Temperaments: Comprehensive Study Notes

1. Introduction

• Temperaments are tunings where some or all concords (octaves, fifths, and thirds) are made slightly impure so that many or all keys can be used more practically.
• Equal temperament (12 uniform semitones per octave) is the modern standard in Western music, especially outside early-music specialists.
• The article traces temperaments’ history in performing practice and harmony, with cross-references to related topics: Tuning, Pythagorean intonation, Just intonation, Microtones, Mean-tone, Well-tempered Clavier, Equal temperament, Interval.
• Core issue: pure intervals (particularly 5ths and 3rds) are not all simultaneously achievable; temperaments adjust these to enable a workable keyboard/music system.
• Key background concepts (brief):
  • Pure octaves and the circle of fifths are mathematically incompatible when all are exact; tempering distributes small commas across the scale.
  • The notion of a “diesis” (different small comma-like adjustments) and how they interplay with major/minor 3rds and circle-of-fifths tuning.
• Important historical takeaway: temperaments arose from practical limits on intonation where instruments (especially keyboard, fretted, organ, clavichord, harpsichord) could not freely intone every note in every context.
• Core historical anchors: Renaissance and Baroque keyboard practice, the balance between stable triads versus flexible keys, and the eventual shift toward equal-tempered systems in late 18th–19th centuries.

2. Quasi-Pythagorean temperament

• Quasi-Pythagorean temperament strand begins with 12-note chromatic systems (fully chromatic keyboard practice) where nearly all fifths are tuned to be pure or nearly pure except one “wolf fifth.”
• Basic idea: in a fully chromatic scheme, 11 of the 12 fifths among naturals can be kept very pure, leaving one fifth to be sour (the wolf fifth) to accommodate a fully chromatic layout.
• The historical consequence: triadic harmony can appear distinctly more euphonious on major triads on D, A, and E, while other triads on B♭, F, C, G become less stable.
• Singers vs keyboard: singers can improvise pitch inflection, but keyboard tuners must adopt a fixed temperament; the clavichord/organ can be tuned ahead of time to support intended triadic sonorities.
• The notion that a chain of perfect fifths and fourths yields imperfect 3rds/6ths as a feature, not a bug, is a hallmark of quasi-Pythagorean practice.
• Early examples show triads on certain roots (e.g., D, A, E) treated as more stable, while those containing sharps (e.g., B♭, F, C, G) are less so. This is evidenced in early vocal and instrumental works (e.g., Du Fay, Matteo da Perugia, Guillaume Du Fay).
• The practical size of tempering is extremely small (e.g., < ¼ of a cent) and often went unnoticed by contemporaries; Helmholtz later tempered with many pitches for study purposes.
• Contextual examples cited:
  • Ex.1 Du Fay, “Mon chier amy” (early 15th c. evidence for triadic stabilization on certain roots)
  • Ex.2 Matteo da Perugia, “Pres du soloil” (wolf fifth discussion; Ramis de Pareia’s broader mean-tone scheme)
  • Ex.3 Paumann (D-Mbs Cim 325b) (mean-tone formulations discussed by later theorists)
• Conceptual link: quasi-Pythagorean temperaments foreshadowed some early mean-tone thinking, while remaining distinct from later regular mean-tone and strict 1/4- or 1/5- comma schemes.

3. Regular mean-tone temperaments to 1600

• Regular mean-tone temperaments aim to make major triads very pure while allowing tempered 5ths to avoid a harsh or discordant temperament overall.
• Core principle: temper the fifths so that major 3rds become pure or nearly pure, even if that means the 5ths/4ths deviate more from their pure sizes than in equal temperament.
• Historical mechanics: a family of tunings where 5ths are tempered to achieve clean triadic sonorities, especially for cadences, while keeping overall circle-of-fifths structure recognizable (though not perfectly closed).
• Ramis de Pareia (Musica practica, 1482) and Lanfranco (1533) contribute key mean-tone ideas; Ramis provides an explicit monochord-based model for plainchant and a practical set of rules to produce “good” cadential 6ths/3rds. Spataro (early 16th c.) corroborates Ramis’s position.
• Wolf fifth concept evolves here: the wolf fifth is moved into a position among naturals (G–D region in Ramis’s scheme), which allows the rest of the scale to follow mean-tone logic.
• Practical harpsichord/organ tuners in late 16th–early 17th centuries (Cerone, Santa María, Aaron, etc.) discuss how to realize mean-tone on keyboard/monochord instruments; many tacitly acknowledge the wolf fifth’s placement and the need for practical fingerings and cadential behavior.
• Why it mattered: mean-tone temperaments produced more sonorous major triads and allowed for cadential progressions that players and audiences found satisfying in many contexts.
• Illustrative musical evidence and commentary:
  • Ex.2 Ramis de Pareia discusses wolf fifth placement and “good” cadential 6ths/3rds.
  • Ex.3 Paumann indicates mean-tone types (−comma, ¼-comma, −/− etc.) with various configurations; 1571 Zarlino describes ¼-comma mean-tone in mathematic terms. Cerone’s 1613 attribution to organ builders reinforces regular mean-tone as a widely used practical scheme.
• Takeaway: by 1600, multiple mean-tone variants were in practice, with a spectrum from −comma to ¼-comma mean-tone; the exact choice varied by geography, instrument, and repertoire. Regular mean-tone established itself as a dominant heritage in Renaissance/early Baroque keyboard practice.

