Ptolemy

Circa 1480

Pythagorean tuning is altered or tempered to avoid the problem of wide thirds

Meantone Temperaments

• Overcame limitation of wide thirds in Pythagorean tuning
• Pietro Aron: Fifths are flattened by $ \frac {1}{4} $ of a syntonic comma to create pure thirds
• Pure intervals were restricted to certain keys
• Aron's system was inexact, relying on the ear

Meantone by Equal Division

• Invented by Vicentino in 1555
• Divides the octave into 31 parts with wholetones divided unequally
• Same as Aron's temperament but semitone sizes were explicitly measured
• Allowed for pure intonation with a fixed-pitch instrument, if you have 31 of them

Vicentino's 31-Part Chromatic Octave

Other Multipart Systems

Allows us to calculate any semitone in any meantone temperament

Fretting Sources Consulted

• Pythagorean tunning: Oronce Finé and Bonaventure des Periers
• Hans Gerle and John Dowland
• Silvestro Ganassi
• Vihuela soures: Luis Milan and Juan Bermudo
• Equal fretting: Vicenzo Galilei and Marin Mersenne

Pythagorean Sources

• All used perfect fifths and fourths
• 9:8 wholetones

Reading Fretting Instructions

“Take a straight-edge that is thin or else a flat piece of wood like a ruler, and make it of such a length that at the top it touches the piece of wood that the strings lie on and also touches the bridge that the strings lie on, and when you have made the ruler so that it touches at both ends (don’t make it too short; it must touch as I have said), mark the bottom part at the bridge with an a, and the top part with a b, so that you will know which end belongs to the bridge. Then lay the ruler on a table, and take a compass and find the middle of the ruler. Mark it with a point or little dot and put an m there.”

from, Hans Gerle (1533)

Gerle and Dowland

• Perfect fifths and fourths
• First fret has a ratio of 33:31
• Mark Lindley, in Lutes, viols, and temperaments, identifies this as a diatonic semitone in sixth-comma meantone

Calculating Meantone Semitones

Octaves move by powers of 2: 2, 4, 8, 16 or ... $ 2^1, 2^2, 2^3, 2^4, $

An equal semitone within one octave: $ 2^\frac{1}{12} $

A diatonic semitone in 1/4 comma: $ 2^\frac{3}{31} $

A diatonic semitone in 1/6 comma: $ 2^\frac{5}{55} $

Gerle/Dowland ratio $ \frac{33}{31} = 1.065 $

$ 2^\frac{5}{55} = 1.065 $

Gerle was using a sixth-comma diatonic semitone

But what about the others?

Ganassi

• Viol treatise
• Begins with Pythagorean ratios, but then adjusts each fret
• Measurements taken from Richard Bodig

Vihuela sources

• Milan speaks of adjusting the fourth fret
• Bermudo gives us three fretting guides (Espinosa):
  1. Pythagorean with adjustments by ear
  2. Pythagorean with detailed adjustments
  3. Very close to equal temperament

Comparing Fretting Schemes

First Frets

Bermudo

Ganassi

Unique Frets

Bermudo's Third Fret

Ganassi's Sixth Fret

The Verdict?

• Meantone ... sort of
• Inconsistent usage
• Intervals from different temperaments
• Retaining Pythagorean intervals is problematic:
  1. Open courses - pure fourths with a major third?
  2. Second fret

As a whole, most historical fretting schemes do not work

"Equal" Temperament

• Can't split 9:8 into equal ratios (requires logarithms)
• You can split 9:8 visually, Dowland and Gerle's 6th
• Other approximations:
  1. Bermudo's third fretting scheme
  2. 18:17 Rule from Mersenne and Galilei
• Potential option for solo music or limited ensembles
• Ignores the ideal of pure thirds

How do we make good music with historically appropriate temperaments and be happy with it?

Meantone Realized on the Lute

Meantone Strategies

• slanted frets
• tastini
• double frets
• left-hand techniques
• use another temperament

Praetorius, Syntagma Musicum II

Thus the semitones cannot be either major nor minor, but are, perforce, "intermediate" if anything. For I reckon that each fret [...] contains four- and-a-half commas, whereas the major semitone contains five and the minor semitone only four. Since the error is only half a comma either way, the ear hardly notices it with these instruments [...] Major and minor semitones are both produced by the same fret, both sound in tune, [...] especially since by particular applications of the finger to the string, over the fret, it is possible to have some control over the pitch of the note produced.

Demonstration

$ \frac{1}{4} $ Meantone

vs.

$ \frac{1}{6} $ Meantone

Lute Song

"Come, Heavy Sleep" from Dowland (1597), The First Booke of Songs or Ayres

The Theorbo

Theorbo Temperament Strategies (Continuo)

• Re-entrant tuning
• Slanted frets or tastini
• Chord voicings
• Omission of notes

Practical

• Effective $ \frac{1}{4} $ comma is impossible without instrument modifications
• $ \frac{1}{6} $ comma is a nicer alternative
• Equal-ish temperaments are acceptable, though not ideal
• Mixed semitone sizes or customized temperaments

Playing with Ensembles

• Playing in tune, versus using the same temperament
• It's the destination, not the journey
• Many solutions exist

Philosophical

• Fretted instruments are different
• stable, but alterable (Bottrigari)
• Uniformity is a modern idea
• We can be too precise
• Ears versus electronics