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CHAPTER 5:
How Keys and Modes REALLY Work
  
5.4 Tuning, Temperament, and Transposing

 
PAGE INDEX
  

5.4.1 “How Come I Can’t Tune this #@*&!% Thing?”

5.4.2 Don’t Lose Your Equal Temperament

5.4.3 “It’s Too Low (or High) for My Voice”: Transposition

5.4.4 How Transposing Instruments Work

 

~ • ~ • ~ • ~


5.4.1

“HOW COME I CAN’T TUNE THIS #@*&!% THING?”

 

If you write a piece of music in a single key, you'll likely have no problem with musical unity. The arrangement of intervals in the diatonic scale ensures a strong tonal centre. A nice, small assortment of six related notes (scale degrees 2 through 7) all point to the tonic note.


     Moreover, assuming your song has words, you'll likely organize the words into verses and choruses, each sung to the same musical phrases. This reinforces musical unity.


     But too much musical unity ain't necessarily a good thing. It can get boring.


     To get some variety happening, you have the option of changing keys partway through the song. And, since every key has its own tonal centre, you preserve unity at the same time as you create variety.


     Several hundred years ago, when musicians and musical theorists were experimenting with changing keys within a piece of music, a nasty problem kept bedevilling them. Whenever they tried to switch to a new key, their instruments sounded out of tune. Hellish out of tune.


     The problem was the dang Pythagorean comma. As discussed in Chapter 4, if you tune an instrument using exact Pythagorean 3:2 frequency ratios, you end up with an octave that is slightly bigger than it ought to be. About a quarter of a tone too big.


     For example, Middle C has a frequency of 261.6 Hz. So the frequency of the C above Middle C ought to be exactly double: 523.2 Hz.

But if you use exact Pythagorean fifths, you end up with C above Middle C having a frequency of 530.3 Hz. Noticeably too sharp.


     If instead you tune in perfect 2:1 octaves, then the other notes derived from simple frequency ratios such as 3:2, 4:2, and so on, end up either too sharp or too flat.


     What to do?


   

5.4.2

DON’T LOSE YOUR EQUAL TEMPERAMENT

 

Musicians and theorists tried all sorts of variations on the theme of just intonation, methods of tuning with simple whole-number frequency ratios. Such tuning systems—and there are many—enable the tuning of an instrument so that it's playable in one key, or perhaps even several keys. But if you try to play in keys the instrument isn't tuned for ... forget it. Doesn't work.


     Eventually, one of the numerous tuning solutions attempted over the centuries emerged the clear winner. The solution was to:

 

     1.  Stick to an exact 2:1 octave, despite the Pythagorean comma.

 

     2.  Divide the octave into 12 exactly equal semitones.


     This system is called equal temperament (from temperare, the Latin root meaning to mix or mingle).


     To get the frequencies for each semitone:

 

        Start with the first note of the scale and multiply its frequency by the 12th root of two.

 

        Take that frequency and multiply it by the 12th root of two, which gives you the frequency for the next semitone up.

 

        Repeat until you get to the next octave.


     The 12th root of 2 is the number 1.05946 (rounded off). So the ratio of any semitone to the semitone below is 1.05946:1.


     Table 29 shows the frequencies of all the notes from Middle C to the octave above Middle C, with each successive frequency multiplied by the 12th root of two:

 



TABLE 29  Equal Temperament Frequencies for Tones from Middle C to C Above Middle C, and Associated Simple Frequency Ratios


Note

Equal

Temperament

Frequency

(Hz)

Interval

with

Middle C

Simple

Freq.

Ratio

(SFR)

Associated

SFR

Frequency

(Hz)

Middle C

C

D

E

E

F

F

G

A

A

B

B

C  

261.6

277.2

293.6

311.1

329.6

349.2

370.0

392.0

415.3

440.0

466.1

493.8

523.2

Unison

Minor 2nd

Major 2nd

Minor 3rd

Major 3rd

Perfect 4th

Tritone

Perfect 5th

Minor 6th

Major 6th

Minor 7th

Major 7th

Octave

1:1

16:15

9:8

6:5

5:4

4:3

45:32

3:2

8:5

5:3

16:9

15:8

2:1

261.6

279.0

294.3

313.9

327.0

348.8

367.9

392.4

418.6

436.0

465.1

490.5

523.2


  

     So, when you're playing in any given key, only the two octave notes are in an exact 2:1 simple frequency relationship. Every other note is slightly out of tune, compared with the simple frequency ratio expected from the harmonic series.


