Ski-hill Graph Pedagogy Meter Fundamentals: Mathematical Music Theory for Beginners

By Andrea M. Calilhanna

This book shows teachers and students how meter fundamentals are taught through Ski-hill Graph Pedagogy, the three-step psychoacoustic mathematical music theory approach developed by music educator-researcher Andrea M. Calilhanna, inspired by contemporary meter theory of Battell Professor of the Theory of Music, Yale University, Richard Cohn.

The ski-hill graph enables students to visually represent meter fundamentals mathematics through a soundbased approach experienced from listening to music in the first lessons!

Students taught the meter as time signatures and beats grouped in measures understand meter as the notation. However, the ski-hill graph is a solution for understanding meter because music is acoustics (sound) and listening is central to Cohn’s sound-based theories.

To apply accurate meter mathematics from the ski-hill graph to music preparation means students save time later in rehearsals from a solid start to decode their work. Visualising meter through the ski-hill layout as a summary of all pulses and all meters from listening assists students to understand their meter experiences and its mathematical aspects.

Students listen, clap, tap and map with mathematics: meter beat-class, first through the ski-hill, then they apply the ski-hill mathematics to annotate, practice and compose music through other representations such as linear and circle graphs. In this way, students not only become aware of new information, but they also understand their new knowledge. Knowing and understanding mathematical elements of meter means the theory can apply to performance to improve timing, inform expression, sight-reading and much more!

Without skills to analyse meter from listening to music, many important details are left out because they are hidden by notation-based understandings of music analysis. Cohn’s theories of meter, however, offer solutions to understand each pulse and meter as cycles to decode music performed and listened to. The book works through small cycles to grow listeners’ awareness of mathematical aspects of meter: mathematical music theory.

The Ski-hill Graph Pedagogy approach provides students with several benefits for meter fundamentals pedagogy, including development of mathematical knowledge and practical skills to understand musical timing and expression, and increased performance confidence through more secure performances from critical thinking and metacognitive processes.

Ski-hill Graph Pedagogy is suitable for most teaching styles, and provides inclusive, ethical music theory for diverse music education. Suitable for teaching meter fundamentals with students of all ages.

• Language: English
• Publisher: Xlibris AU
• Release date: Jan 3, 2024
• ISBN: 9798369490075

Andrea M. Calilhanna

Andrea M. Calilhanna, MMus; GradDipT (Mus); DipMus; LMusA; AMusA; MMTA; FCSME; MAAS; PhD Candidate, the University of Adelaide. Andrea teaches a three-step psychoacoustic approach inspired by the music theories of Richard Cohn (Yale). Her primary research interest is the pedagogy of the fundamentals of meter and rhythm concerning the Ski-hill graph. She has taught music since 1991 to today in both classroom and music studio settings and her students are beginners ages five years to undergraduate students. Andrea is a member of several professional organisations and presents papers at conferences locally, interstate and overseas and has several publications.

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Copyright © 2023 by Andrea M. Calilhanna. 846284

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Cover illustration features Stereo Metronome (Fan & Calilhanna, 2021, beta) an Android app, digitised version of the ski-hill graph including, ski-hill, circle, and linear views, and other features (illustrations throughout the book).

Library of Congress Control Number: 2023903526

Rev. date:   03/08/2024

Contents

List of Figures

List of Tables

Preface

Acknowledgements

Introduction

What is a Meter and Why does Meter Matter?

The Ski-hill Graph and Meter

Ski-Hill Graph Pedagogy

The Benefits of Ski-Hill Graph Pedagogy

Stereo Metronome

The Two Types of Meters: Duple Meter and Triple Meter

Duple Meter and C2

Minimal Duple Meter C2 Lesson

Deep Pure Duple Meter C4

Deep Pure Duple Meter C4 Lesson

Deep Duple Meter C8

Triple Meter C3

Minimal Triple Meter C3 Lesson

Deep Triple Meter C9

Mixed Meter C6 and C12

C6 Polymeter and the 2:3 Hemiola

C8 Additive Meter and Tresillo

Swing Quavers C12

Epilogue

Glossary

Bibliography and Further Reading

About the Author

List of Figures

Figure 1 Ski-hill graphs (a) triple meter; (b) duple meter; (c) duple and triple meters

