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I know that guitar is an imperfect music instrument as in standard tuning the intonation is slightly off. When playing a given chord, one or more notes will be slightly out of tune.

I wanted to know if, despite that fact, some keys were actually "perfectly intonated" for that particular key (as in just intonation).

I am strictly speaking about standard tuning here but I will accept slight modification in the tuning to get a key with a temperament that comes closer to just intonation for that key.

  • By "perfect" do you mean just intonation or equal temperament? – Todd Wilcox Jun 3 '17 at 15:16
  • I mean true temperament. – Joulin Nicolas Jun 3 '17 at 15:55
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    True Temperment is a brand name for a system of products that can be used to build new guitars or modify existing guitars in order to have more control of the intonation of each fret. It's not a temperment in the sense of being a tuning compromise intended to deal with the Pythagorean comma. I'm not familiar with a temperment in the latter sense called perfect temperment or perfect intonation. – Todd Wilcox Jun 3 '17 at 18:14
  • Intonation is a word that describes tuning an instrument to play a particular tone or note of a particular frequency. Intonation of a guitar involves adjusting the scale length to affect the point on the string where the length of the portion of the string allowed to vibrate will result in production of a given sound frequency note with a given string tension. Intonation on a piano can describe how the piano is tuned. The word temperament describes a process where just intonation (perfect for a given key) is adjusted or tempered, to allow a single instrument to play in different keys. – Rockin Cowboy Jun 3 '17 at 19:35
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A piano or other keyboard instrument can be tuned to play notes in a given key as close as practically possible to just intonation. But of course attempting to play in a different key on that keyboard would sound noticeably out of tune to varying degrees. So the concept of equal temperament was devised where each note was equally adjusted in a manner that allowed all 12 potential musical keys to be played without sounding too out of tune.

Fretted instruments such as a guitar, are set up to be capable of being in tune with a piano or other keyboard to the extent possible so that if you play a middle C on piano and guitar, they will be close enough to sound the same. In fact you can tune a guitar's open strings with a piano and get them exact.

But after you leave the open strings on a fretted guitar, it will typically be difficult to achieve any type of perfect or just intonation or even consistent matching of notes with an equally tempered keyboard. If you play any fretted note on your guitar you will notice that you can change the pitch simply by pressing down harder (or not as hard) on the fret. So playing style will cause the fretted notes on the same guitar to sound different depending on who is playing the notes.

Another factor that affects a guitars ability to stay in tune from open string to fretted note - would be the string height above the fret board (as Tim alluded to). A guitar with higher action will intone slightly differently between fretted and open notes than one with a lower action. In standard tuning you have 5 notes with 6 frequencies (low E, high e).

On almost any guitar, if you tune the open strings to a piano perfectly, the fretted notes will be slightly off, even if you could guarantee the exact same application of pressure to the string when pressing it down on a fret. It's a further compromise that a guitar must make because you have one string with a given gauge and tension that is expected to play 20 notes. On a piano, each piano key is tied to a unique set of strings that can be tuned the way the open strings on guitar are tuned.

Bottom line is that instruments such as keyboards and fretted instruments such as guitars, ukuleles, bass guitars, banjos, mandolins, etc. are inherently incapable of playing just intonation in multiple keys without retuning the instrument for each different key.

However, a fretless guitar or fretless bass, or instruments such as violin or viola, allow the musician to make the necessary adjustments in finger placement to play in just intonation in whatever key they choose. Thus a string ensemble or quartet can play in any key and adjust the notes to just intonation accordingly. But if you add a fretted or keyboard instrument into the mix, the strings must adjust to the fixed tuning system of the piano or fretted guitar. Otherwise one or the other will sound "out of tune". So they must all be slightly "out of tune" to sound in sync.

For more on the subject of Just Intonation read the enlightening answers to this question on Stack Exchange Music Why Just Intonation is Impractical

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It can be tuned so that one particular chord shape is spot on, but when you finger a different shape, that won't be. Choose any shape you like for the 'perfect'.

  • is that true for just a single chord or also every notes in the keys that chord fits ? – Joulin Nicolas Jun 3 '17 at 15:55
  • I believe it's true for the shape. So to play I, IV and V chords in a song, the same shape , for example an E shape, would have to be moved up and down the neck. – Tim Jun 3 '17 at 16:27
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If the guitar is tuned to equal temperament, every key is equally far from just intonation. That's a consequence of the definition of equal temperament- every key has the same intervals between the notes.

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Simply, no. What makes True Temperament unique is that it "calibrates each of those 132 fretted notes individually". What took me some time to realize is that TT is not its own intonation like Just Intonation or Equal Temperament. It's actually a method of realizing an intonation system, and in TT's case, it is Thidell Formula 1 .

An intoned guitar with a regular fretboard will actually not be truly equal-tempered because of the change in string tension caused by bending a string from its resting position towards the fretboard. True Temperament frets compensate for this.

If you want to properly realize a key in a given intonation system on a regular guitar, tune each string based on the frets on that string that you want to be completely accurate to the intonation system. At most, you can get two "perfect" notes per string by adjusting its tuning (via the pegs) and its scale length (via neck relief and saddle position).

  • While the OP may not know the correct terminology, It is plain they are asking about intonation, not temperament – Doktor Mayhem Jun 5 '17 at 10:25
  • Well, excuse me for assuming that those first two comments on the question implied that OP meant to talk about True Temperament. – Sarkreth Jun 5 '17 at 23:00
  • No worries. Loads of people don't really get the difference. Here it was his wording in the question that indicated intonation rather than temperament. – Doktor Mayhem Jun 6 '17 at 6:32

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