Changes

'''Equal Temperamenttemperament''' is a method of tuning an instrument, so that each distinct note is equidistant and all notes share the same set of interval ratios. Thus, it could be considered that every key on an equally-tempered instrument is ''equally out-of-tune '' compared to ideal pure-tuning.

Let's go back to the example above of how moving around the circle of fifths does not end on an octave of the original note. Well, what if we slightly lessen a 5th interval from 3:2 such that we reach that 7040 HZ 7040hZ A8 when moving in 5ths? This would be 55 55hZ x ___^12 = 70407040hZ. Moving things around, it is ___ = (70407040hZ/5555hZ)^(1/12), which equals ~1.49831. Thus, we need to slightly flat the 5th - this is called a '''tempered''tempered'' fifth'''. (The numbers here and above are slightly off due to rounding and limiting the number of significant digits.)

Equal temperament can be based on any number of distinct non-octave tones. We are mainly concerned with '''12-Tone Equal Temperamenttone equal temperament''' (12-TET), which is how most modern guitars are designed - the 12 tones correspond to the 12 notes of the classical chromatic scale.

12-Tone Equal Temperament tone equal temperament can be derived by saying that each semitone increases the pitch by x, and 12 semitone movements should reach the octave. In other words,

Let's compare this tuning scheme against Just Intonationjust intonation.

For the case (specific key) where Just Intonation just intonation is "perfect", Equal Temperament equal temperament is off by a maximum of 17.5 cents on the tritone interval which is dissonant regardless. The 3rds and 6ths are a bit off, but most people go through life without ever realizing they have more consonant Just Intonation just intonation counterparts. For most musicians and listeners, the intervals are close enough to work without distracting from the music. Compositionally, it allows modulation into any key without increasing dissonance to the point where things sound out of tune. From the perspective of instrument-builders and musicians, it simplifies the experience, rather than having to worry about adjusting tuning or technique to compensate when using alternate keys.

Now let's compare Equal Temperamentequal temperament's intervals to Just Intonation just intonation when we play in an alternative key. Again, we use the case of an instrument in just intonation for C Major, but actually playing E Major.

You can see quite a big difference now. What appear to be wolf intervals for C Major Just Intonation just intonation are now simply mildly detuned intervals.

Above, we have discussed 12-tone Equal Temperament equal temperament (12-TET). Other common equal temperament tunings are 19-TET and 31-TET. These simply divide the octave into 19 and 31 tones (excluding the octave), just like 12-TET. Adding more possible pitches gives players more choices, able to use certain pitches for a note in one context but different pitches in another. Guitars built on 19 and 31-TET exist, and appear to have an extreme number of equally-spaced frets.

'''''Microtonal''''' music refers to music that uses intervals outside of the 12-tone chromatic scale, such as by subdividing semitones/half-steps. This can be from using notes in an equal temperament with a number of notes greater than 12 that do not align with the 12 intervals associated with Western music. For example, in 31-TET above, 150.43 HZ would be a note between D and D{{s}} - a ''micro-tonal'' note. Also, some forms of music developed outside of Western influence use such intervals and can be considered micro-tonal (even if their scales consist of 12 or less notes). Additionally, the blue notes in ''Blues'' can be considered micro-tonal, often expressed on guitar using quarter bends, with no strict definition on the exact interval relationships of the blue notes to the tonic. ''Jazz'' also typically employs micro-tonal techniques, including blue notes.