Fortunately or unfortunately we can’t definitely answer : it depends on context.
The great french violinist Jacques Thibault was sometimes criticized for his « out of tune » intonation but, as pointed out by one of his famous students,Ivry Gitlis , : »playing the flat very low » he added color
In the art of violin , Laurent Korcia likes the slightly hoarse sound of Isaac Stern who « played out of tune but sounded in tune ».
Mind you, we are discussing international soloists, not amateur fiddler . Yet,their intonation was sometimes dissentious
So,let’s try to make things clearer
Main acoustic systems have been discussed in Acoustics and are summarized below but
first let’s remind the deaf acoustician Joseph Sauveur’s discovery in 1701 on harmonics sounds ,
Most of the time sounds are compound sounds made of a strong sound named fundamental and other weaker sounds called harmonics which frequency is a integer multiple of the fundamental frequency
|Fundamental sound multiplied||By||gives|
|5||The Major third|
(division by 2 may be necessary to remain in a octave span)
Violinist ,violist ,cellist and double bassist have three choices
- The Pythagorean System
based on a string length division
- by 2 that yields the octave
- 3 which produce the Fifth
(actually a 12 th divided by 2 to get a fifth )
12 fifths slightly overtake 7 octaves ( B# is higher than C)
- The octave is perfect (pure) :it is the only common interval for all systems
- Fifths are perfect (except one)
- Major third is higher than Natural major third (harmonics 5)
it consist of
- one sort of tone
- one sort of diatonic semi tone which is very narrow and therefore strongly attractive.
- one sort of chromatic semi-tone larger than the diatonic semi-tone.
- one sort of comma.
- The Zarlino’s system
adds division par 5 to the Pythagorean system to get natural Major third of the harmonic series .
This addition brought great disturbances into the system but enhanced the Major and minor chords (with no beats)
However this advantage est limited to 3 thirds the primary chords in major keys
In this system
- Octaves are pure
- so are the thirds in a given tonality
- only 4 fifths are pure
- There are two kinds of tones
- Two kind of diatonic semi-tones
- Two kind of chromatic semi-tones (both are smaller than diatonic semi-tones )
Globally semi-tones are very large and therefore without or little attractive power
- Both tetrachords are different
- Sharps and flats have no fixed value which depends upon the chords progression but ,in any case,sharps are lower than Pythagorean sharps ( and flats higher )
- B# may be higher or lower than C according to the chord progression.
- About 10 sorts of comma are available.
These inequalities make the system improper to modulations
- The tempered system
We saw that pure intervals but octave impede the circle closure (B can’t go to C)
Therefore the octave were divided into 12 equal parts where every interval is a bit out of tone but bearable by the ear. Any modulation is possible but to the detriment of color.
There is only one sort of tone and semi-ton (a with two orthographies: diatonic/chromatic) and ,therefore, no comma.
These 3 systems are different from theory with its 4 commas diatonic semi-tone and its 5 commas chromatic semi-tone but values are near Pythagorean ones .
It must be pointed out that those systems are mainly based upon mathematics,not on physics.
From this summary ,we can draw some evidences
- In solo playing ,Pythagoras is the most natural since the tuning is based upon the fifths (or fourths for double bass).In another hand, the attractive power of the diatonic semi-ton allows an expressive melodic line .
- In a duet with piano, the tempered system is obviously the best choice since piano is tuned in equal temperament to restrict the number of keys ( C#Db is the same key for example)
- In a quartet Zarlino is partly an option since viola-violin form a sixth (13 th) C-A too low and a third(17th) C-E too high with open strings, furthermore music is partly chordal -but also melodic!
So « partly »is here an important word since we can’t use this system at any time .
Threfore, performer has permanently to make a choice
- strategic choice according to playing set up (solo-duet and so on).
- stylistic choice: for example choose Zarlino for its no attractive semi-tones to play a modal melody or Pythagoras to play a brilliant melody.
- technique choice : Paganini tuned the G string a quarter tone higher to soften the thirds/sixths (opening of Caprice 21 for instance) oh ! we are getting out of the subject.
But choice also means criticism
- Choosing tempered system is the safest but without personification : Again in the art of violin,Isaac Perlman seems to regret the former generation of soloists who sounded all differently.
However , acoustic systems in the intonation work is relative
- These systems are based on calculation that produces different results according to the reckoning approach.
- The ear work s on logarithmic function and endures small frequencies difference.The discrimination is around one Hertz for a trained ear but dissonance only occurs beyond a 5 Hertz span.
- Quasi systematic use of Vibrato obliterates the gap between frequencies and most kind of commas which enter the rear discrimination zone.
- Dynamics may theoretically spoil the intonation but it’s negligible in the scope of usual melody frequencies.
Besides acoustical systems ,the most important thing to discuss.
- harmonics:Usage of open string is common to control intonation. Natural harmonics can be used in the same way but I would like to point out that harmonics of a given string produce a just intonation major chord since harmonics come from division of the string length. A given note played in tune with an harmonics sounds sympathetically with other string. See intonation exercicesNotice only four notes out of twelve ( with octave skips since fifth of a string is root of the next string) are missing but the
- Tartini’s third sound technique makes good the lack.
It may be useful to to play different interval each after other, especially alternating sixth -third to perceive the ghost tone . This Leopold Mozart’s advise demonstrates the hardness to hear it. The double stops progression for a given ghost tone reminds us the Sevcik.book 6
As for harmonics 7, Tartini’s third tone is not really scientific since it is not easily reproducible but it’s a mean to search a clean intonation and a pretty sound.