Equal Temperament, Alternate Tunings, and Microtonality
Before now, we've always taken for granted exactly what pitch was associated with each note. We tuned our open strings with our tuner and assumed our frets were spaced correctly to give us the desired pitches. However, this is a much deeper subject. There are many different tuning systems that define which fundamental pitches belong to each note, changing how chords and melodies sound.
Equal temperament is a tuning system that divides each octave into a certain number of equally-spaced pitches on a logarithmic scale. In other words, each semitone movement changes pitch by the same relative amount for all notes. The most common equal temperament is 12-tone equal temperament (12-TET), and this is the most common modern tuning system, found on virtually every mass-produced guitar. This goes beyond the pitches that the strings are tuned to, involving the scale length of each string and the location of where each fret contacts the string. It is not something that can be adjusted, at least not without replacing the fretboard.
The main advantage of equal temperament is that it is key-neutral. The pitch ratio between any two notes for a type of interval are consistent for all possible keys based on the 12-tone scale. These intervals will deviate slightly from their "ideal" (most harmonic) pure tuning intervals found in a just intonation tuning system. When we deviate from the just or pure intervals, we call this a tempered interval. Various other temperaments exist, using different methodologies to determine how the intervals should be tempered and what their relative pitches are, as seen below.
This article is just a taste of the depth that all the math actually produces. See the Wikipedia page for more info.
Contents
Cents
Cents are a measure of relative pitch between notes, based on a logarithmic scale (so that octaves appear on a linear rather than exponential scale). Each equal-tempered semitone between two notes is divided into 100 cents. Each octave contains 1,200 cents. These are often used to measure how flat/sharp a note is. For example, being 50 cents sharp for C♯ is equivalent to being 50 cents flat for D♭. Cents can also be used to specify intervals. For example, 700 cents is the perfect fifth interval in equal temperament, while ~702 cents is the perfect fifth for just intonation. This provides a useful unit of measure between how different tunings systems define the common 12-tone intervals.
Cents can be calculated using the following formula, if two pitches are known:
n = 1,200 * log base 2 (b/a) which can be rewritten as follows so you can plug it into a calculator... n = 1,200 * log (b/a) / log (2)
To get the pitch of a note given an initial note pitch and the cent difference:
b = a * 2^(n/1200)
GSU provides a handy webpage to calculate cents based on two pitches.
Generally, humans cannot differentiate a difference in pitch when the difference is < 5 cents. Greater than ~12 cents produces a clearly audible detuned effect, with noticeable beating as it gets to about 20 cents.
Background
By the end of this page, it may seem like 12-TET is clearly the best choice as well as the simplest to derive. Yet historically, it seems like it was one of the last tuning systems to be developed (outside of China), with Pythagorean Tuning, Meantone Temperament, Just Intonation and Well-Temperament all being adopted before 12-TET. Here we explore why these tunings were developed in such order.
Pythagorean tuning was developed in ancient Greece (~500 B.C.?), long before any common understandings of music and human perception of sound had been established. Pythagoras noted that pitches using small integer ratios produced harmonious sounds. He used the perfect fifth's 3:2 ratio to derive the entire chromatic scale by traversing the circle of fifths. This produces dissonant 3rds and at least one fifth that sounds quite out-of-tune, but this conformed with how ancient Greece composed and listened to music. There were no big orchestras filling rich harmonies - only a few reed and string instruments in addition to vocalists. Instruments had quite limited ranges compared to modern instruments such as a grand piano.
Meantone Temperament was not developed until about 2,000 years later (~1,500 A.D.), using the same methodology to derive the chromatic scale as Pythagoras, but using a tempered rather than just 5th. The degree to which the fifth was tempered determined various flavors of meantone, with quarter-comma meantone becoming the most popular. This sought to slightly alter Pythagorean tuning so that the resulting 3rd intervals were more consonant (the M3 in Quarter-Comma Meantone is a 5:4 ratio) and the dissonance of the wolf fifth was reduced. Although it still had some dissonant 5th intervals between certain notes, it allowed more complex harmonies to find a consonant expression, opening up the amount of harmonic texture a composer could use.
