The recent thread about True Temperament is quite technical, the first post of this thread is intended to be a more easily understood explanation and summary.
Later posts could perhaps be a place for people to ask questions about it.
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True Temperament's well known fretting system allows playing in a tonal system known as a 'Well Temperament' (WT).
'True Temperament' (TT) is a brand name, it is not the name of a tonal system. The phrase is described by the TT website to mean the experimentally-based alterations to fret shapes such that fretted pitches are as close as possible to the intended tonal system.
About Well Temperament
--------------------------------
WT was used in Baroque and Classical music, it was used by J.S.Bach and Mozart. It was the type of tonal system used before the dominant modern tonal system called '12 Tone Equal Temperament' (12TET).
A WT tonal system can be described as being 12TET with each of the 12 tones sharpened or flattened by a small number of cents (a cent is one hundredth of a semitone, 0.01 semitones). Below are the pitch offsets in cents from 12TET for TT's particular WT:
C +2
C# -4
D +2
D# -4
E -2
F 0
F# -4
G +4
G# -4
A 0
A# -4
B -1
Due to these offsets, the size of a particular interval (for example, the major third) varies according to which tone is used as the root note.
The intention of a WT is to alter the size of certain intervals such that, when played on the more commonly used root notes, they become closer to the perfect harmony of 'Just Intonation' (JI) intervals.
A much appreciated side effect of a WT is that each key of a particular scale (C major, D major etc.) each has a slightly different character and mood, because all the intervals slightly change in size. This effect is called 'Key colour'.
About JI
----------
A JI interval will be heard as a perfect harmony to the ear. Any interval tuned by ear will be tuned to a JI interval. JI intervals are 'precise harmony' because the waveforms of the 2 pitches are precisely 'in phase' in a special way.
However, surprisingly, 12TET is not composed of JI intervals. The main reason is that it is not possible to freely modulate to different keys in JI, as doing so changes all the interval sizes. This is why historical tonal systems progressed from JI to 'Meantone Temperament' to WT and finally to 12TET, each step increased the ability to modulate and moved further away from perfect harmony.
Only the octave is an exact JI interval.
12TET fourths and fifths are very close at 2 cents away from JI.
The 12TET major second is a barely noticeable 4 cents away from JI.
12TET Thirds, sixths and sevenths, both major and minor, are all significantly away from JI, by around 15 cents, roughly one-sixth of a semitone. This is why a guitar tuned precisely to 12TET will play chords that sound out of tune, especially noticeable when using distortion.
Major thirds in detail
--------------------------
A WT is designed primarily to improve the harmony of major and minor triads, because these were so heavily used in Baroque and Classical music.
Because 12TET fifths are already close to JI, the focus of a WT is therefore improving the major and minor thirds.
Looking at major thirds as an example:
The 12TET major third is, of course, 4.00 semitones on all root notes.
The JI major third interval is 3.86 semitones, 14 cents flat of the 12TET major third.
Consider a major third with it's root note on D, the higher note will be F#.
Looking at the tone offsets of TT's WT:
D +2
...
...
...
F# -4
The major third has become 6 cents smaller, 3.94 semitones, almost halfway towards the JI major third. This will be heard as a significant improvement of it's harmony.
However, an improvement of one major third will cause a major third on a different root note to have worse harmony.
To show this, here's a table of the sizes of major thirds in TT's WT, with the root note on each of the 12 tones:
Root note. Major third (semitones)
C 3.96
C# 4.04
D 3.94
D# 4.08
E 3.98
F 4.00
F# 4.00
G 3.95
G# 4.06
A 3.96
A# 4.06
B 3.97
Reminder: The JI major third is 3.86 semitones.
Of the 12 major thirds:
6 are improved by 2-6 cents (perfect harmony would be an improvement of 14 cents), on the more commonly used root notes C D E G A B.
2 are unchanged.
4 are worse by 4-8 cents.
Re-arranging these major thirds into a circle of fifths shows how the improvements are focussed around the central D of the 'white piano key' notes FCGDAEB. The improvement is greatest for D and decreases in both directions away from it:
D# 4.08
A# 4.06
F 4.00
C 3.96
G 3.95
D 3.94
A 3.96
E 3.98
B 3.97
F# 4.00
C# 4.04
G# 4.06
TT's fretting system is designed to be able to play with other instruments that use 12TET, so the pitch offsets cannot be too large, they have been limited to +-4 cents. This limits how much the harmony of the thirds can be improved.
