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In this Discussion
- Andrea Calilhanna January 2019
- Elizabeth Marvin March 2017
- Carson Farley April 2017
- Dimitar Ninov April 2017
- John Paul Ito March 2017
- John McKay April 2017
- Nicolas Meeùs April 2017
- Richard Cohn April 2017
- Stephen Soderberg April 2017
- Stan Kleppinger March 2017
- SMT Discuss April 2017
- William van Geest April 2017
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teaching musical meter
When textbooks define meter, they refer to patterns of strong and weak, or accented and unaccented beats, implying that these paired terms are categorically absolute. But all musicians understand that the third beat in common time, and the fourth beat in compound time, are both weaker than the downbeat and stronger than the immediately flanking beats, implying that accentual strength is contextual and gradated.
I would like the perspectives of colleagues on any of the following questions: (1) Are you aware of any textbooks that acknowledge and theorize the tension between the absolute definition and the gradated experience? (2) How do you address the tension between these two conceptions in your own teaching? (3) How do you respond to students who recognize this tension and would like you to address or resolve it?
---Rick Cohn
SMT Discuss Manager: smtdiscuss@societymusictheory.org
Comments
Hi Rick -- I guess I'll chime in by saying that our textbook (Clendinning and Marvin, The Musician's Guide to Theory and Analysis) does try to represent the accentual strength of beats in a way that gets at some of what you describe.
In the third edition, page 35, we say "Within each measure, the downbeat (beat 1) is strongest. In duple meter, the beats alternate strong-weak; in triple meter, the accents are strong-weaker-weakest; and in quadruple meter, strongest-weak-strong-weak. Strong beats in a meter are heard as metric accents." Then on page 69, we say "At the beat level, metrical accents in compound meters are the same as in simple meters: compound duple (6/8) is strong-weak, compound triple (9/8) is strong-weaker-weakest, and compound quadruple (12/8) is strongest-weak-strong-weak."
Hope that helps!
Betsy Marvin, Eastman School of Music
Rick: I don't immediately think of any textbooks that theorize the distinctions you're describing (any more than Betsy and Jane's does so, at least). But, unless I'm completely misunderstanding your question, certainly GTTM offers a framework for doing so. In "common time," quarter-note tactus pulses are paired into alternating strong and weak beats, creating a half-note level of pulse; and those half-note pulses are themselves paired into strong and weak beats, creating a whole-note level. Downbeats are thus metrically strong at three different levels (whole, half, and quarter), while beat three is strong at two levels (half and quarter). Beat three is thus metrically stronger than either quarter-note beats two or four, but weaker than a downbeat.
When I'm reintroducing meter to graduate students in my review course, I always begin by throwing pulse streams on the blackboard in sync with moderate-tempo music being played simultaneously so as to illustrate the multiple levels of pulse and their interactions. I find that doing so, without any recourse to rhythmic notation at first, helps to clarify for students what time signatures do and don't show, how simple and compound meters differ, and (to your question) the additional complexity that's actually required for the ubiquitious construct of common time. (A similar comparision of metrical levels would elucidate the relative strengths of the pulses of a compound-meter tactus.)
Stan Kleppinger
University of Nebraska-Lincoln
Hi Stan, thanks for this response. Certainly music theorists have understood that there are many levels to the meter hierarchy, all the way back to John Holden in 1770 Glasgow.
Let me ask the question in a more provocative way. Every musician knows that, in a 4/4 measure, there are three kinds of beat: strong,less-strong, and weak. Musicians read in their textbooks that there are two kinds of beat, strong and weak. Some music students are smart enough and courageous enough to say ""I see three things, why are you telling me there are only two?" How do teachers anticipate, or deal with, with this kind of question?
Okay my first comment at SMT ... I'd love to see all music textbooks from tertiary level through to texts for young children address this definition and the experience associated with it. I can't say I ever taught about the experience of musical meter with students very well at all until I learned about Ski-hill graphs and the research behind this approach to teaching meter. Not sure why SHG's are not included in every music textbook along with mod cycles to represent meter...at least this way students would have the opportunity to graph (see visually) what they hear (experience) and then if questions arise about what they are experiencing the teacher can help steer the student towards tracking the pulses to discuss what happened in the music that might have triggered this choice of pulse. There are so many reasons why people choose certain pulses right? And to do this there has to be more than one pulse there in an inclusion relationship either faster or slower 'real or imagined'. I've been using SHG's successfully since 2015 in my studio with advanced musicians to beginners. And the contextual element of musical meter is really fascinating especially, for example, when the student and teacher hear different pulses because they're listening for different things...Additionally, I don't talk much about SWMW for eg. in 4/4 just teach about duple meter, the pulse relationships, why those are present for the listener and if the 3rd pulse in 4/4 is softer for the listener it's probably due to the duple pulse relationship and because it's a second projection or a further entrainment of the same pulses, so in a sense, there's less energy moving it along to the next 'accented' pulse where things take off and move forward again.