Just Intonation came shortly after, with the realization that one can create a scale of small whole number ratios for each interval by building the scale off of alternating M3 and m3 intervals, using ratios of 5:4 and 6:5. 5:4 (M3) and 6:5 (m3) combine to make a 3:2 (P5) ratio. Thus, this method quickly grabs all the notes of the diatonic major scale:
| F | A | C | E | G | B | D |
|---|---|---|---|---|---|---|
| P4 | M6 | U | M3 | P5 | M7 | M2 |
| 4:3 | 5:3 | 1:1 | 5:4 | 3:2 | 15:8 | 9:4 |
However, this still produced some undesirable features, as seen in the next section. This was also about the time that trade with the Far East opened up. China had done extensive research on equal temperament and was likely brought to Western culture around this time. It was adopted slowly as Baroque composers began incorporating more and more modulations into their tunes, and keyed instruments such as the harpsichord became prominent. The Romantic period experimented much more with modulations and chromaticism, securing a widespread adoption. 20th century atonality and serialism went so far as to not only favor but require it.
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 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...
| Movement | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Note | A1 | E2 | B2 | F♯3 | C♯4 | G♯4 | D♯5 | A♯5 | E♯6 | B♯6 | F 7 |
C8 |
G8 (= A8)
|
| Pitch | 55 | 82.5 | 123.75 | 185.63 | 278.44 | 417.66 | 626.48 | 939.73 | 1409.59 | 2114.38 | 3171.58 | 4757.37 | 7136.05 |
7136.05 Hz is not 55 Hz multiplied by some power of 2. A8 should be 2x A7, 4x A6, 8x A5...128x A1. 55 Hz x 128 = 7040 Hz. 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 (3/2)^x = 2^y (but no power of 2 can have 3 in its prime factorization, which a power of 3/2 must have) x * ln(3/2) = y * ln(2) x * ln(3/2) / ln(2) = y ln(3/2) / ln(2) = y/x log2(3/2) = y/x 2^(y/x) = 3/2 (either y or x must not be a whole number for this to be true) 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 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 downward: A 440, D 293.33, G 195.56, C 130.37, F 86.91, Bb 57.94, Eb 38.63, Ab 25.75
These pitches were realized that the octave was a consonant 2:1 ratio, so the pitches could be normalized inside of one octave by multiplying or dividing by 2:
upward: A 440, B 495, C# 556.88, D# 626.48, E 660, F# 742.5, G# 835.31 downward: A 440, Bb 463.54, C 521.48, D 586.67, Eb 618.05, F 695.31, G 782.23
Note: upward movement produces the Lydian mode of the root note, while downward produces the Locrian mode.
Other octaves were simply 2x or 1/2 the frequencies for that octave. This avoids out-of-tune octaves, but introduces a new issue. Since the ratios don't work out perfectly, we get one P5 interval that's way off:
Wolf Intervals
| First Note | Second Note | Pitches | Ratio | Cents |
|---|---|---|---|---|
| A | E | 660 / 440 | 1.5 | 701.96 |
| Bb | F | 695.31 / 463.54 | 1.5 | 701.96 |
| B | F# | 742.5 / 495 | 1.5 | 701.96 |
| C | G | 782.23 / 521.48 | 1.5 | 701.96 |
| C# | G# | 835.31 / 556.88 | 1.5 | 701.96 |
| D | A | 880 / 586.67 | 1.5 | 701.96 |
| Eb | Bb | 927.08 / 618.05 | 1.5 | 701.96 |
| E | B | 990 / 660 | 1.5 | 701.96 |
| F | C | 1042.96 / 695.31 | 1.5 | 701.96 |
| Gb | Db | 1113.76 / 742.5 | 1.5 | 701.96 |
| G | D | 1173.34 / 782.23 | 1.5 | 701.96 |
| Ab | Eb | 1236.1 / 835.31 | 1.47981 | 678.49 |
The A♭ - E♭ interval above is called a wolf fifth, a very noticeable divergence from the desired ratio - it "howls", with noticeable beating between the notes. For the above tuning scheme, this interval is present in 6 of the 12 major/natural minor keys based on each note of the 12-tone scale.
Depending on the tuning scheme, intervals other than the perfect fifth may also deviate widely from the pure tuning ratio, creating a dissonant harmony. These are often called wolf intervals. They may be present in every key rather than just a subset of those possible based on the 12-tones.
