1. Dr. Jason Luke Thompson, Indiana Wesleyan University
Fine Tuning the orchestra How Understanding Temperaments Can Improve Your Orchestra’s Intonation
Dr. Jason Luke Thompson, Indiana Wesleyan University

3. Where does my first finger Go?

4. General Advice for fixing intonation
Practice with a tuner
Practice with drones
Lower your thirds and raise your fifths
“Expressive Intonation”
Keep your half-steps small
Raise your leading tones
Notes with sharps should be raised
Notes with flats should be lowered

6. The Pythagorean Comma
ca cent discrepancy

7. The Syntonic Comma (chromatic diesis, comma of Didymus, Ptolemaic comma, diatonic comma)
21.51 cent discrepancy

8. Why can’t we simply tune just?
Syntonic Comma is built into the open strings.
Which note is correct? The root? The bass? How do you determine?
What do you do when the same pitch changes function?

9. Equal Temperament
Equally distributes the Pythagorean comma over twelve 5ths
Fifths are good (1.9 cents narrow of just)
Major Thirds are not good (13.19 cents wide of just)

10. 1/4 Comma Mean-Tone
Popular in the 16th and early 17th centuries (and appropriate for music of that time)
Divides the syntonic comma by four and distributes it over eleven 5ths
Thirds are good (essentially just).
Fifths are not as good (5.38 cents narrow of just).
The remaining fifth (called the wolf) is unacceptably wide.
Not possible to use enharmonic equivalents.

11. 1/6 Comma Mean-Tone Compromise between 1/4 Comma and Equal Temperament
Divides the syntonic comma by 6 and distributes it over eleven fifths
Thirds are okay (7.2 cents wide) – about half-way between ET and just
Fifths are okay (3.56 cents narrow) – about half-way between 1/4 comma and just
Remaining fifth (wolf) is not as bad as 1/4 comma
Still can’t use enharmonic equivalents

12. Temperaments aren’t just for Keyboards
Ross Duffin cites three different non-keyboard treatises from the eighteenth century in How Equal Temperament Ruined Harmony (and Why You Should Care).
Peter Prelleur, The Modern Musick-Master: V. The Art of Playing on the Violin (1730–31)
Pier Francesco Tossi, Opinioni de’ Cantori (1723)
Johann Joachim Quanz, On Playing the Flute (1752)

13. Peter Prelleur Fingerboard Diagram from The Modern Musick-Master: “The Art of Playing on the Violin” (1730–31)

14. Pier Francesco Tosi, Opinioni de’ Cantori (1723)
[The teacher] ought to make [the student] sing the semitones according to the true rules. Not everyone understands that there is a major semitone and a minor semitone, because the difference cannot be demonstrated on an organ or harpsichord if it doesn’t have split keys. A whole tone is divided into nine almost imperceptible intervals which are called commas, five of which constitute the major semitone, and four the minor semitone An understanding of this matter has become very necessary, for if a soprano, for example, sings D-sharp at the same pitch as E-flat, a sensitive ear will hear that it is out of tune, since the latter pitch should be somewhat higher than the former.

15. Johann Joachim Quantz, On Playing the Flute (1752)
A keyboard player who understands the division of the whole tone, also understands that D-sharp and E-flat are differentiated by a comma, and therefore cause, because of its lack of split keys, some inequality of intonation upon this instrument [that is, the keyboard] as compared with other instruments on which these notes are produced in their true ratios.