Math in Motion SAMPLE FIRST STEPS IN MUSIC THEORY. Caleb Skogen

Math in Motion FIRST STEPS IN MUSIC THEORY Caleb Skogen

2 Math in Motion: First Steps in Music Theory C lassical onversations MULTIMEDIA Caleb Skogen, Math in Motion: First Steps in Music Theory 2015 Classical Conversations MultiMedia, Inc. All rights reserved. Published by Classical Conversations MultiMedia, Inc. P.O. Box 909 West End, NC 27376.ClassicalConversations.com.ClassicalConversationsBooks.com Cover design by Classical Conversations. Cover image: Music and Literature, William Michael Harnett, 1878. Courtesy of the Albright-Knox Art Gallery, public domain. All Scripture quotations, unless otherise noted, are taken from the King James Version of the Bible. Printed in the United States of America ISBN: 978-0-9904720-9-4 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form by any means, electronic, mechanical, photocopy, recording, or otherise, ithout prior permission of the author, except as provided by USA copyright la. Honoring copyright: it is a matter of integrity! Classical Conversations orks diligently to keep costs don and provide helpful resources for your family s home school. This book is copyrighted, hich means that unauthorized reproduction or sharing of the pages is prohibited. This includes sharing copies ith other families or groups, even ithin your on community. Rather, encourage your friends ho ould like this resource to honor copyright and invest by purchasing it for their family. Permission is granted to make copies ithin your on home for your family.

3 Table of Contents Introduction...5 Acknoledgments...7 CHAPTER 1 Music Grammar Part 1: The Keyboard and Notation...9 CHAPTER 2 Music Grammar Part 2: Symbols and Notes...27 CHAPTER 3 Introduction to Rhythm...41 CHAPTER 4 Time Signature and Simple Meter...57 CHAPTER 5 Major and Natural Minor Scales...69 CHAPTER 6 Key Signatures...83 CHAPTER 7 Scale Degrees and Transposing...101 CHAPTER 8 Intervals...119 CHAPTER 9 Triads and Triad Qualities...139 CHAPTER 10 Triads: Roman Numeral Analysis...153 CHAPTER 11 Triad Inversions...169 CHAPTER 12 Score Analysis...181 CHAPTER 13 Revie...197 What Have I Learned?...207 BONUS CHAPTER Compound Meter...209 Helpful Tools...225 In-Class Activity Anser Key...233 Anthology of Scores...249 For Further Reading...267 Project Help...269 Glossary...275

4 Math in Motion: First Steps in Music Theory

5 Introduction In ancient and medieval philosophy, music as considered to be one of the four mathematical sciences of the quadrivium along ith arithmetic, geometry, and astronomy. These sciences ere called mathematical because they ere based on the study of number: arithmetic is the study of pure number, geometry is the study of number in space, music is the study of number in time, and astronomy is the study of number in both time and space. Music is not to be experienced on merely a listening level or performance level. It is a science to be studied in order to learn more about number, time, space, and the orld. Moreover, hen e treat music as a science, e re able to incorporate studies in music history, music philosophy, and music theory to anser big questions like Ho is music beautiful? and What does it mean to create beautiful orks? This book ill discuss foundational concepts in music theory and ho these concepts relate to ritten pieces of music. This curriculum ill concentrate on a specific part of music theory, called score analysis, using figured bass. It is important to note that there are aspects of music theory that this text ill not cover fully. For example, hile rhythm is an extremely important element of music, the scope of this text prevents a full treatment of it. The student may be left ith questions about rhythm, but that is the point. When questions pique a student s curiosity, he or she ill be more likely to take learning into his or her on hands. Such is the goal of any classical curriculum. It is helpful if students either begin playing or continue playing an instrument hile going through this book. That ay, the forms of music theory can be discovered (or rediscovered) ithin actions of music. Hoever, the text of this curriculum should be accessible to those ho do not have any prior knoledge of music. If you do not have any musical training, do not orry this book is for you too!