Demonstration of Issues with Just Intonation
It is possible to make an instrument such that for some note, all the intervals against that note closely reflect their pure tuning intervals - this is called just intonation. The issue is that it can only be constructed this way for a single note - for other notes, the intervals will actually be further from their pure tunings.
| Note Name | Pitch | Interval Name | Just Intonation Ratio |
|---|---|---|---|
| C | 130.8 | P1 | 1:1 = 1 |
| C♯ | 139.52 | m2 | 16:15 = 1.0666... |
| D | 147.15 | M2 | 9:8 = 1.125 |
| D♯ | 156.96 | m3 | 6:5 = 1.2 |
| E | 163.5 | M3 | 5:4 = 1.25 |
| F | 174.4 | P4 | 4:3 = 1.333... |
| F♯ | 183.12 | tt | 45:32 = 1.40625 |
| G | 196.2 | P5 | 3:2 = 1.5 |
| G♯ | 209.28 | m6 | 8:5 = 1.6 |
| A | 218 | M6 | 5:3 = 1.666... |
| A♯ | 232.53 | m7 | 9:5 = 1.8 |
| B | 245.25 | M7 | 15:8 = 1.875 |
| C | 261.6 | P8 | 2:1 = 2 |
Take those same pitches and see what the actual intervals are in another key:
| Note Name | Pitch | Interval Name | Just Intonation Ratio | JI Cents | Actual Ratio | Actual Cents | Cent Difference from Just Intonation |
|---|---|---|---|---|---|---|---|
| E | 163.5 | P1 | 1:1 = 1 | 1 | 0 | 0 | |
| F | 174.4 | m2 | 16:15 = 1.0666... | 111.73 | 1.0666... | 111.73 | 0 |
| F♯ | 183.12 | M2 | 9:8 = 1.125 | 203.91 | 1.12 | 196.20 | -7.71 |
| G | 196.2 | m3 | 6:5 = 1.2 | 315.64 | 1.2 | 315.64 | 0 |
| G♯ | 209.28 | M3 | 5:4 = 1.25 | 386.31 | 1.28 | 427.37 | +41.06 |
| A | 218 | P4 | 4:3 = 1.333... | 498.04 | 1.333... | 498.04 | 0 |
| A♯ | 232.53 | tt | 45:32 = 1.40625 | 590.22 | 1.4222... | 609.68 | +19.46 |
| B | 245.25 | P5 | 3:2 = 1.5 | 701.96 | 1.5 | 701.96 | 0 |
| C | 261.6 | m6 | 8:5 = 1.6 | 813.69 | 1.6 | 813.69 | 0 |
| C♯ | 279.04 | M6 | 5:3 = 1.666... | 884.36 | 1.70666... | 925.42 | +41.06 |
| D | 294.3 | m7 | 9:5 = 1.8 | 1,017.60 | 1.8 | 1,017.60 | 0 |
| D♯ | 313.92 | M7 | 15:8 = 1.875 | 1,088.27 | 1.92 | 1,129.33 | +41.06 |
| E | 327 | P8 | 2:1 = 2 | 2 | 0 | 0 |
Yikes, half of the intervals are off from the ideal pure tuning ratios by quite a bit. That will definitely be noticeable. Above 6 cents is audible, and 12 cents is noticeably dissonant. Above 20 cents will sound quite out of tune. 40 cents is off by almost a half of a semi-tone! Wolf intervals galore!
Equal Temperament
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 7040 Hz A8 when moving in 5ths? This would be 55 Hz x ___^12 = 7040 Hz. Moving things around, it is ___ = (7040 Hz/55 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.
12-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,
currentPitch * (1 + x)^12 = 2 * currentPitch.
This reduces to
x = 2^(1/12) - 1 = ~.05946.
Each semitone increases the pitch by just under 6%.
Let's compare this tuning scheme against just intonation.
