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== Inconsistencies between 12 Tones and Pure Intervals ==
In the idealized world, intervals are very harmonic ratios between pitches. For example, a perfect 5th is a 3:2 ratio. This doesn't exactly match up to how we define the 12-tone chromatic scale, where each 5th is another note inside the chromatic scale, but each octave is 2x the pitch of a note. An A note at 55 HZ Hz should follow the circle of fifths, through 12 movements to arrive back at A. But if we multiply the pitch by 3:2 each time...
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7136.05 HZ Hz is not 55 HZ Hz multiplied by some power of 2. A8 should be 2x A7, 4x A6, 8x A5...128x A1. 55 HZ Hz x 128 = 7040 HZHz. We are sharp by 1.364%. What gives? ''This is simply an anomoly of trying to marry the mathematically incompatible concepts of pure-tuned intervals to the 12-note chromatic scale.'' This is highlighted clearly when trying to marry the 3:2 ratio 5th interval with the 2:1 ratio octave:
n * (3/2)^x = n * 2^y
2^(y/x + 1) = 3
Older tuning methods typically started at a certain note and derived other notes by moving up and down from it using some small whole number ratio. For example, moving up and down from A 440 HZ Hz in fifths defined as 3:2...
upward: A 440, E 660, B 990, F# 1485, C# 2227.5, G# 3341.25, D# 5011.88, A# 7517.81
'''Equal temperament''' 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 7040hZ 7040 Hz A8 when moving in 5ths? This would be 55hZ 55 Hz x ___^12 = 7040hZ7040 Hz. Moving things around, it is ___ = (7040hZ7040 Hz/55hZ55 Hz)^(1/12), which equals ~1.49831. Thus, we need to slightly flat the 5th - this is called a '''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 temperament''' (12-TET), which is how most modern guitars are designed - the 12 tones correspond to the 12 notes of the classical chromatic scale.
=== Micro-Tonality ===
'''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 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.
Microtonal music can be played on guitar by using alternate tunings, [[#Just-Intonated and Special Instruments|specialized guitars]], [[Bending|bends]], and [[Slide Guitar|slide guitar]].
! Harmonic !! 1st !! 2nd !! 3rd !! 4th !! 5th !! 6th
|-
! Harmonic Series for A Pitches (HZHz)
| 110 || 220 || 330 || 440 || '''550''' || 660
|-
! Harmonic Series for C{{s}} Pitches (HZHz)
| 137.5 || 275 || 412.5 || '''550''' || 687.5 || '''825'''
|-
! Harmonic Series for E Pitches (HZHz)
| 165 || 330 || 495 || 660 || '''825''' || 990
|-
! 12-TET Harmonic Series for C{{s}} Pitches (HZHz)
| 138.59 || 277.18 || 415.77 || '''554.36''' || 692.95 || '''831.54'''
|-
! 12-TET Harmonic Series for E Pitches (HZHz)
| 164.81 || 329.62 || 494.43 || 659.24 || '''824.05''' || 988.86
|}
Let's now look at a whole triad. If the fundamentals of each note form the nice pure tuning intervals (5:4 and 3:2 for M3 and P5 respectfully), then we see a few overtones line-up perfectly. The C{{s}} series shares 550 HZ Hz with A and 825 HZ Hz with E.
Compare these to equal temperament. We know the C{{s}} is sharp by ~0.79%, which is fairly audible as a M3 when using a sine wave with no overtones whatsoever. This gets worse with overtones - notice the 550 in the A series now matches up to 554.36 in the C{{s}} series. This produces a clear detuned effect. Things are even worse between the C{{s}} and E - since the C{{s}} is slightly sharp and E is slightly flat, the intervals between them are even sharper. Instead of 825 HZ Hz on each, we have 831.54 and 824.05, which is ~0.9% pitch difference. This may venture past a simply detuned sound and create noticeable beating between the pitches.
So, when playing distorted rhythm guitar, it's often best to play power chords, and let the harmonic series fill in the M3 tonality to the chord. Playing the M3 is sure to create beating. Playing a minor triad can be particularly nasty, as the m3 fundamental will clash with the M3 overtone of the root note. On the other hand, sus2 and sus4 chords tend to sound quite nice. For "bigger" sounding chords when using distortion, try to use as many octaves as possible. 2 notes is the lazy approach. 3 is almost just as easy. 4 is a bit more difficult but really starts to get that big tone and has a balance between the root and 5th. 5 and 6 note power chords are usually more difficult and for specialized positions (although alternate tunings like Drop D make it much easier) but really flesh out the guitar sound.