reason to see it here where it doesn't seem very logical at first or is it logical to hear the progression as modulating ever farther away from C minor upon repetition?
@MdeVelde: I think that you are wrong in believing that notation may imply any type of tuning. The choice of writing D♯ or E♭ in the progression mentioned by Scott Murphy is fully independent of the type of tuning or temperament involved. The choice is of some importance even in equal temperament, and you were right to argue that the logical way of writing was C–A–F♯–D♯ (instead of E♭) if things were to stop there. This merely is a matter of voice leading and has nothing to do with the tuning involved. If the chain of thirds was pushed one step further, however, the question of ending on C rather than B♯ would have arisen, which may have involved enharmonic voice leading. But, once again, this has nothing to do with the tuning.
Your idea that Pythagorean corresponds to "how our brain 'functionally' handles the intervals" seems wishful thinking to me. I don't think any musical "functionality" is involved in any of this (but I am quite willing to hear your arguments). Your point about "the progression 'keeping touch' with for instance Cminor" is precisely the reason why Sauveur considered his just intonation unusable. He wrote, refering to Huygens, that "the ear of the musician, maintaining the idea of the first sound ut, necessarily falls back on it in the end, by an imperceptible change of those intervals that are therefore rendered slightly altered, which marks the necessity of a tempered system". This would apply to Pythagorean tuning as well as any other.
You cannot reasonably claim that 1/4 comma meantone sounds out of tune and that Pythagorean tuning sounds in tune, while meantone is about the only tuning sysem used from the late 16th to the early 18th century. You cannot claim that "all theorists [you]'ve read on 5-limit tuning, including Ramos, Zarlino, Rameau, Helmholtz etc. were hopelessly naive in this regard". Not at your age (whatever your age). I am afraid that this discussion opposes that you think is "the truth" to what I consider is historical fact. This obviously results from the fact that you are much younger than I am. Don't argue against me that "[You]'ve performed for about 10 years now precise tuning comparisons". I am very sorry (really ;-)) to have to say that I did so for now more than 40 years... This does not mean that I am right, of course, merely that I have a more overarching view of the matter.
@Nicolas
In our current notation system the D# or Eb difference is not fully tuning independent as our notation system implies a chain of fifths and octaves. So one could argue that it works for meantones and 12tet and Pythagorean (tuning the fifths and octaves pure as 2/1 and 3/2) and even something like 17tet. But it does not correspond to 5-limit.
The theory (and music) in this system is very much tied to the chain of fifths and octaves in such a way that we can really throw out most theory if this chain of fifths and octaves has no merit in our actual perception of music. I think it should be very clear that our music theory does say very meaningful things about music which correspond to our perception of it, including enharmonic distinction between a D# and Eb which stems from this chain of fifths and octaves. Take for instance a German sixth chord and a dominant seventh. We truly hear that diminished sixth as a different interval than the minor seventh in the context of the music even when they are both tuned the same in 12tet.
So yes it was an overstatement to say our notation is linked to only Pythagorean as the fifths (and even octaves) can be wider or smaller than a 3/2 - 2/1. But it is linked to fifths and octaves not higher harmonics and does exclude tuning systems like 5-limit.
I'm not sure what you mean with that there is no musical "functionality" involved in any of this. You do agree that there is a real "functional" difference between a diminished sixth and a minor seventh, or an augmented second and minor third in how our brain perceives these interval right? If not, then what is much of music theory? Merely a difficult spelling excersize and we could just as well do away with the enharmonic difference and notate just 12 notes per octave? I'm assuming this is not your position, but I'm not sure what it is you do mean.
As for the "keeping in touch" with C minor argument. The thing is that the way this works in 5-limit is completely different from how it works in Pythagorean. In 5-limit we get problems in simple things like circle progressions (and many many other situations) while in Pythagorean there are none (so not the problem of "ut" etc like you describe in Pythagorean, it instead does occur but in locations that are acceptable. For instance play the following triads, Cmajor - Fminor - Emajor - Aminor where the melody goes G, Ab, G#, A. Here it must rise from Ab to G# and is very natural in doing so, much more so than in 12tet where they play the same pitch).
The question of "keeping in touch" with a certain tonic is not a problem but a valid question I think. Both the solutions to the progression I gave are not unacceptable, I just don't know which one is more likely to correspond to how we hear it (if I had to make a guess I'd still go for the infinitely modulating one though in this case). But Pythagorean tuning has no real problem here. If you can notate it in our system you can tune it as such in Pythagorean. The only argument against Pythagorean tuning on a "functional" level would be to argue that enharmonic equivalence is real and corresponds to our brain's perception of music (in which case I argue you can throw out much of music theory as it would be meaningless to differentiate between an Eb and D# etc)
My conclusion about Pythagorean tuning being the correct just intonation is based on my personal original research which I've performed full time for the past 10 or so years (it was not meant in and of itself as an argument against you :). It is the only thing I do in life ;) No other work. I've not yet published my research precisely because of these type of arguments (which I do enjoy though!). We are merely skimming the surface now. To get to the deeper unleying points and resolve the differences would require a much deeper discussion which is difficult to graps also for experts. Therefore I will publish my theories with musical examples which argue the case much more accessibly. One type of musical examples I'm preparing now has precisely to do with the enharmonic difference. It is Turkish makam music, where I interpret for instance the Rast tetrachord as G, A, Cb, C. In Pythagorean the Cb is about 24 cents lower than a Pythagorean B corresponding to Turkish tuning practice for Rast. And I've developed a theory for harmonizing such truly chromatic and enharmonic melodies (normal voice leading largely breaks down here. Try for instance for fun to harmonize the melody G, Gb, G, A, Cb, A, G with major and minor triads). My results are musically completely "normal" for our brain while expressing truly new "colours". I could not have arrived at these results without Pythagorean based theory nor could, I think, anybody explain the obvious musically correct results any other way. (Here theory becomes predictive instead of just looking at the past. As far as I know nobody has previously succeeded in harmonizing Turkish makam modes, they always interpret the modes in a western way like G, A, B, C for Rast and then merely tuning the B lower which makes nothing but an out of tune B to our brains when the supporting chord indicates that B).