| Interval Name | Just Intonation Ratio | Equal Temperament | Just Intonation Cents | 12-TET Cents | Difference |
|---|---|---|---|---|---|
| m2 | 16:15 = 1.0666... | 1.05946 | 111.73 | 100 | -11.73 |
| M2 | 9:8 = 1.125 | 1.1224555 | 203.91 | 200 | -3.91 |
| m3 | 6:5 = 1.2 | 1.1891967 | 315.64 | 300 | -15.64 |
| M3 | 5:4 = 1.25 | 1.2599063 | 386.31 | 400 | +13.69 |
| P4 | 4:3 = 1.333... | 1.3348204 | 498.04 | 500 | +1.96 |
| tt | 7:5 = 1.4 | 1.4141888 | 582.51 | 600 | +17.49 |
| P5 | 3:2 = 1.5 | 1.4982764 | 701.96 | 700 | -1.96 |
| m6 | 8:5 = 1.6 | 1.5873640 | 813.69 | 800 | -13.69 |
| M6 | 5:3 = 1.666... | 1.6817486 | 884.36 | 900 | +15.64 |
| m7 | 16:9 = 1.777... | 1.7817454 | 996.09 | 1,000 | +3.91 |
| M7 | 15:8 = 1.875 | 1.8876880 | 1088.27 | 1,100 | +11.73 |
For the case (specific key) where just intonation is "perfect", 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 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 temperament's intervals to 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.
| Note Name | 12-TET Pitch | Interval Name | Just Intonation Ratio | Actual Ratio | 12-TET Cent Difference | JI Alt Cent Difference |
|---|---|---|---|---|---|---|
| E | 163.5 | P1 | 1:1 = 1 | 1 | 0 | 0 |
| F | 173.22 | m2 | 16:15 = 1.0666... | 1.059 | -11.73 | 0 |
| F♯ | 183.52 | M2 | 9:8 = 1.125 | 1.122 | -3.91 | -7.71 |
| G | 194.44 | m3 | 6:5 = 1.2 | 1.189 | -15.64 | 0 |
| G♯ | 206 | M3 | 5:4 = 1.25 | 1.260 | +13.69 | +41.06 |
| A | 218.25 | P4 | 4:3 = 1.333... | 1.335 | +1.96 | 0 |
| A♯ | 231.22 | tt | 7:5 = 1.40625 | 1.414 | +17.49 | +27.17 |
| B | 244.97 | P5 | 3:2 = 1.5 | 1.498 | -1.96 | 0 |
| C | 259.54 | m6 | 8:5 = 1.6 | 1.587 | -13.69 | 0 |
| C♯ | 274.97 | M6 | 5:3 = 1.666... | 1.682 | +15.64 | +41.06 |
| D | 291.32 | m7 | 16:9 = 1.777... | 1.782 | +3.91 | +21.51 |
| D♯ | 308.65 | M7 | 15:8 = 1.875 | 1.888 | +11.73 | +41.06 |
| E | 327 | P8 | 2:1 = 2 | 2 | 0 | 0 |
You can see quite a big difference now. What appear to be wolf intervals for C Major just intonation are now simply mildly detuned intervals.
Equal Temperament for over 12 Tones
Above, we have discussed 12-tone 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.
The pitches for these systems follow the same formula:
- 12-TET
currentPitch * (1 + x)^12 = 2 * currentPitch x = 2^(1/12) - 1 = ~5.946%.
- 19-TET
currentPitch * (1 + x)^19 = 2 * currentPitch x = 2^(1/19) - 1 = ~3.716%.
- 31-TET
currentPitch * (1 + x)^31 = 2 * currentPitch x = 2^(1/31) - 1 = ~2.261%.
| Note | Pure Interval Pitch | 12-TET Pitch | 19-TET Pitch | 31-TET Pitch | Pure Interval Ratio |
12-TET Ratio | 19-TET Ratio | 31-TET Ratio |
|---|---|---|---|---|---|---|---|---|
| A | 110 | 110 | 110 | 110 | 1 | 1 | 1 | 1 |
| 112.49 | 0 | 0 | 1.02 | |||||
| 114.09 | 115.03 | 0 | 1.04 | 1.05 | ||||
| A# | 117.33 | 116.54 | 118.33 | 117.63 | 1.07 | 1.06 | 1.08 | 1.07 |
| 120.29 | 0 | 0 | 1.09 | |||||
| B | 123.75 | 123.47 | 122.72 | 123.01 | 1.13 | 1.12 | 1.12 | 1.12 |
| 125.79 | 0 | 0 | 1.14 | |||||
| 127.28 | 128.64 | 0 | 1.16 | 1.17 | ||||
| C | 132 | 130.81 | 132.01 | 131.55 | 1.2 | 1.19 | 1.2 | 1.2 |
| 134.52 | 0 | 0 | 1.22 | |||||
| C# | 137.5 | 138.59 | 136.92 | 137.56 | 1.25 | 1.26 | 1.24 | 1.25 |
| 140.67 | 0 | 0 | 1.28 | |||||
| 142 | 143.85 | 0 | 1.29 | 1.31 | ||||
| D | 146.67 | 146.83 | 147.28 | 147.11 | 1.33 | 1.33 | 1.34 | 1.34 |
| 150.43 | 0 | 0 | 1.37 | |||||
| D# | 154 | 155.56 | 152.75 | 153.83 | 1.4 | 1.41 | 1.39 | 1.4 |
| 158.43 | 157.31 | 0 | 1.44 | 1.43 | ||||
| 160.87 | 0 | 0 | 1.46 | |||||
| E | 165 | 164.81 | 164.31 | 164.51 | 1.5 | 1.5 | 1.49 | 1.5 |
| 170.42 | 168.23 | 0 | 1.55 | 1.53 | ||||