If all instruments were using WT, a WT with larger offsets and more significant improvements in harmony could be used.
Compared to historical WTs, TT's system is quite subtle.
Summary
------------
TT's fretting system moderately improves the harmony of major and minor thirds on 6 out of the 12 root notes, at the expense of thirds on the other root notes.
The intention is for the musician to play music that is careful to only use the root notes that improve, or do not worsen, the thirds. Also, chords should be mostly major and minor triads.
Being restricted in this way was not much of a problem for Baroque and Classical music, as those often used rather limited and predictable key modulations and chord progressions, and chords were mostly major and minor triads. Mainstream pop and rock music also does so.
Musicians who have a more unconventional use of chords, key modulations and chord progressions will not experience an overall improvement in harmony, because gains cause equal losses elsewhere.
However, 'Key colour' may still make a WT worth using for these musicians.
Misunderstandings
------------------------
It does not result in perfect harmony chords, at best, minor and major thirds are modified halfway towards perfect harmony.
It does not improve all chords. The focus is on major triads with root notes C G D A E B, or minor triads with root notes A E B F# C# G#. Any gain creates an equal loss elsewhere.
The often stated benefit of 'perfect intonation' does not mean chords have perfect harmony, 'intonation' is used here to mean the fretted pitches are very close to the intended tonal system (in this case a WT) whether or not that tonal system results in perfect harmony is a separate matter (and is actually not the case for a WT).
The lack of information on the TT website, and the very common misunderstandings of guitarists and the music media, may give the impression that TT is actually 'perfectly intonated 12TET'. It is actually intentionally slightly out of tune with 12TET.
To improve the harmony of chords you have to be out of tune with 12TET, because 12TET inherently has imperfect harmony.
Disadvantages
-------------------
Due to a WT having offsets for each tone, only standard tuning EADGBE can be used, or transpositions that maintain the relative tuning (for example Eb standard, D standard, F standard).
Due to TT being per-string, per-fret fret offsets determined by experiment to compensate for common string gauges, only a narrow range of gauges can be used, from 9-46 to 11-50.
Due to the unusual and variable fret shapes, pitch response to string bending will be a little more unusual and a little less consistent.
Later posts could perhaps be a place for people to ask questions about it.
/////////////////////////////////////////
True Temperament's well known fretting system allows playing in a tonal system known as a 'Well Temperament' (WT).
'True Temperament' (TT) is a brand name, it is not the name of a tonal system. The phrase is described by the TT website to mean the experimentally-based alterations to fret shapes such that fretted pitches are as close as possible to the intended tonal system.
About Well Temperament
--------------------------------
WT was used in Baroque and Classical music, it was used by J.S.Bach and Mozart. It was the type of tonal system used before the dominant modern tonal system called '12 Tone Equal Temperament' (12TET).
A WT tonal system can be described as being 12TET with each of the 12 tones sharpened or flattened by a small number of cents (a cent is one hundredth of a semitone, 0.01 semitones). Below are the pitch offsets in cents from 12TET for TT's particular WT:
C +2
C# -4
D +2
D# -4
E -2
F 0
F# -4
G +4
G# -4
A 0
A# -4
B -1
Due to these offsets, the size of a particular interval (for example, the major third) varies according to which tone is used as the root note.
The intention of a WT is to alter the size of certain intervals such that, when played on the more commonly used root notes, they become closer to the perfect harmony of 'Just Intonation' (JI) intervals.
A much appreciated side effect of a WT is that each key of a particular scale (C major, D major etc.) each has a slightly different character and mood, because all the intervals slightly change in size. This effect is called 'Key colour'.
About JI
----------
A JI interval will be heard as a perfect harmony to the ear. Any interval tuned by ear will be tuned to a JI interval. JI intervals are 'precise harmony' because the waveforms of the 2 pitches are precisely 'in phase' in a special way.
However, surprisingly, 12TET is not composed of JI intervals. The main reason is that it is not possible to freely modulate to different keys in JI, as doing so changes all the interval sizes. This is why historical tonal systems progressed from JI to 'Meantone Temperament' to WT and finally to 12TET, each step increased the ability to modulate and moved further away from perfect harmony.
Only the octave is an exact JI interval.
12TET fourths and fifths are very close at 2 cents away from JI.
The 12TET major second is a barely noticeable 4 cents away from JI.
