The Equal Tempered Scale

And How to Sing In Tune with

”Vocal Coach”

What’s the

tempered scale?

C3 C4

• The tempered scale is a system of 12 notes included within the interval of a perfect octave (or eighth).

Example: the 12 notes between C3 and C4

• It’s widely used in modern western music, with instruments like the piano, electronic keyboards, and fretted instruments like guitars.

• In other words, what are the frequencies of the notes that form a tempered scale?

• Quite simply, there are only two steps involved…

How

is it

built?

• Take a note

• Double its frequency

• You have an octave

For example, C3 - C4 is an interval of an octave because the

frequency of C4 (262 Hz) is two times that of C3 (131 Hz).

C3 C4

First step defining

the octave

C4 = 262 Hz

C3 = 131 Hz

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In the second step we need to define the frequencies of all the other

11 notes

Let’s start from

three

basic facts

1. There are 12 notes in the scale. I’ll explain why exactly 12 in another presentation. For now, let’s take this number for granted.

2. Equal temperament means that the scale is made of 12 equal intervals…

3. … and intervals are perceived by the ear based on the ratio of the two composing frequencies, not their difference.

For example the eighth (octave) is defined by a ratio of 2

• This means that the ratio between any two adjacent notes of the scale must be the same.

For example the ratio between F3 and E3 must be the same as the ratio between F3# and F3.

C4 = 262 Hz

C3 = 131 Hz

x R

x R

x R

x R

x R

x R

x R

x R

x R

x R

x R

Let’s call this ratio “R”, we’ll calculate its value later.

What does it all

mean?

• This ratio is the fundamental building block that defines all the notes of the scale. From this number we can derive all the frequencies of the composing notes of the scale.

• The name of the interval associated to the fundamental ratio R is called…

…the semitone

a.k.a. the minor second, or m2.

One

ratio

fits all

Math advisory

• We’ll do the math to calculate the value of this ratio in the next slides. You can skip it if not interested.

Math

C4 = 262 Hz

C3 = 131 Hz

x R

x R

x R

x R

x R

x R

x R

x R

x R

x R

x R

RCC 33#

233#3 RCRCD

3333# RCRDD

1234 RCC

1212 23

4

C

CR

Finally, the value of the ratio between any two adjacent notes in the scale is equal to:

1.059463212 R

Now we know how to build the scale. The frequency of each note is calculated by multiplying the frequency of the previous note by 1.059.

C4 = 262 Hz

C3 = 131 Hz

x 1.059 = 140Hz

x 1.059

x 1.059

x 1.059

x 1.059

x 1.059

x 1.059

x 1.059

x 1.059

x 1.059

x 1.059

We built the tempered scale… Now we need to make a good use of it. So the question is:

“How to play or sing well in tune?”

I’ll leave the answer to your music teacher, but I can explain here a way to measure intonation. After all, you can only correct what you see (or better, hear).

100 cents

One hundredth of a semitone is small enough even for the most sensitive ear. So we split the semitone in 100 equal parts called cents.

We need to measure small variations of intonation…

we need to measure much smaller intervals than the semitone.

• When you play an instrument with fixed intonation (like a piano or a guitar for example), the ratios between any two notes is guaranteed to be exact (provided your piano was well tuned, of course).

• But when you sing, or play another type of instrument, YOU, the singer or the player, are responsible to guarantee the good intonation.

• With cents we can measure the accuracy of intonation. While a single cent is such a small distance that’s virtually unperceivable, at distances of about 10 cents intonation problems start becoming noticeable.

Splitting the semitone:

Why do we

do that?

cents

0 50 100

A 50-cent interval corresponds to half a semitone, or

a quarter of a tone.

Smaller

than the

semitone

Software Tuners

• A tuner is a device that measures

• the pitch of a sound.

• Now conveniently available also on mobile platforms, such as iPhones and Android smartphones.

• Vocal Coach is a tuner in the form of an app for Android devices. It’s specifically designed for singers and measures (at the precision of 1 cent) your ability to sing in tune. You can download it at:

https://play.google.com/store/apps/details?id=com.basisfive.vocaltrainer

A screenshot of “Voice Coach” for Android