| 172.03 | 0 | 0 | 1.56 | |||||
| F | 176 | 174.61 | 176.75 | 175.92 | 1.6 | 1.59 | 1.61 | 1.6 |
| 179.9 | 0 | 0 | 1.64 | |||||
| F# | 183.33 | 185 | 183.32 | 183.97 | 1.67 | 1.68 | 1.67 | 1.67 |
| 190.13 | 188.13 | 0 | 1.73 | 1.71 | ||||
| 192.38 | 0 | 0 | 1.75 | |||||
| G | 195.56 | 196 | 197.19 | 196.73 | 1.78 | 1.78 | 1.79 | 1.79 |
| 201.18 | 0 | 0 | 1.83 | |||||
| G# | 206.25 | 207.65 | 204.52 | 205.73 | 1.88 | 1.89 | 1.86 | 1.87 |
| 212.12 | 210.38 | 0 | 1.93 | 1.91 | ||||
| 215.14 | 0 | 0 | 1.96 | |||||
| A | 220 | 220 | 220 | 220 | 2 | 2 | 2 | 2 |
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 would be a note between D and D♯ - 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, specialized guitars, bends, and slide guitar.
Alternate Tuning Systems for Guitar
Buzz Feiten Tuning System
The Buzz Feiten Tuning System (BFTS) is a patented design to help try to get better intonation throughout the fretboard. Specifically, it tries to eliminate two issues: sharp notes on the first 3 frets and inconsistent harmonic perception of intervals between different chord shapes.
The first issue is solved by placing the nut slightly closer to the first fret. The increased tension necessary to fret those frets is what makes them sharp, so shortening the scale length compensates for this. Existing non-BFTS guitars can be retrofitted (by Buzz Feiten or licensed luthiers) with the BFTS without having to recut the neck by using a shelf nut that extends a bit on top of the fretboard.
The second issue is solved by using an alternate method of intonating each string according to patented formulas he worked up. This boils down to bridge saddle adjustments. These deviate from the standard methods of intonating a guitar and may seem as though you will get the wrong results. But apparently the magic isn't about exactly what pitches you get, but how the pitches in any chord relate to each other. The human ear has tolerance for certain intervals being out-of-tune more than others, with more intracacies when multiple intervals are occurring in chords.
Here's the FAQ on the Buzz Feiten site. Buzz describes the need for the tuning system in this video:
I have not used this system myself, and I don't know if I ever will. I do not like the shroud of mystery surrounding the process and the need to use an authorized luthier to do the adjustment, which are byproducts of his patent and trying to extract royalties from it. It seems like the more cooperative and helpful thing to do would have been to publish his findings and let anyone freely use his system, whereas he is more interested in making money over propagating his idea. And so the BFTS is not widely employed throughout the guitar market. Those who adopted seem to like it greatly, but there's tons of legendary players that do not feel the need to adapt.
True Temperament Fretting Systems and Compensated Nuts
The True Temperament Fretting System uses "squiggly" frets, so that each string will have a slightly different length for each fret (ignoring different bridge saddle adjustments). The makers of this system claim that string pitch is incorrectly simplified as a linear equation using string diameter and length - there is more advanced physics going on, and this system compensates for that.
The big mistake is that people assume this is a just intonation, designed for a specific key. No, this is still equal temperament - it is just a supposed improvement to intonate better towards an ideal equal temperament than traditional straight fretting. However, they do provide a few variations that move more towards just intonation for a few keys.