Figure 2 Stereo Metronome (Fan, S. and Calilhanna, A., 2021. Beta version)

Figure 3 Stereo Metronome

Figure 4 Ski-hill graph: Minimal duple meter C2 <2> 2:1

Figure 5 Ski-hill graph: Minimal duple meter C2 one level of duple meter

Figure 6 Ski-hill graph: Deep pure duple meter C8 <2, 2, 2> or <2>

Figure 7 Twinkle, Twinkle, Little Star mm. 1-4

Figure 8 Ski-hill graph: Minimal duple meter 2:1 C2 <2> and beat-class

Figure 9 Circle graph: C2 2:1 <2> minimal duple meter beat-class polygons

Figure 10 Circle graph: C2 minimal duple meter beat-class polygons, and fractions

Figure 11 Twinkle, Twinkle, Little Star (mm. 1–4) with accompaniment

Figure 12 Ski-hill graph: Duple meter C4 three pulses and two metric levels deep

Figure 13 Twinkle, Twinkle, Little Star (Unaccompanied, mm. 1–4)

Figure 14 Linear graph: C4 duple meter beat-class volumes (annotations)

Figure 15 Twinkle, Twinkle, Little Star (mm. 1–4) C4 beat-class and volumes

Figure 16 Twinkle, Twinkle, Little Star annotations

Figure 17 Circle graph: C4 duple meter beat-class polygons

Figure 18 Circle Graphs (a)–(f)

Figure 19 Ski-hill graph: Deep duple meter C8 <2>

Figure 20 Linear graph: C8 duple meter beat-class and volumes (timepoints)

Figure 21 Twinkle, Twinkle, Little Star (mm. 1–4) C8 beat-class and volumes

Figure 22 Circle graph: C8 pure duple meter beat-class and polygons

Figure 23 Circle graph: C8 pure duple meter

Figure 24 Ski-hill graph: Duple meter C8 including 2- and 4-measure hyperpulses

Figure 25 Linear graph: C8 meter cycle - five pulses and four levels of duple meter

Figure 26 Circle graph: C8 duple meter beat-class and polygons

Figure 27 Ski-hill graph: Minimal triple meter C3 <3>

Figure 28 Ski-hill graph: Minimal triple meter: C3 <3>

Figure 29 Ringing the Bells (Calilhanna)

Figure 30 Ski-hill graph: Minimal triple meter and beat-class.

Figure 31 Ringing the Bells (Calilhanna) annotated with beat-class and volumes

Figure 32 Circle graph: C3 minimal duple meter 3:1 <3> as beat-class and polygons

Figure 33 Circle graph: C3 minimal triple meter 3:1 <3>

Figure 34 C9 or <3, 3>

Figure 35 C9 or <2, 3, 3>

Figure 36 Happy Birthday (arranged by Andrea Calilhanna) ski-hill annotation.

Figure 37 Ringing the Bells (Calilhanna)

Figure 38 Ski-hill graph: Mixed meter: C6 or <3, 2>

Figure 39 Circle graph: C6 mixed meter <3, 2> ski-hill meter beat-class and polygons

Figure 40 Linear graph: C6 <3, 2> mixed meter cycle

Figure 41 Mixed meter: C6 or <2, 3> includes 2-measure hyperpulse.

Figure 42 Linear graph annotation of C6 <2, 3>, beat-class counting, and volumes

Figure 43 Circle graph: C6 or <2, 3>

Figure 44 Ski-hill graph C6 <2, 2, 3, 2>

Figure 45 Annotation of Ringing the Bells (Calilhanna)

Figure 46 Mixed meter: C12 or <2, 2, 3, 2> includes 2- and 4-measure hyperpulses.