12TET Thirds, sixths and sevenths, both major and minor, are all significantly away from JI, by around 15 cents, roughly one-sixth of a semitone. This is why a guitar tuned precisely to 12TET will play chords that sound out of tune, especially noticeable when using distortion.
Major thirds in detail
--------------------------
A WT is designed primarily to improve the harmony of major and minor triads, because these were so heavily used in Baroque and Classical music.
Because 12TET fifths are already close to JI, the focus of a WT is therefore improving the major and minor thirds.
Looking at major thirds as an example:
The 12TET major third is, of course, 4.00 semitones on all root notes.
The JI major third interval is 3.86 semitones, 14 cents flat of the 12TET major third.
Consider a major third with it's root note on D, the higher note will be F#.
Looking at the tone offsets of TT's WT:
D +2
...
...
...
F# -4
The major third has become 6 cents smaller, 3.94 semitones, almost halfway towards the JI major third. This will be heard as a significant improvement of it's harmony.
However, an improvement of one major third will cause a major third on a different root note to have worse harmony.
To show this, here's a table of the sizes of major thirds in TT's WT, with the root note on each of the 12 tones:
Root note. Major third (semitones)
C 3.96
C# 4.04
D 3.94
D# 4.08
E 3.98
F 4.00
F# 4.00
G 3.95
G# 4.06
A 3.96
A# 4.06
B 3.97
Reminder: The JI major third is 3.86 semitones.
Of the 12 major thirds:
6 are improved by 2-6 cents (perfect harmony would be an improvement of 14 cents), on the more commonly used root notes C D E G A B.
2 are unchanged.
4 are worse by 4-8 cents.
Re-arranging these major thirds into a circle of fifths shows how the improvements are focussed around the central D of the 'white piano key' notes FCGDAEB. The improvement is greatest for D and decreases in both directions away from it:
D# 4.08
A# 4.06
F 4.00
C 3.96
G 3.95
D 3.94
A 3.96
E 3.98
B 3.97
F# 4.00
C# 4.04
G# 4.06
TT's fretting system is designed to be able to play with other instruments that use 12TET, so the pitch offsets cannot be too large, they have been limited to +-4 cents. This limits how much the harmony of the thirds can be improved.
If all instruments were using WT, a WT with larger offsets and more significant improvements in harmony could be used.
Compared to historical WTs, TT's system is quite subtle.
Summary
------------
TT's fretting system moderately improves the harmony of major and minor thirds on 6 out of the 12 root notes, at the expense of thirds on the other root notes.
The intention is for the musician to play music that is careful to only use the root notes that improve, or do not worsen, the thirds. Also, chords should be mostly major and minor triads.
Being restricted in this way was not much of a problem for Baroque and Classical music, as those often used rather limited and predictable key modulations and chord progressions, and chords were mostly major and minor triads. Mainstream pop and rock music also does so.
Musicians who have a more unconventional use of chords, key modulations and chord progressions will not experience an overall improvement in harmony, because gains cause equal losses elsewhere.
However, 'Key colour' may still make a WT worth using for these musicians.
Misunderstandings
------------------------
It does not result in perfect harmony chords, at best, minor and major thirds are modified halfway towards perfect harmony.
It does not improve all chords. The focus is on major triads with root notes C G D A E B, or minor triads with root notes A E B F# C# G#. Any gain creates an equal loss elsewhere.
The often stated benefit of 'perfect intonation' does not mean chords have perfect harmony, 'intonation' is used here to mean the fretted pitches are very close to the intended tonal system (in this case a WT) whether or not that tonal system results in perfect harmony is a separate matter (and is actually not the case for a WT).
The lack of information on the TT website, and the very common misunderstandings of guitarists and the music media, may give the impression that TT is actually 'perfectly intonated 12TET'. It is actually intentionally slightly out of tune with 12TET.
To improve the harmony of chords you have to be out of tune with 12TET, because 12TET inherently has imperfect harmony.
Disadvantages
-------------------
Due to a WT having offsets for each tone, only standard tuning EADGBE can be used, or transpositions that maintain the relative tuning (for example Eb standard, D standard, F standard).
Due to TT being per-string, per-fret fret offsets determined by experiment to compensate for common string gauges, only a narrow range of gauges can be used, from 9-46 to 11-50.
Due to the unusual and variable fret shapes, pitch response to string bending will be a little more unusual and a little less consistent.