I would really like to play one of these guitars to see if the supposed benefits are clearly noticeable as well as if there is any trade-off on playability - it seems like bends might feel a bit weird, and there's a possibility some chords will be harder to finger. I am also curious about maintenance - I do not know if the frets can be crowned using traditional methods.
Compensated nuts are nuts that vary the point at which the string leaves the nut for each string. Thus, if your bridge saddles are all set to the same spot, you would still have different string lengths for open strings. This applies the true temperament concept above to only the nut. So it can help for open position chords to sound in-tune regardless of which chord shape you use; but beyond open position, there are no benefits.
Just-Intonated and Special Instruments
Some guitars are designed to have just intonation for some key, assuming a certain tuning (most likely Standard E). One way this is accomplished by using "squiggly" frets similar to True Temperament, varying the distance from fret to bridge per string for each fret.
Another way is to use a special fretboard where all frets are divided into 6ths (or however many strings are on the guitar) and adjustible forwards or backwards per string. This allows the guitar to be setup to be just-intonated for any specific key, but it would be difficult to try to adjust in the middle of a song.
A variation on the above is to use an equal-tempered fingerboard with continuous frets in their normal position, but allow adding single-string-sized frets between the normal frets to give additional pitch options for certain notes. This could be setup so you can get pure intervals in a certain key or possibly even multiple keys. You could also resort to micro-tonal melodies.
And similar to that are equal temperament guitars that use more than 12 tones, such as 31-tone equal temperament (31-TET). This allows the player to play in any key with more options of exactly what pitches he wants for certain notes, potentially allowing him to play closer to just intonation for any key than a 12-TET instrument.
EverTune Bridge
EverTune makes the EverTune Bridge, which uses springs and levers to maintain constant tension throughout all the strings, preventing them from slipping out-of-tune. It is not necessarily a tuning system and should work with various tuning paradigms as well as intonation settings. I just wanted to mention it here to prevent confusion.
Worth It?
Guitars have been used for a long time before any of the above tuning/intonation systems were invented. A luthier even told me that Gibson was using an incorrect formula to place their frets along the fretboard, so they were impossible to get anything close to true equal temperament; yet players still enjoyed them and made them a popular brand.
It all depends on your perception and the style of music you play. For styles where you have to play lots of bigger chords, better intonation is going to be more noticeable. Also, the differences will be more apparent on brighter and more distorted guitar tones than dark and clean, since the more overtones add more noticeable harmonic divergences from ideal and the beating that occurs in distortion makes even slight harmonic issues quite obvious.
However, for distorted lead playing, there would likely be little noticeable difference.
EVH Tuning
I've read that Eddie Van Halen likes to tune his G and B strings to a pure-tuned M3. This ensures that the open A shape for chords gets a nicer sound to them. EVH uses M3's between the G and B strings pretty often, so it works well for him.
The downside is that m3 intervals between those strings will now sound further away from the pure-tuned m3. Also, the B and E strings will be out-of-tune, or you have to tune the B and E strings together, but then octaves using the G and E strings or D and E strings will be out-of-tune. So it's a compromise, not a holistic solution, and whether it's worth incorporating depends on your style of play. It requires using certain chord shapes and only using certain notes within those chord shapes to sound best.
I find it works well when you are using lots of A Major chord shapes and smaller chords or intervals on the G and B strings, which is more typical for rock playing. If you find the M3 between the G and B strings are creating lots of dissonant beating, particularly with distortion, it is worth checking out. If you need to play larger chords, such as in Jazz, this approach will likely lead to frustration.
Bending One Note in an Interval to Compensate
If you are just playing M3 or m3's on adjacent strings, you can bend one of the notes to get closer to the pure-tuned m3 or M3. For a major 3rd, you bend the lower note slightly, decreasing the ratio between notes. For a minor 3rd, you bend the higher note slightly, increasing the ratio between them.
You'll have to use your ear to find just how much bend to use. Also keep in mind that in a band context, if you are bending the same note that another equal-temperment instrument is playing without bending, those notes may now sound out-of-tune, likely moreso than the equal temperament m3 and M3. This technique works best for distorted rock guitar where there's no piano or synth playing the same notes underneath. If you are just playing over bass, I recommend trying this approach. It works well for solo'ing around the 8th fret or higher on the G and B strings - giving you enough distance from the rhythm players.