6 Math in Motion: First Steps in Music Theory Music is enough for a hole lifetime but a lifetime is not enough for music. Sergei Rachmaninoff, The Goal Many students experience basic music theory merely as assignments detached from the real goal of understanding music. While e ant to understand music theory so e can better perform and practice our instruments, there is also a conceptual side of music theory that e must not overlook. What is going on in the music? This important question provides the central theme for this textbook. Math in Motion seeks to teach the foundations of music through an increased analysis of hymn scores to help students apprehend hat is going on behind the music they hear. Exercises 1956 Each chapter is designed to introduce and teach a ne concept of music theory. The in-class activities ill help students understand the material and prepare for the exercises they ill do for homeork. At the end of each chapter there are exercises split into four separate days. The students ill complete the exercises on each day prior to the next community class day. Many of the exercises use excerpts from the six scores that are found at the end of the book. These scores are important tools in teaching score analysis and build toard the end-ofsemester project. Get to kno these scores! Terms Important terms are ritten in bold hen the term is first introduced so the student may find the term more easily hen revie is needed. Helpful Tools In the back of the book there are tools that ill be helpful for completing exercises and for revieing core concepts. Included in this section are blank staff paper, keyboards, note names, note values, and the circle of fifths diagram. Let s Get Started Be curious about the possibilities of music. You cannot live in this orld and truly escape music. You experience it in millions of different settings, you are moved by music in many different ays, and you ill continue to find music surprising throughout your life. Be curious about this art, this science, this gift that has become and continues to become pivotal in your life and the lives of those around you. A onderful opportunity lies ahead of you in the pages of Math in Motion.

9 1Music Grammar Part 1: The Keyboard and Notation Music is a language and, like all languages, it is meant to communicate ideas. In order for any language to ork, hoever, e as listeners must understand the rules that describe it, or its grammar. We begin the study ith this grammar. The most basic element of musical grammar is music notation. Even though there are many different styles of instrumentation, music notation is key to both playing and understanding music. Music notation tells us many things about a piece of music, but the most foundational information it gives us is about pitch. Pitch is the attribute of a musical tone produced by the number of vibrations generating it. Pitch can be high or lo. The notes on the keyboard [Music] takes us out of the and the notes on the staff signify certain highness or loness of pitch. actual and hispers to us dim secrets that startle So, hat are the names of the notes on a musical staff? What is a musical staff? What is a our onder as to (hat) treble clef? What is a bass clef? This chapter aims to anser these and other foundational ho e are and for hat, questions through activities and exercises so that you can see the additional music theory hence, and hereto. All concepts introduced later, in all their orderly beauty. the great interogatories, like questioning Angels, float in Notes on the Keyboard on its aves of sound. Of all the great variety of instruments, the keyboard is the most helpful for learning the Ralph Waldo Emerson concepts of music theory, especially notation, since it helps the student to visualize the structure of music in a ay that aids memory. Even if you do not have a keyboard or piano in your home, you can still learn about music theory. For this curriculum, you ill need only the diagrams on the page, though you may ish to add a practical component by using a physical keyboard or one of the digital keyboards readily accessible online.

10 Math in Motion: First Steps in Music Theory Figure 1.1 C# D# F# G# A# Db Eb Gb Ab Bb C# D# F# G# A# Db Eb Gb Ab Bb C# D# F# G# A# Db Eb Gb Ab Bb C D E F G A B C D E F G A B C D E F G A B In Figure 1.1, you can see every note that exists in modern Western music theory. The notes of a keyboard have higher pitches hen moving right (up) and loer pitches hen moving left (don). A keyboard or piano has black and hite keys. The hite keys are named ith the first seven letters of the alphabet: A, B, C, D, E, F, and G. These notes are called natural notes. Natural notes repeat themselves in alphabetical order up and don the keyboard. The groupings of the black keys in tos and threes help the eye see the differences beteen the locations of each hite key. For example: C is alays immediately belo (left) the first black key in the group of to. The F is alays immediately belo (left) the first black key in the group of three. Refer to Figure 1.2 belo. Figure 1.2 2 3 2 3 2 3 C F C F C F