I don't teach from a textbook, and here at CMU freshmen learn about meter in eurythmics class, so I'm not worried about the level of nuance they get!
That said, when I have taught meter in freshman theory, I usually start out by showing far more than three levels of hierarchy. I play a Strauss polka that's extremely symmetrical in its phrase/formal organization, and invite them to conduct along. (I've always taught in conservatories, so there's a lot I can assume; I do show the patterns but I know most students will find them familiar.) After talking a bit about the different ways they're conducting, I play the music again, everyone conducting together, and show them how we can shift up to slower and slower beat levels. Then I show a GTTM metrical grid.
If I have time, I also do a quick demo on dynamic attending theory, to show that the beat level we follow makes a difference to how the music sounds. I play the trumpet solo from "Move" from The Birth of the Cool, and I ask them to pay attention to the ride cymbal, and ask whether or not Max Roach is swinging the 8ths. We listen at least twice, conducting the measures in two and in four - so in four following the quarter-note pulse which I think clocks in around 240 bpm. What I find, and what many students find, is either a categorical difference - swung or not swung - or else greater clarity in hearing the swinging when following the shorter pulse -- like standing closer to a painting and seeing more detail. In fact, the 8ths are swung, I measured around a 60/40 division of the quarter-note beat during the trumpet solo.
In some upper-level classes I go beyond talking about beats as possessing graded quantities of accent and introduce the idea of beats within measures or hypermeasures as having different qualia, essentially, showing them the things I do in my hypermeter article in JMT from a few years back. I"ve now also got some fun examples of manipulating heard beats at a metrical level, by the way, which hopefully will make it out of the pipeline before too long.
So whatever the level of the students, if I'm talking about beats, I'm never talking about a binary choice of strong or weak.
Rick: I think I understand your question better. To answer it, then, in a "more provocative way," I guess I'm echoing John's post: I think he and I both anticipate the question by refusing to set up a simple strong-vs-weak dichotomy in the first place. If we introduce meter as a multi-leveled phenomenon from the start, the "beat-three objection" that you're articulating doesn't ever emerge.
At the risk of seeming impertinent, I'd like to reframe your question as: Why do we/textbooks still teach meter as a single-leveled alternation of strong and weak beats when (as you point out) we've known better for a couple of centuries?
Dear Rick,
I teach meter in a totally different manner than the mainstream textbooks do. I use a simple theory of meter that is widely spread in Europe. Its roots stem outside of Leonin and Perotin's medieval mensural system. Perhaps that is Francisco Salinas's legacy from the 17th century which found a solid ground in many European countries, including Germany, and influencing the old American school of music theory in the face of Percy Goetchius. In the spring of 2012 I delivered a paper named "A New Theory of Meter: Towards Removing of Discrepancies" (at a CMS regional conference in Brownsville, Texas). Mathew Royal has a wonderful article on Francisco Salinas in MTS 34/2 Fall 2012.
According to the simple theory I was taught in, the only note values which are capable of generating a time signature are those that divide naturally in two parts: the whole, the half, the quarter, the eight and the sixteenth note. A dotted note value is considered an extended beat (compound beat) which represents a group of three basic beats. In other words, two dotted quarters in six eight time denote two simple meters of 3/8 as organic parts of the larger 6/8, rather than two basic beats. Indeed, 3 + 3 equals 6, not two (!) as some textbook massively postulate, confusing students and teachers alike! No matter how fast the tempo is, a waltz in 3/8 always has three beats, even if you conduct it in one gesture. A piece in 6/8 always has six beats, even if you conduct two beats per measure in fast tempi. Each of your conducting gestures represents a group of three basic beats corresponding to a simple 3/8 meter.
Those who claim that the dotted quarter is the basic beat in 6/8 time shall make the following experiment: let them devise a melody which only consists of dotted quarters and larger note values and give it as a dictation to their students without announcing the meter. The result will be very disappointing to the teacher, for the students will copy the melody in either 4/4 or 2/4. Why is that? Because the listeners have been deprived from hearing the basic pulse that generates the meter: the eight note.