For example, in Table 29 above:

 

        The frequency for the note G would be 392.4 Hz if it were tuned in an exact 3:2 ratio with Middle C. But the equal temperament frequency of G is 392.0 Hz (slightly flatter).

 

        The frequency for the note E would be 327.0 Hz if it were tuned in an exact 5:4 ratio with Middle C. But the equal temperament frequency of E is 329.6 Hz (slightly sharper).

 

        The frequency for the note C♯ would be 279.0 Hz if it were tuned in an exact 16:15 ratio with Middle C. But the equal temperament frequency of C♯ is 277.2 Hz (slightly flatter).


     Similarly, in equal-temperament tuning, all of the other notes are either slightly flat or slightly sharp, compared with their simple-frequency-ratio counterparts.


     The equal temperament solution works. Your brain accepts the small “pitch errors”—slight deviations from simple ratios—when they're equally distributed over all 12 semitones. Since every semitone interval is exactly equal, you can construct diatonic scales using any of the 12 semitones as the tonic note, and the octave notes will always have a frequency ratio of exactly 2:1. Equal temperament makes something called modulation possible (coming up shortly).


     Consequently, equal temperament has been the norm for about three centuries in Western music.


     Equal temperament works only because the pitch errors are small—so small that your forgiving brain processes them as though they were simple frequency ratios.


     When you try to tune a guitar or other stringed instrument using harmonics from string to string, it doesn't quite work out because you're not using equal temperament. That's why the best tuning device is a digital tuner, with equally-tempered frequencies built into the electronics that are accurate to many decimal places.


 

Get That Man a Digital Tuner

 

Some people think equal temperament is a Bad Thing because every single note between the octave notes in any key is slightly dissonant. Others think equal temperament is a Good Thing for two main reasons:

 

1.  It solves the dang tuning problem, already; and

 

2.  Every single note between the octave notes in any key is slightly dissonant—and therefore music played on equally-tempered instruments sounds more colourful and interesting than it would if all the notes were exactly in tune.


Obviously, J. S. Bach agreed with the latter view. Nobody had a keener ear. Bach would surely have been able to easily hear the out-of-tuneness of equal temperament. Yet he famously celebrated equal temperament by composing The Well-tempered Clavier, a two-book masterwork of 24 preludes, one in each major and minor key, and 24 fugues, one in each major and minor key.

 


     One other tuning-related problem took even longer to solve: what to do about a reference frequency. One note and its associated frequency needs to serve as a standard to derive the frequencies for all the other notes, using equal temperament.


     After centuries of hair-pulling and fang-gnashing, everybody agreed in 1939 that the note A above Middle C would always be tuned to exactly 440 Hz, and would therefore serve as the reference pitch for setting all the other pitches. (Then World War II started.)


     This tuning pitch is called Concert A or A-440.


  

A Free Emergency Digital Tuner

 

When you're lost in Juarez in the rain and you don't have a digital tuner with you but you must tune your guitar, what can you do?


Why, just reach in your pocket and whip out your trusty cell phone. Or wander around until you find a pay phone. Get a dial tone, and you've got your reference note. The dial tone is F. Specifically, it's the F on the first fret of the low E-string of your guitar, the F that's one and a half octaves below Middle C.



5.4.3

“IT’S TOO LOW (OR HIGH) FOR MY VOICE”: TRANSPOSITION

 

It happens to everybody. You swagger into the Wrong Ranch Saloon and start singin' a tune and everything's goin' along fine until you get to the lowest notes (or the highest notes), and you can't hit them.


     People turn and laugh at you. Especially the dusty cowpoke, the one who out-drew Billy Joe. You laugh along with them, vainly attempting to hide your humiliation. Tears stream down your face. It’s no use. They know, they know. Yes, they know you started singing in a key that did not match your vocal register for that particular song.


     But it's too late. Marshal Puma senses trouble brewing and allows as how you might live to see tomorrow if you get outta Dodge tonight. So you stumble out of the saloon into the dusty main street. Sadie and Ellie Sue offer you a fresh horse and away you go to join Ex-Marshal McDillon in exile.