Chapter 1 Music Grammar Part 1: The Keyboard and Notation 11 The black keys on a keyboard can be identified by to important symbols: a flat sign (b ) or a sharp sign (# ). Sharp and flat signs are called accidental signs. An accidental is a symbol placed to the left of a note used to alter the pitch of that note by one semitone going either up (#, called sharp ) or don ( b, called flat ). The names of the black notes have the same letter names as the hite notes along ith these additional accidental signs (either flat or sharp). Refer to Figure 1.3 to see examples. Another important accidental is the natural sign (n ). The natural sign is used to cancel out either a sharp or flat so as to return the note to its natural state, signifying that is it is neither sharp nor flat. Figure 1.3 Db Eb Gb Ab Bb C# D# F# G# A# semitone semitone C D E F G A B C hole tone A semitone is the shortest distance beteen to notes. The distance beteen En and Eb is one semitone. The distance beteen Bn and Cn is one semitone. A semitone is more commonly referred to as a half step. A hole tone is the distance of pitch equal to to semitones. For example, the distance beteen F n and Gn is a hole tone. A hole tone is also referred to as a hole step. A flat sign indicates the note is loered (moved left) one semitone. For example, the note Eb is positioned one semitone aay to the left (don) from En. The flat sign indicates the shift donard to the next semitone. A sharp sign indicates the note is raised (moved right) one semitone. For example, take a look at the position of F# on the keyboard in Figure 1.3. Its position immediately to the right of Fn indicates that it is one semitone higher than Fn. You may have noticed in Figure 1.3 that the black key just above (to the right of) C has to names: C# and Db. In fact, each of the black keys has to different names. Why?

12 Math in Motion: First Steps in Music Theory When to notes have the same pitch or tone, but different names, the names are said to be enharmonic equivalents of each other. For example, C# is one semitone higher than Cn, so it can be called C#. Hoever, it is also one semitone loer than D n, so it can also be called Db. In theory, every note has an enharmonic equivalent. Math in Motion ill only deal ith enharmonic equivalents associated ith black keys. Activity 1.1 In-class activity a) Write in the name of the notes on the blank lines you see on the keyboard in Figure 1.4. b) If all the notes in Figure 1.4 ere loered one half step (flatted), hat ould the C names of the notes be? # c) If all the notes in Figure 1.4 ere raised one half step (sharped), hat ould the names of the notes be? Figure 1.4 Db a b c d Notes on the Staff Musical notes are ritten on a five-line staff (Figure 1.5). The notes can be either on the line or in the space beteen the lines on the staff. These lines and spaces represent different notes. Figure 1.5

Chapter 1 Music Grammar Part 1: The Keyboard and Notation 17 Daily Exercises for Chapter 1 v EXERCISES FOR DAY 1 Re-read chapter 1 and complete the folloing exercises: Exercise 1.1 Provide the letter names for each specified key marked on the keyboard. D Exercise 1.2 Provide the name for each black note marked on the keyboard and include its enharmonic equivalent name. C # D b

18 Math in Motion: First Steps in Music Theory Exercise 1.3 Use arros to sho motions of hole or half steps. H H H W H W W W W H W H W H Exercise 1.4 On the staff belo, rite flat signs in front of notes in the first to measures, sharp signs in front of the notes in the second and third measures, and natural signs in front of the notes in the final to measures. & 4 4 bh h h h h H h H H h H H

169 11 Triad Inversions Most music is not ritten in a series of perfectly stacked triads in root position. Very often the notes are rearranged, played in succession, or stacked differently. Up until no you have learned triads in root position because this is their simplest form, but in this chapter you ill learn ho to identify and create inversions of major and minor triads and analyze them ith Roman numerals. There are to different inversions for any triad, depending on the bottom note. In this chapter you ill learn these to inversions and ho to identify, notate, and analyze them in a variety of situations. In identifying types of triad inversions, you are using hat you have learned in chapters 1 10. In these last fe chapters, e are approaching the goal and culmination of this book. Stay focused and determined as e near the finish line! Introduction to Inverted Triads: Root Position An inverted triad is hen the notes are transposed so that the root is no longer on the bottom. If you remember a triad in terms of its intervals, this can be easy to learn. The triad is a three-note chord ritten in thirds consisting of the root, third, and fifth. Figure 11.1 & b 4 4 Root Bass 3rd Bass 5th Bass Root Position 1st Inversion 2nd Inversion