Departing from the understanding that only basic not values generate a meter, we can classify the meters as simple and compound - not according to beat divisions - but according to the number of accented beats per measure.
Simple meters are those which have only one accented beat per measure. These are all meters whose upper number is 2 or 3 (2/1, 2/2, 2/4, 2/8, 2/16; 3/1, 3/2, 3/4, 3/8, 3/16).
Compound meters represent intrinsic combinations of two or more simple meters and therefore they have more than one accented beat per measure. These are all meters whose upper number is 4 or larger (4, 5,6,7,8,9,10,11,12,13,14,15, etc.) Thus in 4/4, which is an intrinsic combination of 2/4 and 2/4, the downbeat will be strong, and the third beat will be relatively strong, for it is the latent downbeat of the second 2/4 meter within 4/4. A six-beat meter will be a combination of two simple triple meters, and therefore the downbeat will be strong, and the fourth beat will be relatively strong, because it is the downbeat of the second group of three. 9/8 and 12/8 of classical type will have their accented beats on the beginning of every simple triple meter within the compound one.
What about irregular meters? All of them are compound and represent various combinations of simple duple and simple triple meters. It is the beginning of each simple meter within the irregular one that will take an accent. The group of three naturally receives a stronger accent than any group of two, because it is longer (like a long syllable in poetry) - even if it is not placed as first group in the measure.
By the way, there are a few simple booklets on elementary theory, written by American authors, which explain the theory of meter almost exactly as I do. They may be found in any bookstore which has a music isle. Besides, most applied faculty teach meter along the principles exposed above.
Yes, tempo alters perception. But a dotted note value never generates a meter. The medieval mensural theory does not provide a solid platform for a contemporary meter theory. No, 6/8 is never about " one-la-li, two-la-li" - it is always about "one-two-three, four-five-six" - no matter what the tempo is! In fast tempi we conduct whole groups of beats in one gesture, but this does not mean that this gesture represents the basic beat which generates the time, no! This compound beat is a group, it is a simple meter with the compound one!
In terms of fast tempo in 3/4 or 3/8, some people talk about "compound single" which is non-sense. Even the fastest waltz on the planet never has one beat - it has three. Try to dance in one, and you will lose the arsis...(ha-ha)
Last but not least. This simple theory allows a uniform interpretation of the way all time signatures are labeled. One shall bravely postulate: The top number of any naturally devised time signature (no matter simple of compound) always reveals the number of basic beats per measure, and the bottom number always reveals the true beat which generates the time signature. Knowing this, students will write dictations in 6/8 very easily, for their criterion will be the eight note, and even the most complex rhythms with dotted eight notes and syncopations will be easier to decipher. Try to do this with the dotted quarter as a point of reference...and you will enter a dark forest of confusion :)
Best regards,
Dimitar
Dr. Dimitar Ninov
Texas State University
Dear all,
To Rick's original question, I would echo Stan's invocation of GTTM. In my view, Lerdahl and Jackendoff capture elegantly 1. the intuition that many share of metrical "vocabulary" being binary by means of the dot: either a particular metrical position is strong, and thus receives a dot, or is weak, and doesn't; but also 2. relative strength of one metrical position compared with another by means of their hierarchical levels, whereby these dots accumulate vertically, and the more dots, the stronger a beat.
I'm also on board with Stan on the point that the problem may lie in representing metrical accent as single-leveled. While admittedly dot notation and notions of metrical level may be a bit abstract for a beginning theory class (then again, maybe not?), I wonder if at least the notion of metrical level may be made clear simply by taking a piece that's regular on multiple metrical levels, both hypermetrical and submetrical and speeding it up, on one hand, and slowing it down, on the other; at some point, where the tactus gets too quick, we're forced to seek a new tactus by backing down (or jumping up) to a neighboring metrical level. It's sort of like zooming in and out of the little ticks on a measuring tape: from too far away, the little ticks mean nothing to us, but from too close, they're spaced inadequately far from one another.
To Stan's objection, I'd add another (and I'm not the first to point this out): that we teach 4/4 as a thorough-going meter. Obviously, there are advantages to this from the standpoint of repertoire: there are scads of works in this meter. But if we want to eliminate theoretical noise (which GTTM does so admirably on meter), 4/4 is of course nothing other than two bars of 2/4 conceptualized from one metrical level higher. In this sense, I think Dimitar's point about conceiving any meter whose numerator exceeds 3 as compound makes plenty of sense: at a numerator of 4 or higher, we need to recourse to multiple metrical levels. Of course, it's reasonable that composers have relied on 4/4 so extensively, since that many bar-lines would be fussy.
