Posts Tagged: Pythagoras

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
	2	The octave
	3	The fifth
	4	The octave
	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

1. 1. The Pythagorean System

based on a string length division

• • by 2 that yields the octave

and by

•  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.

NOTICE
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

Since

• 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

Conclusion

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.