Figure 47 Linear graph annotation of Ringing the Bells (Calilhanna) C12

Figure 48 Ski-hill graph: simple hemiola 2:3 C6 or <2, 3> and <3, 2> fractions.

Figure 49 Ski-hill graph: simple hemiola 2:3 C6 or <2, 3> and <3, 2> staff notation

Figure 50 Linear graph: simple hemiola 2:3 C6 or <2, 3>and <3, 2> beat-class

Figure 51 Linear graph: simple hemiola 2:3 C6 or <2, 3> and <3, 2> beat-class

Figure 52 Circle graph: simple hemiola 2:3 C6 or <2, 3> and <3, 2>

Figure 53 Linear graph: C8, d3 {0, 3, 6} tresillo

Figure 54 Ski-hill graph: C8 Deep duple meter

Figure 55 Ski-hill graph: Mixed meter and simple hemiola 3:2

Figure 56 Linear graph: tresillo meter beat-class annotation

Figure 57 Circle graph: C8 duple meter

Figure 58 Circle graph: C8 duple meter and tresillo {0, 3, 6}

Figure 59 Swing quavers C12 exercise

Figure 60 Ski-hill graph: C12 <2, 2, 3> fractions

Figure 61 Linear graph: C12

Figure 62 Ski-hill meter mathematics annotation Swing quavers

Figure 63 C12 <2, 2, 3>

Figure 64 Circle graph C12 <2, 2, 3> rhythm practice

List of Tables

Table 1: Metric Equivalence C2 <2>

Table 2: Metric Equivalence C4 <2>; <2, 2>

Table 3: Metric Equivalence C8 <2>; <2, 2, 2>

Table 4: Metric Equivalence C3 <3>

Table 5: Metric Equivalence C6 <3, 2>

Table 6: Metric Equivalence C6 <2, 3>

Table 7: Metric Equivalence C12 <2, 2, 3, 2>

Table 8: Metric Equivalence C12 <2, 2, 3>

Ski-hill Graph Pedagogy Meter Fundamentals

Mathematical Music Theory for Beginners

001_a_aa.jpeg

Five-year-old student mapping the meter through a

ski-hill graph at Cherrybrook Music Studio

Preface

Ski-hill Graph Pedagogy is a three-step listening approach for teaching meter fundamentals, inspired by the contemporary meter theorist, Richard Cohn, Battell Professor of the Theory of Music at Yale University.¹ Ski-hill graphs information of meter engages students in processes learning meter because ski-hill meter mapping enables them to acknowledge mathematical meter details to observe, relate, and apply to their performances. From listening to music, students first clap each pulse, tap meters, and map visualisations of all pulses and each meter through the ski-hill graph layout.

Meter fundamentals through ski-hill graphs provides students listening skills foundations with information and knowledge for performance timing and expression and much more!

Ski-hill Graph Pedagogy to teach meter fundamentals, adopts Cohn’s meter theory, originally designed for university students, and adapts it as curricula suitable for beginners.

The idea for Ski-hill Graph Pedagogy to apply Cohn’s meter theories with school-age students for meter fundamentals began during my studies in 2015, at the Sydney Conservatorium of Music, The University of Sydney, where I attended Cohn’s lectures on musical Meter. The class focus on listening to music to represent all pulses and meters mapped through ski-hill meter pathways to visualise meter, challenged conventional time signatures’ theories and augmented untheoretical descriptions of meter as notation. Including listening also means to acknowledge subjective and psychoacoustic listening experiences of duple meter 2:1 and triple meter 3:1 mathematical aspects of meter.

Ski-hill graph mapping to represent meter opened my eyes and ears! From ski-hill graphs I learned meter is not notation. Listening is not part of meter signatures’ requirements, conventional meter theory through time signatures is notation-based and describes a notation of a beat grouped in a measure. The top number, or numerator, is meant to ‘tell’ you ‘how many beats’ there are in each measure, and the number below, the denominator ‘what kind of notes’ are notated as a group. Yet, beginners know a time signature doesn’t represent all the pulses and meter they experience, and their experiences of meter don’t match the meter of time signatures. Cohn’s meter theory helped me to understand how problematic time signatures are to represent meter, how they provide little helpful information for performance, and how ski-hill graphs provide solutions!