But I also wonder if this is an indication that conventional Western metrical practice, reflected in our notational technology and our everyday vocabulary for metrical phenomena, is in fact built on the bar rather than on, say, pulse, or (more theoretically precise) time-point. The latter would be much more theoretically clean; the former, like the notion of tactus, is ready-to-hand and theoretically hazy--as London's (and others') acknowledged vagueness on the upper and lower thresholds for length of bars or speed of tactus indicates. This should be no surprise, of course: presumably our notation and vocabulary were designed in part for ease of use, not just theoretical efficiency. Anyway, I'm not quite sure what it implies if our system is bar- vs. time-point-based, but it strikes me that I haven't seen this distinction in the literature yet.
William van Geest
University of Michigan
Many 18th century theorists argued that "essentially there are two kinds of measures to which all others are related — the duple and the triple" and that "The measure of four beats is nothing but a two-beat measure doubled." (Monteclair, quoted in George Houle's Meter 1600-1800, p. 36). This argument essentially invested meter in the perception of musical sound, rather than in the meter signature. But Loulié's signature-oriented definition won out, and we've been using it ever since.
Perhaps this is too elementary a response for such a complex question, but don't the common textbook designations of duple, triple, and quadruple meters already contain an implicit differentiation of hierarchy (something William already hinted at in his note about 4/4)? Many intro textbooks (like Betsy's) not only use this classification, but also make the point of the downbeat always being the "strongest" (regardless of other potential "strong" beats in a measure). These two descriptions together seem to already create enough of a distinction to avoid confusion in many students -- quadruple meter has an extra hierarchical level of accent built into the nomenclature. Fundamentals textbooks that include conducting patterns further reinforce these distinctions through kinetic and visual representations.
I also don't put too much emphasis on any categorical distinction into "strong" vs. "weak" as absolutes. For example, from the very first time I'd talk about quadruple meter, I'd reinforce the "downbeat is the strongest" notion. (Just as an aside: I'd also go back a lot further in time to see the undermining of that categorization, perhaps even to the 14th century and Johannes de Muris's deconstruction of the longa/brevis dichotomy to allow multiple divisions across metrical levels, which also implicitly carried stress in a lot of cases.)
To my mind, the harder question to answer is from a student who wants to know about choice of time signature, e.g., whether to use 4/4 vs. 2/2, or why to use 4/4 rather than more measures of 2/4. What does a quadruple meter get you over duple in this instance? The simplistic answer for the first distinction is probably to talk about where we "feel the beat" (i.e. tactus) and point out common tempo differences between the two. But of course it's not so simple -- and why could we "feel the beat" in some pieces at half note=60, but in other pieces at quarter note=120? To my mind, the traditional notational differences then have to do with consistency of metrical hierarchy, which is often shaped by a lot of rhetorical elements (placement of cadences and strong harmonic resolutions, textural and orchestration choices that reinforce different metric levels, and within the "rhythm" proper: use of agogic accents, consistent subdivisions, etc.). That's probably too much for a fundamentals class audience, but I do bring this up in upper-level classes when we talk about what can create a feeling of hypermeter or disrupt it, as well as instances of notated meter that conflicts with aural perception.
Yes, and if you mean that we typically use 4/4 instead of (2/4 + 2/4) or 6/4 instead of (3/4 + 3/4), or 6/8 instead of (3/8 + 3/8), I agree. Practically, however, we must be aware of two important principles:
1. The basic metrical unit which generates a meter is a note value which naturally divides in two parts. It alone can create the impression of any meter, while a dotted note value cannot do this by itself. Example: you can express 4/4 by playing quarter notes; you can express 6/8 by playing eight notes but you can never express 6/8 by playing dotted quarter notes. This is the proof that a dotted value is a result of grouping of basic beats rather than a basic beat itself. If it ever were a basic beat, it alone would be capable of fully outlining meters whose numerator is 6.
2. If you play in the same tempo the following series of meters: 3/8, 4/8, 5/8, 6/8 and 7/8, for example - you will realize that there is no difference in the perception of the basic unit which is the eight note. That 3/8 is a simple meter and the rest are compound (including 4/8) does not affect the manner in which the mind perceives the single pulse. This is why the statement "3/8 has three beats, while 6/8 has two beats" cannot be more erroneous (I will spare a harsher word, although it deserves). Indeed, 3+3 never equals 2, even in music!