For fun I could also advice you to try the difference between 1/4 meantone and Pythagorean with more chromatic late romantic music. The difference grows larger the longer the chain of fifths and becomes more noticeable. Tristan und Isolde prelude is one of my favorites, to hear it in 1/4 comma meantone is truly horrible, in Pythagorean it sounds fantastic on the other hand (and even noticeably improved upon 12tet even though the difference is small. I often use sampled piano sounds they give a good sense of relative pitch height, much more so than something like a sine or sawtooth wave).
As for the before mentioned authors being naive in certain aspects. Well for instance they did not even begin to touch upon the problems of 5-limit.. Maybe naive isn't a nice word, but how else to call that?
Hello Scott and others!
This is an intriguing progression, and it's possible to make a graph that be used to map the path that this progression takes:
http://www.mtosmt.org/issues/mto.09.15.5/rockwell_ex12.html
The bad news from me, unfortunately, is that when looking into these questions I wasn't able to find examples of this progression being used with any consistency or thoroughness. As you say, there may be some instances in the jazz literature, but I didn't find them. Like you, though, I'd also love to hear from anyone who does.
Best wishes,
Joti
@MdeVelde. You write:
Agreed. But the difference that we hear can only arise from the subsequent voice leading. And this would remain true even if the notation was wrong, if the the +6 was written as a 7th, or inversely. Functionality does not depend on a notation, but on the succession of harmonies.
You may conclude that Pythagorean tuning is just, that it is the best tuning available, etc.; but "Just intonation" is a name with a history and I don't think it wise to change its meaning in this way. You are aware that "Just intonation" often is used today, especially on Internet, to denote any prime-limit tuning, and you may agree that this is wrong. But reducing just intonation to 3-limit is no solution either. Ellis, who may have been responsible for the introduction of the expression "Just intonation" in English, clearly meant 5-limit tuning, neither 3- nor 7-limit.
You may be aware that Safi al-Din al-Urmawi, in the 13th century, tried to justify the micro intervals of maqam music by an extended Pythagorean tuning, before abandoning the idea because it failed. As to your Turkish Rast, I wonder why you have to hear the third degree as a Pythagorean C♭, instead of a "just" B, which would boil down to the same (a just B being a syntonic comma, 22 cents, lower than a Pythagorean one).
Well, Sauveur certainly did and recognized it as unusable in practice. As to the others, I'd need to know what you mean by "the problems of 5-limit".
But there is little point in continuing this discussion if we base it on matters of faith rather than facts...
@Nicolas
After 10 years of full time original research into just intonation (in all its potential forms) and the conclusions of this I can safely say that my statements regarding Pythagorean tuning are far more fact based than any "faith" one may place in 12tet, 5-limit, meantones, or other tuning systems. I do not expect you to take my word for it and my proofs / facts on this are too large to put in a post here. But it is not correct to asume that my statements are merely faith based.
I was not aware however that Urmawi abandoned his 17-tone Pythagorean tuning as one of his descriptions of the makam modes (I am aware that this was not his only discription of the intervals). Do you have a source for this? It does not really matter (I am not an historian) but I'd still be interested in reading about it. As far as I thought the current discription of Turkish modes still stem from his 17-tone Pythagorean model even though they've dropped the Pythagorean ratios and gone for Arel-Ezgi-Uzdilek 53tet comma descriptions (which is basically the same thing).
The point about the Pythagorean Cb vs B tuned to 5/4 above G is that although if tuned accordingly they are almost imperceptably the same pitch, the Cb is harmonized completely differently than a B. Even if we tune the B to 5/4 the context of the chords / voice leading will still indicate the B to our brain. Turkish music has a makam which uses the western major scale, it is called Cargah and it would be correct to harmonize it normally with a B a major third above G. However the whole point of Rast is that it expresses an interval disctinctly different from the major third of Cargah. The difference is made clear in monophonic music by tuning it lower (in Arabic music it is most often tuned even lower, up to a full comma, than a Pythagorean Cb to emphasize the difference from a B). To harmonize Rast with a major third above its tonic destroys the unique Rast feeling and expression and turns it into a common major scale. To harmonize it as a diminished fourth above the tonic however preserves (I can even say enhances) the Rast expression. And Pythagorean theory not only predicts that a diminished fourth is the correct interval to express for Rast (something which in for instance 1/4 comma meantone would be the opposite as it would be even higher than a major third), it also gives me the insight into how to harmonize this diminished fourth above the tonic in a way that is completely natural and musical to our brain (and again, completely distinct from a major third). These statements I make are not faith based, if you want facts I'll gladly send you a musical example.