     If only you had thought to start over, in a different key.


     Transposition refers to moving a whole group of notes (such as the entire melody of a song) up or down in pitch.

 

        If you play guitar, you can do this easily without even changing chord fingering. All you do is move your capo up or down the fretboard.

 

        On the piano, it's not so easy. You have to change the way you finger the melody and chords for every dang key you play in.


     You can use tables 24, 25, 26, and 27 for transposing. They show you, row by row, the scales for each key. If you're singing in the key of C major and you want to know what notes you'd be singing if the tune were transposed to E major, just go to Table 24 and compare the C Major row with the E major row. For instance, if you want to transpose the notes C, D and E in the key of C major to the key of E major, the equivalent notes would be E, F♯, and G♯.


     It's that simple.


     One important thing to keep in mind at all times with respect to key changes and transposing: There's no such thing as a “high key” or a “low key.” A key is just an interval order with respect to a key note or tonic note. The key of E major is neither “higher” nor “lower” than the key or C major or any other key.


     The way a songwriter or composer has arranged the intervals of a particular melody determines which key you will be able to sing it in, without the tune being too high or too low for your voice.


     You can sing some songs easily in the key of C major, but not in the key of G major. You can sing other songs easily in the key of G major, but not in C major. The determining factor is not the key. It's how the melody itself is structured. The key of C major is not inherently “higher” or “lower” than the key of G major.


     That goes for all the keys, major and minor.



5.4.4

HOW TRANSPOSING INSTRUMENTS WORK

 

If you happen to read music notation, the idea of a “transposing instrument” will make more sense than if you don't happen to read music notation.


     Most musicians don't read music notation, which is why this book has no music notation. Still, even if you don't read music, you might find a brief description of the meaning of “transposing instrument” mildly entertaining. George Martin, the classically-trained producer of the Beatles, once tried to explain the workings of transposing instruments to John Lennon, who did not read a note of music. Lennon thought it was all pretty daft.


     A true transposing instrument (as opposed to an octave transposing instrument—more on the distinction in a minute) is a wind instrument (aerophone) for which the musical notes on the page differ from the notes the instrument makes. You see a note on the page, you finger the instrument to play that note, and a different note comes out of your instrument.


     What's going on?


     Any given musical instrument is constructed so that it can handle only a certain range of pitches. The guitar, for instance, only has a certain number of frets, limiting the upper and lower range of the instrument.


     This applies to wind instruments, like any other. So it's common to have “families” of wind instruments—families of clarinets, flutes, and saxophones, for instance—of varying sizes. The smaller-sized instruments handle higher pitches, the larger ones, lower pitches.


     For instance, each of the four common sizes in the saxophone family—soprano, alto, tenor, and baritone—is good for a certain range of pitches, from a high-pitched range (soprano sax) to a low-pitched range (baritone sax).


     All saxophones use the same fingering for a particular written note. So, if you learn to play, say, alto sax, and you decide to switch to another sax in the same family, you don't have to learn a whole different way of fingering.


     Problem is, because each instrument is built for a different pitch range, when you finger the alto sax to play, say, the written note C, the note you actually hear coming out of your horn is E♭, 9 semitones below C. On the tenor sax, when you finger the instrument to play C, the note that comes out is B♭, more than an octave below the C written on the page.


     Therefore, composers and orchestrators must notate the music so that it accounts for the difference between the notes that come out of the transposing instrument and the notes on the page.


     Suppose the composer wants the sound coming out of the alto saxophone to be in the key of C. The composer needs to notate the music nine semitones higher (an interval of a major sixth) on the page—in the key of A. The alto sax player sees an A on the page, fingers the horn to play A, and out comes the sound of the note C, nine semitones lower—as the composer intended.


     So, written music for the alto sax must be transposed up by an interval of a major sixth (all notes!), in order to sound the way the composer intended.


     This all seems pretty odd, but it makes a lot of sense for wind players who read music. They don't have to cope with learning new fingerings for each instrument in a family. Instead, it's up to the composer or orchestrator to ensure that the music is transposed on the page properly for the intended instrument and the intended sound.


     Some instruments are “octave transposing” instruments. The guitar, for instance. Notated music for the guitar is written an octave higher than it sounds when you play the music. When you play the note Middle C from the page, you still hear the note C, but it's the C an octave below Middle C.