Some theorists try to "reconcile" this discrepancy by saying: "well, in a slow tempo 6/4 and 6/8 each have six beats, but in a fast tempo they have only two beats per measure." If I used their logic, I could claim that "in a slow tempo 3/4 and 3/8 each have three beats but in a fast tempo they only have one beat per measure". Does anyone believe there is a naturally devised key signature which only has a thesis and has no arsis? Does any one dance a waltz "in one" because of a fast tempo?
The meter signature is true as it exists now and, according to it, there is no difference between the meaning of a numerator in a simple meter and the meaning of numerator in a compound meter. In both cases, that is the true number of beats which generate the respective meter. As for the denominator - it is always the true basic beat in both simple and compound meters. It is another matter that when you combine time signatures with different denominators (such as 4/4 and 6/8, for example) it is practically preferable to think at the level of the smaller denominator (the eight note) so that you link these meters smoothly in the same tempo.
Dimitar
The question "do you feel the beat as quarter = 120 or half = 60" is equivalent to the question of whether Lite Beer tastes great or is less filling. It sets up a false opposition. Meter requires at least two beats. Why does one of these need to the "the beat?" Yes, a composer might choose one through which to communicate the tempo. So too does the map maker decide to wrote "1 inch = 100 miles" or "1/2 imch = 50 miles."
Rick et al... With all due respect... Seriously.... As the question is framed and as this discussion is going – as if everyone knows what they are talking about
..... THIS IS THE ISSUE????
It's like assuming a universal diatonic by pointing to the white notes & black notes on a piano & triumphantly declaring QED... and then teaching it, thereby reinforcing it for the next generation.
I want to curl up and die.
Stephen unless I have misunderstood your comment would you agree that is it about time music textbooks for all ages and levels were re-written to include a re-definition of meter? One that not only explains what meter is but how to teach it according to the 'decades of research mainly from North America' that prove 'what meter is and where meter exists'? To do this, though, music theory teaching resources, tools, language and approaches to teaching meter would also need to reflect this new definition of meter in practice (which is precisely what is happening)... Hence my interest in Ski-hill graphs and mod cycles, which, when used according to this approach, reflect the listener's temporal experience of meter in a graphic representation...'according to this approach' also refers to 'not telling the student how to hear' because it's physically impossible to hear for someone else :)
Thank you, Andrea! I most likely agree with what you say & I should have replied in support of your previous post. (I would delete your word 'prove' here - but you may have intended it ironically.) I don't have experience with 'Ski-hill graphs' in the specific sense you seem to intend (and would like to know more) but the term is suggestive enough that I think I get the gist and, if I do understand what you mean by it, would agree with it. FWIW, I consider a 'time signature' less a structural clue than a performance cue - how to keep things together on stage. This is not to say (absurdly) that many works do not have a fundamental repetitive metric structure, but only that teaching 'one-e-&-a' or whatever as THE basis for undertanding rhythmic (and deeper) structure (quite different from 'metric') - which I read here but which others will certainly deny - is going to keep students forever at an elementary analytical level. Since people here are talking about undergrad level, not about elementary school, I find this whole discussion really disturbing.
Thanks Stephen. I find the discussion fascinating! But there is a great deal of 'inertia' out there where teaching meter is concerned. What strikes me as fundamental is that many music teachers aren't aware or at least don't teach that music can be 'written in a number of different ways to be made to sound the same' and that the time signature and barlines are in a sense 'arbitrary.'
@RichardCohn
Rick, I absolutely agree that to some degree the selection of tactus level is arbitrary. Which is why I framed my statement about 2/2 vs. 4/4 about the "traditional notational differences," not as some abstract idea of meter. There are, as I'm sure you're well aware, compositional traditions and conventions that dictate when to use different time signatures which might be "equivalent" in some abstract sense (and which sometimes might affect preferred tactus level). And when introducing meter to students in fundamentals (which was the initial topic of this thread), part of our pedagogical responsibility to teach conventions of notation. Whether you or others think the question of tactus level is just a matter of arbitrary definition (or compositional discretion), I do think that there are frequently conventional stylistic issues that might be helpful to discuss with students when making these notational distinctions.
