- Personal journey: What? How?
- What does it mean for a lute player to use a temperament?
- What
*exactly*is quarter-comma meantone? - How do we make good music with historically appropriate temperaments and be happy with it?

- Lute, and all fretted-instrument temperament, is multifaceted, multivariate and completely individual
- Players will execute the same temperament in different ways
- Even in the same ensemble, a lute's temperament can be different than the other instruments
- Historical criteria for temperaments applies differently to lutes
- It is not historically
*wrong*to use a quasi-equal temperament

- A brief history of tuning and temperament
- Surveying and measuring historical fretting sources
- Executing temperaments for solo and ensemble performance
- Conclusions and questions

- Credited with developing the first tuning system
- Relied on ratios of string lengths
- Used only unisons, octaves, fifths, and fourths
- Created a comma and very wide thirds

- Created the just intonation system
- Any super-particular ratio is acceptable, e.x. 5:4, 6:5
- All intervals are pure
- Required two different sizes of semitone
- Impractical for instruments with twelve fixed pitches

- Rejected the use of ratios
- Divided strings into multiple small parts
- Pitches were grouped into multiple parts

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

- 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

- 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

Wholetone | Diatonic | Chromatic | Temperament |
---|---|---|---|

5 | 3 | 2 | 1/4 comma meantone |

7 | 4 | 3 | 1/5 comma meantone |

9 | 5 | 4 | 1/6 comma meantone |

11 | 6 | 5 | 1/7 comma meantone |

13 | 7 | 6 | 1/8 comma meantone |

Allows us to calculate any semitone in any meantone temperament

- 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

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

“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)

- 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

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?

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

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

Bermudo

Ganassi

Bermudo's Third Fret

Ganassi's Sixth Fret

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

**As a whole, most historical fretting schemes do not work**

- Can't split 9:8 into equal ratios (requires logarithms)
- You can split 9:8
*visually*, Dowland and Gerle's 6th - Other approximations:
- Bermudo's third fretting scheme
*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?

- Alternating chromatic and diatonic semitones
- Same semitone for all courses
- Not all courses agree
- What if you need two differently-sized semitones on the same course?

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

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.

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

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

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

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