~ • ~ • ~ • ~

 

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~ • ~ • ~ • ~

 

You are reading the FREE SAMPLE Chapters 1 through 6 of the acclaimed 12-Chapter book, How Music REALLY Works!, 2nd Edition. Here's what's in Chapters 7 through 12. 

 

To order the book, click here:

    

  

 

 

 TABLE OF
 CONTENTS

  

 PART I

 The Big Picture    Introduction

   1. W-5 of Music
  
2. Pop Music
   
    Industry

  
 PART II
 Essential
 Building Blocks
 of Music
   3.
Tones/Overtones
   4. Scales/Intervals
   5. Keys/Modes
 
 PART III
 How to Create
 Emotionally
 Powerful Music
 and Lyrics
   6.
Chords/
  
      Progressions

   7. Pulse/Meter/
  
      Tempo/Rhythm

   8. Phrase/Form
   9. Melody
 10. Lyrics
 11. Repertoire/
     
  Performance

  

 PART IV
 Making a
 Living In Music
 12.
Business of
   
     Music

 
 Appendixes

   

 Notes

   

 References

  

 Index
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   Top

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 TABLE OF
 CONTENTS

  

 PART I

 The Big Picture    Introduction

   1. W-5 of Music
  
2. Pop Music
   
    Industry

  
 PART II
 Essential
 Building Blocks
 of Music
   3.
Tones/Overtones
   4. Scales/Intervals
   5. Keys/Modes
 
 PART III
 How to Create
 Emotionally
 Powerful Music
 and Lyrics
   6.
Chords/
  
      Progressions

   7. Pulse/Meter/
  
      Tempo/Rhythm

   8. Phrase/Form
   9. Melody
 10. Lyrics
 11. Repertoire/
     
  Performance

  

 PART IV
 Making a
 Living In Music
 12.
Business of
   
     Music

 
 Appendixes

   

 Notes

   

 References

  

 Index
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   Top

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 TABLE OF
 CONTENTS

  

 PART I

 The Big Picture    Introduction

   1. W-5 of Music
  
2. Pop Music
   
    Industry

  
 PART II
 Essential
 Building Blocks
 of Music
   3.
Tones/Overtones
   4. Scales/Intervals
   5. Keys/Modes
 
 PART III
 How to Create
 Emotionally
 Powerful Music
 and Lyrics
   6.
Chords/
  
      Progressions

   7. Pulse/Meter/
  
      Tempo/Rhythm

   8. Phrase/Form
   9. Melody
 10. Lyrics
 11. Repertoire/
     
  Performance

  

 PART IV
 Making a
 Living In Music
 12.
Business of
   
     Music

 
 Appendixes

   

 Notes

   

 References

  

 Index
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   Top

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 TABLE OF
 CONTENTS

  

 PART I

 The Big Picture    Introduction

   1. W-5 of Music
  
2. Pop Music
   
    Industry

  
 PART II
 Essential
 Building Blocks
 of Music
   3.
Tones/Overtones
   4. Scales/Intervals
   5. Keys/Modes
 
 PART III
 How to Create
 Emotionally
 Powerful Music
 and Lyrics
   6.
Chords/
  
      Progressions

   7. Pulse/Meter/
  
      Tempo/Rhythm

   8. Phrase/Form
   9. Melody
 10. Lyrics
 11. Repertoire/
     
  Performance

  

 PART IV
 Making a
 Living In Music
 12.
Business of
   
     Music

 
 Appendixes

   

 Notes

   

 References

  

 Index
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   Top

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 TABLE OF
 CONTENTS

  

 PART I

 The Big Picture    Introduction

   1. W-5 of Music
  
2. Pop Music
   
    Industry

  
 PART II
 Essential
 Building Blocks
 of Music
   3.
Tones/Overtones
   4. Scales/Intervals
   5. Keys/Modes
 
 PART III
 How to Create
 Emotionally
 Powerful Music
 and Lyrics
   6.
Chords/
  
      Progressions

   7. Pulse/Meter/
  
      Tempo/Rhythm

   8. Phrase/Form
   9. Melody
 10. Lyrics
 11. Repertoire/
     
  Performance

  

 PART IV
 Making a
 Living In Music
 12.
Business of
   
     Music

 
 Appendixes

   

 Notes

   

 References

  

 Index
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   Top

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

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