In any case, I think this discussion seems to be going in a few different ways. Some arguments are about the perception of meter (regardless of notation); others are trying to deal with interpretation of notation. As Stephen pointed out in his most recent post, a time signature is mostly a "performance cue," which I completely agree with. But the original question, as framed, is about teaching. I'm not sure how -- in that context -- we can set aside the pedagogical responsibility of teaching the conventional associations and usage of these performance cues for standard notation, regardless of whatever abstract theory of meter we may think is best or what might be a better model of perception for meter.
I'm all in favor of enhancing our discussions of abstract meter and even completely rewriting intro textbooks (as some have suggested), but unless we're also going to jettison conventional notation entirely (and perhaps a large section of the historical canon, which made use of these conventions), we still have to deal with practical questions like time signature choices. And one thing I disagree with in Stephen's post is the opposition of "performance cue" with "structural cue." It implicitly continues this opposition between "analysis" and performance that I don't believe in. Time signatures DO things to performance. Barlines DO things to performance. And by affecting performance, they affect the piece as played (even, as I frequently teach in my upper-level classes, when there's a clear conflict between written and perceived meter, which creates a deliberate tension that will affect the performer). How is that not "structural" to the piece? I disagree with Andrea's assertion that such distinctions are "arbitrary," because they can certainly impact performance choices (both conscious and unconscious). Just because Finale or Sibelius would play "equivalent" notations as if they sounded precisely the same doesn't mean that human performers do.
It seems to me we can have a discussion about practical difficulties encountered in teaching standard meter and the conventions of music notation that most beginning students want to learn in introductory classes, or we could have a discussion about how to drop standard metric notation and teach something else (which might actually be a better theory of meter as perceived or whatever). But a lot of discussion here seems to be people talking past one another because people are trying to do both.
John & Rick,
My fault for trying to be too clever by playing 'performance cue' against 'structural cLue', further obfuscating a subtle concept. Reading in the 'L' may show that I am close to what John replied. The way I approach 'meter' (a term we're stuck with, implying it's a 'thing' although it's actually, IMO, a 'temporal scaling' analogous to 'pitch scaling') is as the time coordinate in music's pitch-time coordinate system. [This turns notation into a board game & it might be taught that way - but another time on that.] I agree that different & not always compatible rhythmic things are possible - both with respect to performance and structure (interpretation & analysis?) - depending on whether the underlying time coordinate is set to 9/16 or 3/8. Certainly 3/4 & 6/8 also can suggest identical or similar or incompatibly distinct structures depending on 'how it all goes along' (composition, performance, hearing). And of course 2/2 & 4/4 are different meters, but so close that it's conceivable that many here have come across an adagio in 2/2 where, while trying to keep 2/2 in mind to discover & express the structure, had to mentally or actually cross out the signature & think 'in 4/4' for performance. But complete interchangeability based on common divisors shows itself as absurd as soon as you contemplate crossing out 8/8 and replacing it with the "'identical'" signature 1/1 --- try performing and/or analyzing that! (And yes, I too have been frustrated by the notation software demand of declaring a time signature before I can do anything -- we're the victims of the success of teaching our assumptions.)
Perhaps I come off as clueless in this conversation because when I see 'pedagogy' as an over-arching topic I don't linger over how to teach 'fundamentals' -- I immediately think about what might be the larger consequences of this or that approach for the individual student and for possible future musics. Is the student given the means to escape the starter-box s/he is necessarily put into? Or will s/he forever see that box as 'the way things are'? So to judge any approach I need to see where it will likely lead.
Admittedly, on this particular topic, I may be tempted even more than usual to throw monkey wrenches into the status quo machine due to being immersed in Messiaen right now. An interesting question for your students: what happens if there are no time signatures, and barlines seem to be placed at random? -- Does that mean or imply that there is no 'pulse'? And then: what is pulse?
@John McKay just to clarify my expression was 'in a sense arbitrary' akin to "quasi-arbitrary". The notation can only 'give us clues as to meter'. As an educator my interest is in historically informed theory (HIT) and in our era we are privileged to have research in music psychology and music theory to draw on which can inform us about meter in theory and practice. This influenced my response to Rick's question.
@S_Soderberg,
I completely agree with your concern about not "boxing students in" with inferior pedagogical choices at the start. I guess I'm also not quite sure how to try to answer otherwise when this question seems about extant textbook pedagogy and an absolute stressed/unstressed dichotomy on a singular metric level that I've never actually encountered in teaching. And most fundamentals textbooks I've actually encountered don't seem to present things that way either. So I've actually never seen a student confused about Rick's question.
