Timing and the Pedagogy and Notation of the Meter

Recently I presented two seminars on timing and the pedagogy and notation of the meter to two groups of colleagues in pedagogy and performance, where I discussed the three-step psychoacoustic process I developed to teach the meter, rhythm, pitch, and harmony through mathematical music theory. First, I focused on the meter by visualising the beat-class to express temporal or spatial relationships through ski-hill, circular and linear, cyclic graphs. The process provides a viable solution for teaching and notating meter, performing music with expression, annotating scores, and analysing music on a level playing field through mathematical music theory.

The papers began with an overview of what the meter is and why the definition of meter has changed and provided examples of music in both audio and notation to demonstrate traditional music theory and new developments in meter theory applied. For instance, the combination of duple meter and tresillo rhythm initiated by Burt Bacharach and Hal David’s ‘I Say a Little Prayer' famously sung by Aretha Franklin can now be compared with traditional indigenous music such as the Igbo tribe of Nigeria with evidence of both the tresillo and duple meter.

My seminars provided a discussion about the background to the present confusion in music pedagogy about the meter; I proceeded to refer to what is probably the world’s first survey and report on the pedagogy and understanding of meter Understanding and Teaching Meter Survey, the Understanding and Teaching Meter Survey Report (Calilhanna, 2017) and the article Teaching time: A survey of music educators’ approaches to the meter (Calilhanna and Webb, 2018 unpublished).

The survey report and article revealed a fascinating mix of not too surprising results. For instance, fascinating because most music educators know there are problems with the pedagogy and notation of meter. For example, several respondents expressed difficulties their students experience where the “fractional” system of notating time signatures is concerned. Yet close to half of all participants expressed their understanding of meter as score-based. Results such as these can be identified as prominent themes in the current discussions about what meter is, where the meter is and what to do about meter pedagogy and notation.

Thankfully, an increasing number of music educators are asking deeper questions that, due to modern meter theory, can now be answered, such as:

Question. Where is the meter - in the notation or the listener?

Answer.  In the listener.

Question. Are meter signatures accurate representations of the meter?

Answer.  They only have the capacity to represent one pulse. The theory surrounding this notational idea is critical to understand for accurate pedagogy.

Yet how many of us really understand deeply what time signatures represent?

For instance, many of us as children and throughout our music education learned the meter as the notated time signature, such as, 3/4 which meant there were three crotchets in a measure. In 6/8, the notation has two dotted crotchets and six eighth notes or quavers per measure. We were not taught to think about what a measure IS or whether a time signature was a fraction, as those who wondered about time signatures as fractions found that idea problematic. We weren’t taught to think deeply about the time signature beyond understanding that the bottom number, such as the four in 4/4, represented crotchets or quarter notes. The number on top told the reader how many quarters notes to read in a notated measure. The traditional notation-based approach to music education was not to question the numbers even if they didn’t represent the note values and didn’t make sense. Questioning could mean the teacher was being questioned, and likely the teacher didn’t have the answer. Questioning students were sometimes made to feel uncooperative, and worst of all, unmusical.

Even though most of the world's music is not notated, notation-based music theory is applied to analyse the same music. Unfortunately, music theory until recently had not caught up with developments in music performance and composition, particularly where the influence of music other than Western flourishes and influences. Yet now that the meter has been redefined and based on recent research in cognitive science about what the meter is and where the meter is located, “A meter is an inclusionary related set of distinct, notionally isochronous time-point sets” (Cohn, 2020), our job is to teach students accurate information and best practices.

As an example, rhythm and meter are not the same things. Rhythms are often studied without context to the intrinsic relationship with the metric structure. Why is this important? Analysing rhythm ignores the structures that they initiate, and this can lead to oversimplification. Important patterns and structures can be overlooked, which can unfairly disadvantage the representation of music such as traditional indigenous music that is not notated. Unlike time signatures, visual representations of meter make this possible. All the pulses and meters initiated by rhythms can be studied with the metric hierarchy through ski-hill graphs:

Unlike time signatures, ski-hill graphs have the capacity to represent all pulses and meters a listener experiences. Each pair of pulses form a minimal meter. Minimal meters are either duple meter 2:1 or triple meter 3:1. The minimal meter is a relation between two pulses of different speeds, the slower of which is included in, the faster.

Each fraction (a division of time) represents a pulse. Edges connect pulses to represent the relationship of ALL pulses and meters as part of the whole metric space. In total, the mapping forms the metric space.

The horizontal red line indicates a hemiola: the pulses are related in a ratio of 2:3, i.e., the division is uneven. Therefore, Hemiolas cannot be accurately represented with time signatures.

Transitioning to music theory which is not notation-based, is a major turning point in music theory and music theory pedagogy history, and teachers need new music theory and accurate visual representations of the meter. That is the visual representation of music in space experienced through the psychoacoustic mind and body process of embodied acoustics.

I’ll share some of the points from my seminar where I discussed how we could teach modern meter theory:

+   Visualise all pulses and all meters through ski-hill graphs.

+   Teach with a psychoacoustic approach to music theory.

+   Teach with a psychoacoustic approach to mathematical music theory.

+   Terms such as simple and compound are being phased out because they don’t represent the listener’s music experience or any clear pattern.

+   Quadruple is being phased out because there is an intermediary pulse: a minim between, for example, a semibreve and crotchet that the listener can detect.

+   Beat: a) A point in time; b) A rhythm has beats; c) Isochronous beats.

+   Pulse: A set of time points. A meter has pulses.

+   Teachers are tired of telling their students the lies they were taught to represent meter with inadequate meter theory – teachers want a theory that makes sense and is true.

+   Teachers want a theory to apply to all music.

+   Teachers need a theory that students find engaging because it helps them reach their goals and encourages independent learning.

+   Learning meter is dynamic listening and requires suitable instruments of mathematical music theory.

 In my next blog post, I will discuss dynamic listening, the three-step psychoacoustic approach to meter pedagogy and notation I developed and why the ski-hill graph offers a portal into new meter theory.