M3 m3 B--5--5---------5--5b5.2*-- G--5--5b5.2*----6--6------- *5.2 is approximate
Harmonics and Overtones
I want to point out that harmonics and overtones, being perfect integer multiples of the fretted/open note, follow the pure interval ratios, diverging from equal temperament.
For example, the first few harmonics on open E are 12th fret E (2:1), 7/19th fret B (3:1), 5/24th fret E (4:1), 3.8th fret G♯ (5:1). We can divide these frequency ratios by 2 (moving an octave lower) until we get a ratio that's between 1:1 and 2:1 - expressing the ratio in terms of the notes being inside a single octave. Looking at the B and G♯, we see 3:2 and 5:4, the pure tuning ratios for a P5 and a M3. These pitches diverge from equal termperament. If we were to play a 3.8th fret G♯ harmonic on the low E string, that should give us roughly the same pitch as playing 4th fret (non-harmonic) G♯ on the high E string. However, if you play both these notes simultaneously, you'll notice that the pitches do not align, sounding out-of-tune and possibly even getting a slight beating between the pitches.
Overtones work the same way. Guitar has plenty of overtones naturally, with tons more once distortion is added. A low open E has overtones that follow the harmonic series, again containing the G♯ 5x the pitch of the E fundamental as seen above. Again, if you play an open E with distortion and play a 4th fret G♯ on the high string, the low E's overtone will clash with the G♯ fundamental.
This becomes particularly apparent when playing chords with distortion. The power chord is usually preferred over full chords for this exact reason. Since equal temperament's 5ths are very close to the pure ratio, there seems to be less clashing
| Harmonic | 1st | 2nd | 3rd | 4th | 5th | 6th |
|---|---|---|---|---|---|---|
| Harmonic Series for A Pitches | 110 | 220 | 330 | 440 | 550 | 660 |
| Harmonic Series for E Pitches | 165 | 330 | 495 | 660 | 825 | 990 |
| 12-TET Harmonic Series for E Pitches | 164.81 | 329.62 | 494.43 | 659.24 | 824.05 | 988.86 |
Notice how both series have 330 and 660 in common. When using equal temperament, the fundamental pitch of E is a bit flat from the 3:2 ratio, but it's quite close (~0.1% difference). The harmonic series are thus quite harmonic, if very, very slightly "detuned-sounding".
| Harmonic | 1st | 2nd | 3rd | 4th | 5th | 6th |
|---|---|---|---|---|---|---|
| Harmonic Series for A Pitches (Hz) | 110 | 220 | 330 | 440 | 550 | 660 |
| Harmonic Series for C♯ Pitches (Hz) | 137.5 | 275 | 412.5 | 550 | 687.5 | 825 |
| Harmonic Series for E Pitches (Hz) | 165 | 330 | 495 | 660 | 825 | 990 |
| 12-TET Harmonic Series for C♯ Pitches (Hz) | 138.59 | 277.18 | 415.77 | 554.36 | 692.95 | 831.54 |
| 12-TET Harmonic Series for E Pitches (Hz) | 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♯ series shares 550 Hz with A and 825 Hz with E.
Compare these to equal temperament. We know the C♯ 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♯ series. This produces a clear detuned effect. Things are even worse between the C♯ and E - since the C♯ is slightly sharp and E is slightly flat, the intervals between them are even sharper. Instead of 825 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.
Other Tuning Systems
Pythagorean Tuning
Pythagorean Tuning was a common tuning system in Western civilization from the time of the ancient Greeks to not long after the Renaissance began. It is based upon using the 3:2 ratio for a perfect fifth and traversing the circle of fiths in both directions from a tonic pitch to derive all other notes. As pitches move upwards in 5ths, they quickly leave the octave of the starting pitch - these pitches are divided by 2 (lowered by an octave) until they fit back into the tonic's octave.
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Notice that we end up with supposedly enharmonic notes (G♯ and A♭), yet they do not share the same pitches or intervals against the tonic note. One can be dropped for the complete 12-tone scale, or the choice of which one to use can be left to context to avoid wolf intervals.
Meantone Temperament
Meantone temperament is a tuning system that follows a similar logic as Pythagorean tuning; however, rather than use a 3:2 ratio for the perfect 5th to derive the other notes, it uses a perfect 5th slightly flat of that ratio, such that a specific M3 interval is reached. This avoids the somewhat dissonant 3rds that Pythagorean tuning produces, while following the same methodology to derive its tones.