### What is 'Just' Tuning and how do I use it?

So
you have spent hours with the tuner trying to get your open G exactly
in tune, and now your teacher wants you to play it 14 cents flat. What's
going on here? Read on to find out.

It's all about the difference between Just tuning and Equal Temperament tuning. The first part of this post is an overview of what Just tuning is all about. The second part explains how to use Just tuning with Tonal Energy Tuner, a popular app for iOS and Android phones.

## Equal Temperament (ET) Tuning in Bullet Points

Intonation
is all about the frequency of the sound waves The standard orchestra
tuning note, A440, has a frequency of 440 Hz. Raising the pitch of any
note by one octave means doubling the frequency, so the next higher A
above A440 has a frequency of 880 Hz. One octave lower A has a
frequency of 220 Hz.

In Western music, the octave is divided into 12 half-steps or semi-tones. Your first thought might be take the difference between 880 and 440 and divide it by 12 to find the frequencies of the semi-tones. Well, sort of. What you do is set up the scale so the percent change in frequency is constant from each half step to the next. The magic number that makes the octave work is about 6% (actually, 5.946 %). The A# frequency is 5.946% higher than A. B is 5.946% higher than A#. Keep doing this and, when you go from G# to A you have doubled your original frequency to make an octave. This system is the "well-tempered scale" which was developed in the 1700's.

5.946% is awkward to work with, so there is an alternate unit called cents. the 5.946% change between semitones is 100 cents. Suppose you are playing a C and your tuner says you are 50 cents sharp. That means you are half way between C and C#, so you are off by 1/2 semitone. There are 100 cents in a semitone, and 1200 cents in an octave.

In Western music, the octave is divided into 12 half-steps or semi-tones. Your first thought might be take the difference between 880 and 440 and divide it by 12 to find the frequencies of the semi-tones. Well, sort of. What you do is set up the scale so the percent change in frequency is constant from each half step to the next. The magic number that makes the octave work is about 6% (actually, 5.946 %). The A# frequency is 5.946% higher than A. B is 5.946% higher than A#. Keep doing this and, when you go from G# to A you have doubled your original frequency to make an octave. This system is the "well-tempered scale" which was developed in the 1700's.

5.946% is awkward to work with, so there is an alternate unit called cents. the 5.946% change between semitones is 100 cents. Suppose you are playing a C and your tuner says you are 50 cents sharp. That means you are half way between C and C#, so you are off by 1/2 semitone. There are 100 cents in a semitone, and 1200 cents in an octave.

The point of Equal temperament is to allow a keyboard instrument play in any key with minimal flaws in intonation.It replaced earlier systems based on more acoustically "pure" systems. (Just tuning is an one of these more pure systems.)

## What is Just Tuning?

The
problem with ET is that, as previously mentioned, the notes sound just
slightly off. Our Western ears are most attuned to intervals with
integer multiples of frequency, like 2:1 for the octave. One of the
most common intervals is the third (C to E in the C scale). In ET, the
ratio of the E frequency to the fundamental C frequency is 1.260. That
interval would sound a bit less discordant if the ratio was 1.25 (or
5:4).

Just
Tuning is just one alternative temperament that adjusts the semi-tone
intervals to produce a system that gives results that "sound better" in
some sense. There are over a dozen alternative temperaments --
Pythagorean and Perfect Fifths to name just a couple of them.

Is Just Tuning better than ET? Or maybe Pythagorean is better than both of them. It's sort of like asking whether a M13 mouthpiece is better than a B45 -- it depends on what you want to accomplish. It is, however, important that everybody in the group is tuning on the same system. If you are playing with a pianist, you want ET tuning. Just tuning is commonly used when playing in a wind or string ensemble.

Is Just Tuning better than ET? Or maybe Pythagorean is better than both of them. It's sort of like asking whether a M13 mouthpiece is better than a B45 -- it depends on what you want to accomplish. It is, however, important that everybody in the group is tuning on the same system. If you are playing with a pianist, you want ET tuning. Just tuning is commonly used when playing in a wind or string ensemble.

An
alternative temperament is typically defined by specifying the
frequency ratio as the ratio of two small integers -- the smaller the
better. For example, here is a comparison of Just and ET frequency
ratios for three common intervals:

Just | ET | |
---|---|---|

third | 5:4 (1.25) | 1.260 |

fourth | 4:3 (1.333) | 1.335 |

fifth | 3:2 (1.5) | 1.498 |

Perhaps the best way to compare Just tuning with ET is to show how much higher (or lower) Just tuning is than ET for the major scale. The difference in cents is usually called cET. It depends on the position in the scale, not the absolute name of the note.

cET | |
---|---|

do | 0 |

re | 3.9 |

mi | -13.7 |

fa | -2.0 |

sol | 2.0 |

la | -15.6 |

ti | -11.7 |

do | 0 |

One of the larger differences is on mi. If you are playing in C major, E=mi, so you should play it 14c flat. However, if you are playing in E major E=do, and you want to be right on pitch. That's assuming your tuner is set for Equal Temperament. The attached table provides a more detailed comparison of Just and ET tuning, including a cET table for the complete chromatic scale.

## Using Just Tuning in Tonal Energy Tuner

If
you are using a mechanical ET tuner, you are stuck using a cET table to
manually adjust the results. But a good tuner app will do it for you.
Here is how to do it in Tonal Energy Tuner. Another of my favorite apps, Tunable, can do the same adjustment.

- Select
**Preferences / Alternate Temperament**

You
can choose from more than a dozen alternate temperaments. Or, if you
don't like those, you can define your own. For purposes of matching the
numbers in this article, choose Just/Pure

- Go to the
**Sound**screen

In
the upper left corner there is a toggle that lets you switch between
Equal temperament and your chosen alternate temperament (Just/Pure)

- One final, very critical step - Tell TE Tuner what key you're playing in.

With
equal temperament, all half steps are the same distance apart. However,
with Just tuning, the spacing of the half-steps varies depending on
what key you are playing in. Double tap a note on the tone wheel to
pick your your key signature.

Now
let's do some exploration of Just tuning. Select Just tuning, and
double tap C to tune for the key of C. When you enable Sustain and tap a
note, the drone will sound. Some things you see on the screen:

**(1)**Reminder to double-tap on the tone wheel to select your tonic key

**(2)**Shows what key has been selected -- in this illustration, key of C.

**(3)**When the drone is playing the tonic note, the note is purple

**(4)**When the drone is playing a non-tonic note, the note is orange.

**(5)**This section shows the cents deviation from Just tuning. Here +0.1 cents is variance from Just tuning, and +0.1 cET is variance from Equal Temperament (ET) tuning. Since C is the tonic note, the two deviations are identical.

**(6)**Illustrates the deviations for a non-tonic note. +0.1 cents deviation from Just tuning, +4.0 cET from ET tuning.

Note:
When reading deviations while using the drone, it's best to use
headphones. You want your mic to hear your clarinet, not the drone
coming through the speakers.

Here is an
experiment to try if you want to hear the difference between Just and ET
tuning. Set the the tonic to C. The note in the C scale where the
difference is largest is A (major sixth). Start the drone on A. While
the drone is playing, toggle back and forth between Just and ET tuning.
You should hear a slight shift in the pitch of the drone as you toggle.

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