An explanation like Betsy's is generally sufficient, with a little reinforcement, to make students realize there are various degrees of stress. Also, from the very beginning of when I introduce the notion of stress in fundamentals, I point out the various ways it can be created other than metric patterning -- articulations, agogic accents, accents produced by melodic or harmonic structure, etc. When these various types of accents line up, other stresses can be enhanced or undermined. I frequently bring in an example of "unmetered" music to get discussion going on that topic. So no student would ever come away from a class with me thinking there weren't gradations of "stress" of every shade of the rainbow, which can be strengthened or weakened by all sorts of musical contextual elements.
On the other hand, I have had students confused about why to notate "equivalent" rhythms in duple vs. quadruple meter, and other practical notational issues, which perhaps gets at the hierarchical problems discussed in the question.
I'm all for trying to present a more nuanced and open perspective that doesn't reinforce bad habits from the start. But we're also really a bit hampered in this if part of the goal is to train students to read and interpret existing music with all of its (bad) assumptions about meter that are built into the notation.
@Andrea ,
Thanks for the reply, and I apologize if I interpreted your statement a bit too strongly; you did indeed put it in quotation marks. And I should have mentioned that I'm all for using different visual representations of meter as you discussed too. I hadn't actually considered using ski-hill graphs in intro classes, but perhaps it could be helpful in parsing hemiola and other metric irregularities.
@ John McKay No problem! I'm sure your students will adapt to Ski-hill graphs very well especially as the pathways will reflect their own hearing of the pulses for the meters they hear. It's great you are so open to new ideas. It may surprise your students how often the 'conflicts' occur with pulses 'directly' and 'indirectly' in all sorts of music.
Isn't it time to introduce Schenker in the discussion? Chapter 4 of Free Composition, on meter and rhythm, is so short and so difficult to understand that we tend to forget it. But what it says (also between the lines) is important. Schenker begins (§ 284) stating that
This, at first reading, is not a very impressive statement about the difference between meter and rhythm. The important point, however, is the comparison with language. This had been a primary (albeit often veiled) concern of Schenker since Der Geist der musikalischen Technik in 1895. And his purpose was not to show the similitudes between language and music, but the differences. He continues:
and soon comes to what he considers (since Der Geist) the distinctive mark of music, repetition. He writes (§ 285):
Meter does not arise from the opposition between strong and weak beats (of which Schenker never speaks), but from their repetition. It strikes me that repetition hardly was mentioned in the present discussion (Stephen wrote a day ago of the "repetitive metric structure"). Schenker refers to his Harmonielehre, § 4, where he speaks of "Repetition as the underlying principle of the motif". Because meter is repetitive, it also is motivic. Meter and rhythm arise, through diminutions, from the utter ametricity and arhythmicity of the background. Schenkers theory of meter, like his theory of form, is a transformational theory.
I don't know whether any of this can be useful in undergraduate teaching, but it certainly is worth some consideration.
Rhythm requires a unit of sound duration. How that duration is utilized, subdivided, patterned, is wholly dependent on context of music style, period, culture, and language. Rhythm has mathematical properties and measurement of infinite groupings. When discussing rhythm with students I prefer to focus on particular musical examples and talk about what the composer has done with the rhythmic elements: motives, figures, sequences, intensification, evaporation, rhythmic groupings symmetrical and non symmetrical, reordering, displacement, the use of space and silence, shapes, repetition, etc. There are four primary predicaments for rhythmic events common to all music - before the beat, on the beat, after the beat, and the absence of sound (which still requires duration) - silence.
@Nicolas
I just remembered you once mentioned having taken a theory course with Messiaen. If so, was it his course on rhythm? And, if so, how did he teach it -- historical/survey or related to his own theories on rhythm?
@S_Soderberg No, Stephen, I never took any course with Messiaen. From what I heard of his courses, he never taught historical surveys; but this is only hearsay, in fact I don't know.
Posted on behalf of @RichardCohn
John, there's no question that music theorists can help students to interpret meter signatures according to the principles of 18th-century tempo giusto, and that these lessons will make them more skillful performers for the Saturday-night dances of perruqued and hoop-skirted courtiers, and the Sunday-morning settings of pietist liturgical texts. But let's address the question of whether what we are teaching them is "a theory of musical meter" as part of "a theory of music."
We are teaching students to interpret "meter signatures" which include some information about: meter; tempo; articulation; microtiming; place of performance; social class.
We are teaching not "music" full stop, but "music of 18th century court and chapel."
And we are teaching the history of musical notation and performance practice. We not teaching a theory of musical meter.