170 Math in Motion: First Steps in Music Theory When the root of the triad is the bottom note, or acting bass note, the triad is in root position. When the third of the triad is on the bass, the triad is in first inversion. When the fifth of the triad is on the bass, the triad is in second inversion. In order to properly invert triads, you must be careful to kno the notes of the triad before you invert it, since the notes of a triad do not change, no matter hat form it is in. The root position of a triad can be identified ith the method of analysis you learned last eek. After the Roman numeral indicating the degree of the triad, certain numbers are used to identify the inversion. The root position of a triad is labeled by the numbers 5 and 3 and is placed to the right of the Roman numeral (Figure 11.2). The reason for the number 5 is because the distance from the bottom note to the top note is a fifth interval in size. The reason for the number 3 is because the distance from the bottom note to the middle note is a third interval in size. Figure 11.2 3rd 5th interval & I 5 3 First Inversion Triads A first inversion triad transposes the root up an octave, leaving the third in the bass. Figure 11.3 & First Inversion Bass In this example, the bottom note of the root position C triad has been raised a full octave. Notice the root at the top of the first inversion. In Figure 11.4 on the next page, the first inversion is symbolized by the number 6 placed to the right of the Roman numeral. The number 6 is used because the distance from the bass (E) to the top note (C) is a sixth interval in size.

Chapter 11 Triad Inversions 171 Figure 11.4 & 6th interval I 6 Activity 11.1 In-class activity Make the folloing root position triads into first inversion triads in the blank measures on the right. Include the Roman numeral and inversion number. & iii 5 3 IV 5 3 ii 5 3 Second Inversion Triads In a second inversion triad, both the root and third of the chord are transposed up an octave, leaving the fifth in the bass. Figure 11.5 Second Inversion & b Bass Notice in this example ho the bass and third note of the root position F triad have been raised a full octave. In analysis ith Roman numerals for a second inversion triad, the numbers 6 and 4 are placed to the right of the Roman numeral because the distance from the bass (C) to the middle note (F) is an interval of a fourth. Figure 11.6 & b I 6 4 4th 6th interval

172 Math in Motion: First Steps in Music Theory & b iii? b I should be sorry if I only entertained them. I ish to make them better. Georg Friedrich Handel Activity 11.2 In-class activity Make the folloing root position triads into first and second inversion triads in the blank measures on the right. Include the Roman numerals and inversion numbers on each triad. 5 3 ii 3 5 V 3 5 IV 3 5 Close & Open Harmony As you might be able to tell, inversions do not alays look as neat and precise as the root position. But all these variations on the triad you have seen so far are examples of close harmony, hich is hen the notes of the triad are as close as possible to the bottom note. The notes of a chord can also be spread further apart, referred to as open harmony. Close Harmony For example, notice belo ho the notes G, B, and D of the G major triad are organized as close as possible to the bass note in both the root and the inversions shon belo in Figure 11.7. Figure 11.7 & # Root Position 1st Inversion 2nd Inversion

In-Class Activity Anser Key 233 In teaching music theory, many concepts are understood best by spending time ith students. Some students ill not grasp these concepts as quickly as other students ill. Take the time to repeat these activities or come up ith additional (similar) activities so that the concepts may sink in ith each individual student. Even if you have students ho learn these concepts quickly, it is good for them to learn to be patient ith others. Perhaps students ho do understand the material can teach those students ho are struggling. In that ay, all students may be invested in the learning of the entire class.

234 Math in Motion: First Steps in Music Theory Chapter 1 Activity 1.1 a) Write in the name of the notes on the blank lines you see on the keyboard in Figure 1.4. b) If all the notes in Figure 1.4 ere loered one half step (flatted), hat ould the names of the notes be? Eb, Gb, Bb, Eb, E, Ab, B, Db c) If all the notes in Figure 1.4 ere raised one half step (sharped), hat ould the names of the notes be? F, G#, C, F, F#, A#, C#, D# d) Name the enharmonic equivalents of each of the black keys, a d, belo. a. A#, Bb; b. D#, Eb; c. C#, Db; d. G#, Ab Figure 1.4 a b c d E G B E F A C D