Cohn’s classes taught there are perceptual gaps experienced learning meter as time signatures. Time signatures disconnect meter experiences of listening from the representation of meter: two numbers cannot represent all pulses and every meter in relation to each other. However, ski-hill graphs connect listening experiences and meter representation to assist students to solve their timing problems. Visualising the meter mathematical relations through ski-hill graphs provides students new information and knowledge to apply to their performance preparation.

Time signature meter theories result in serious implications by denying student agency for their own listening, cultural heritage, and best teaching practices. From as early as childhood beginners learn to believe meter signatures are the meter; since having represented meter through ski-hill graphs I understand how limited time signatures’ theories are to represent meter. Teaching meter as signatures means students learn to dismiss rather than acknowledge their responses to music and instead are to believe meter signatures are meter. Ski-hill graphs are a bridge between experiences of music and acoustics, STEM (Science, Technology, Engineering, and Mathematics) and Arts, and provide a music theory tool listeners access. Ski-hill graphs meter representations of listener’s meter information of mathematical aspects of meter is accessible to study, explain, and apply to performance, interdisciplinary research, curation and more – a tangible representation of knowledge and data.

Cohn’s meter theory was a revelation to me as I had always looked to time signatures in classes, and beyond the information about notation thought, that’s all there is to know and teach. Since learning how to represent the meter through ski-hill graphs, I haven’t taught students meter as time signatures, and I now realise why meter is so ad hoc.

Ski-hill Graph Pedagogy explains how to teach meter fundamentals from listening.

When I first introduced ski-hill graphs with students, my object was to test whether the ski-hill would assist students to solve problems requiring stabilizing timing of performances, I wasn’t disappointed. Once children mapped meter through ski-hill graph visual representations they understood divisions of pulses and how they related as meters and this process enabled new information. I was convinced the ski-hill process is the means through which students learned new information for knowledge needed to solve and understand the timing problems (Hilton, Calilhanna, and Milne, 2018).

Children from around the of age four recognise the ski-hill graph represents more pulses and meters than time signatures. The depth of thinking and thoughtful comments from children astonished me. When children clap, tap, and map mathematical aspects of the meter through ski-hill graph fractions, they show how Cohn’s meter theory augments time signature’s theories. From these first ski-hill graph teaching experiences my motivation to write this book and to pursue a formal study of Ski-hill Graph Pedagogy began.

Before I learnt to visualise the meter through ski-hill graphs the disconnections between what I taught as the meter (time signatures) and what students experience as meter (duple and triple meter) had never occurred to me. I realised that I would need to change how I taught meter. However, the results with students were exciting from the start and this book introduces that story!

This is a book for anyone who would like to know more about meter, for anyone who knows time signatures’ meter fundamentals theories do not represent listening experiences during performance preparation for beginners, and for those who require meter fundamentals which does!

Cohn (2020)’s contemporary meter definition and approach is a turning point for meter theory and pedagogy history. Time signatures were designed to represent notation of European ‘art’ music, pieces such as waltzes and marches. The meter is not time signatures or represented by time signatures. In today’s world where music teachers require meter theory with inclusivity for any metric music, ski-hill graphs offer a viable solution to replace and or significantly change meter pedagogy.

When the polymath Gottfried Leibniz (1646-1716) observed connections between music and mathematics he mentioned pleasure obtained from music comes from counting but without awareness of counting. The thinking suggests that music is not just an art form, but also a mathematical problem that the soul solves unconsciously. Although this is beautiful, a way to describe the complexity of music and its importance for humanity, the downside is we can also romanticise the experience of music and this is problematic for teaching music especially with beginners. Ski-hill graphs assist students to understand more about how and why we count (objectively) to learn music, which usually means performing is more enjoyable because counting unconsciously renders fluent timing and impacts rewarding cognitive and physical skills and can be confidence-building!

I learned from working for years