The question "should we teach meter notation, or should we teach a theory of meter?" is another false opposition. Just like the question "should we teach music theory OR teach music history?" Can't we do both?
--Rick
smtdiscuss@societymusictheory.org
Somewhere in the Universe
How does Japanese court music organize its rhythmical structure? How do African cultures arrange their poly rhythms? How does Haydn use intensification and reduction to control movement and heighten emotion in his music? How do Indian musicians use much longer rhythmic cycles of beats to accent and place complex groupings? How does jazz swing time and syncopation differ from classical meter? I think my perspective is that we can teach rhythm/meter/pulse in a much broader context and view it as "elements of human music." The Japanese have a rhythm not found in the West that is essentially what happens when you drop a rubber ball and each bounce occurs rhythmically faster than the previous - a phenomena that occurs in nature. I prefer a more global approach to music and find it more useful as a composer and thinker.
At least there is an overtone system in nature that is as constant as gravity Stephen! A string vibrates the same no matter where you make pluck it or bow it in the universe. However that might change near the event horizon of a black hole! Why I consider myself a Pythagorian and a Buddhist - you can't fool mother nature.
I don't know if this is helpful at this point. I was looking through my music library and came across "Structural Functions In Music" by Wallace Berry, Dover. Chapter three is "rhythm and meter." Quite an extensive examination of the subject with well over 120 pages dedicated to the subject. I intend to read through it myself.
The bottom line is that dotted note values cannot generate a meter. If a beat is genuine, it is capable of generating a meter without the help of smaller or larger note values. This privilege belongs only to simple note values which divide naturally in two parts: the whole, half, quarter, and sixteenth note. Try to generate a 6/8 meter with dotted quarters and you will fail. That should serve you as empirical evidence that dotted notes represent groups of basic beats - they are not basic beats themselves. Rick, no matter what tempo you perform in, a waltz will always have three beats, and a meter of 6/8 will always have six beats. When you are tapping "in one" for the waltz, or "in two" for the 6/8 meter, you are taking groups of three beats in one gesture. I hope this is understood by all colleagues who teach meter. I also hope that the sentence: "The numerator in compound meters shows the number of the subdivisions, and the denominator shows the value of the subdivision" is erroneous and should not be used in professional sources anymore. The numerator always shows the number of the true beats, and the denominator reveals the true beat which generates the meter, regardless of the tempo.
Just one last comment.
Beethoven, Symphony 9, Scherzo.
Molto vivace 3/4 with 1 measure (dotted half) = MM116
1) Obviously, this is not 'in 3' by the time signature Beethoven chose... no conductor would beat 3 to a bar (but it would be fun to watch someone try) and no one I know of would try to hear it that way.
2) At the beginning: either in-2 or in-4 are 'super meter' possibilities, but 4 is certainly Beethoven's intention by the music's phrasing & gestures (also, see 5 below for Beethoven's own words )
3) mm151-176 (harmonically a long Riemannian sequence first pointed out by @RichardCohn) the meter begins to break up a little bit although still square (2 &/or 4).
4) At m177 Beethoven writes 'Ritmo di tre battute' in the score, emphasizing the (super)meter is now in-3.
5) At m234 Beethoven writes 'Ritmo di quattro battute - (super)meter returns to in-4.
6) m416 begins a stringendo leading to the Presto (m424) in cut time with half-note = MM116.
Regarding dotted notes not being able to generate meter, what about in situations of metric modulation?
Dear Colleagues,
Beating and tapping some times do not have anything to do with the true pulsation of the meter; you all know that - you may beat in one and have three real beats per measure; or you may beat in two and have six beats per measure.
In metric modulation, you find the smallest denominator which allows you to navigate throughout all meters with no change of tempo. If you have to switch from 4/4 to 6/8, you will temporarily think of the eight note as a basis. You may still conduct 4/4 in four and 6/8 in two, but your attention will be attached to the eight note pulsation which links the meters fluently.
In a previous post I mentioned I've been teaching musical meter using modern meter theory through instruments of meter theory such as the ski-hill graph and cyclic graph, to equip my students to articulate their experience of meter. I discuss this here: 'Teaching Musical Meter to School-Age Students Through The Ski-Hill Graph' http://hdl.handle.net/2123/19791 My students find that representing meter through visualizations and sonifications using mathematical music theory assists them to articulate their experience of meter more accurately which results in a more satisfying performance experience. I'd love any feedback about my recent thesis and to continue discussing how you approach teaching meter.