4. Irregular keyboard temperaments to 1680

• Irregular temperaments are those in which not all fifths are treated equally (or not all fifths are uniformly tempered), but where the temperament remains implementable and serviceable across many keys.
• Historical lineage: irregular temperaments emerge in late Renaissance/early Baroque periods and become particularly prominent in the 17th century.
• Key features:
  • 5ths are tempered differently across the circle of fifths; some keys receive milder adjustments, others stronger, which yields a richer diversity of key character and color.
  • The goal was to preserve use of commonly used triads while exploiting the diverse shading available from different tempering to support musical expression.
• Schlick (1511) is a seminal example of an irregular tempering approach: he created a variant of regular mean-tone with major 3rds slightly larger than pure, but tempered some 5ths more than others; his scheme explicitly allows some enharmonic forms to be more usable than others (e.g., G♯, A♭, E♭ relationships).
• Mersenne’s later work (1635–6) hints at irregular temperaments aimed at balancing practical organ/keyboard needs with tonal color—though he largely advocated regular ¼-comma mean-tone in many places, later readers misinterpret his stance due to ambiguous language.
• Ex.8 Schlick (Mainz, 1512) and Ex.9 Schlick (Mainz, 1512) illustrate this approach in practice.
• The Mersenne era shows a tension: the desire for regular mean-tone’s clarity vs. patrons’ demands for expressive flexibility across a broader key palette.
• Other contributors: Andreas Werckmeister, Romieu, and later 18th-century writers discuss irregular schemes that temper 3rds more or less depending on the key region, sometimes using enharmonic modifications (e.g., E♭ vs D♯) to achieve a desired effect.
• Practical musical consequences: irregular temperaments allowed for more expressive cadences and chromatic color across many keys, which was valued in late Baroque and early Classical periods. They could provide a richer palette than strict mean-tone without collapsing into the dissonances that fully equal temperament would later permit.
• Example usage and discussion:
  • Ex.4 John Bull (early irregular mean-tone) shows how non-uniform 5ths can color the sharps and flats.
  • Ex.5 Gabrieli and Ex.6 Valente show practical differences in diatonic vs chromatic semitones and cadential shading under irregular tempering.
  • Ex.7 Cabezón illustrates how regular mean-tone can still yield a variety of effects in organ/tiento contexts.
• Summary: by 1680 irregular temperaments provided a sophisticated middle ground between strict regular mean-tone and later fully irregular systems; they enabled nuanced tonal shading across keys while preserving musical practicality.

5. Equal temperament to 1735

• Concept: equally divide the octave into 12 semitones, so every key sounds the same (in terms of interval sizes) and modulation across keys becomes seamless.
• The core mathematical bridge: ratio 18:17 for a semitone yields 99 cents, nearly identical to the modern 100-cent semitone; early theorists like Vincenzo Galilei suggested this before Renaissance refinements.
• Practical history: long-standing tension between fretted instruments (lute, viol) and keyboard intonation, with early hints that equal temperament might be feasible before formal theoretical arguments emerged.
• Notable early adopters and advocates (timeline):
  • 1530s–1560s: some 16th-century composers and theorists recognized enharmonic advantages of equal temperament in certain works (e.g., Willaert, Orso in Petrarch settings).
  • 1630s–1640s: Frescobaldi endorsed equal temperament for keyboard instruments in some contexts; by 1645, Jean Gallé and others in Paris began teaching/using equal temperament on organs/spinet; Jean Denis shows that harpsichords might tolerate equal temperament.
  • 1640s–1650s: the debate intensifies; some baroque organ builders resisted but a growing camp saw practical merit.
  • 1697–1707: Werckmeister’s writings show a nuanced stance: he approved equal temperament with a caveat—diatonic major 3rds could be left purer than others; his later writings offered both mathematical schemes and practical non-mathematical descriptions.
  • 1703–1707: Mersenne’s tuning procedures on organs/spinet wrestle with the practicalities of equal temperament; Chaumont’s method (1695) shows contemporary awareness without full adoption.
  • 1730s–1740s: Rameau endorses equal temperament in Génération harmonique (1737); Rameau’s authority helps anchor equal temperament in France as a legitimate option for musical theory and practice.
  • 1737–1750s: a French and German intellectual current increasingly supports equal temperament as the standard for practical ensemble playing, especially where multiple instruments participate.
• Central historical claim: While Renaissance and Baroque musicians did not advocate equal temperament as a universal standard, the practical feasibility and musical advantages of equal temperament gained momentum in the late 17th through 18th centuries, culminating in its broader acceptance in the 19th century.
• The Well-Tempered Clavier’s reputation and the broader Baroque impulse towards diverse keyboard tunings vs. strict equal temperament are discussed elsewhere; here the emphasis is the gradual shift toward equal temperament as a viable historical option rather than a radical departure from tradition.
• Notable shifts in opinion: Werckmeister’s later acceptance; Barbour’s and Baro’s studies on Bach’s tuning; and mainstream German and English view shifts toward equal temperament as a normative option in ensemble contexts.
• Practical consequences: equal temperament enabled more uniform transposition and ensemble coherence, later underpinning Romantic-era tonal experimentation and 19th-century orchestration/keyboard design.

6. Regular mean-tone temperaments from 1600

• The late Renaissance church modes fit into a larger 12-note mean-tone palette (from E♭ to G♯ in the fifths chain), provided cadences did not rely heavily on particular modal transpositions.
• Practical devices to adapt to modulation: use split accidentals (e.g., D♯/E♭ and A♭/G♯) to accommodate D and A cadences in different tonal centers; similarly split accidentals can help cadence towards D♯/E♭, etc. The “Praetorianische Temperatur” was a German label for ¼-comma mean-tone in keyboard practice.
• France’s approach (Mersenne, Salinas) relies on the idea of ¼-comma mean-tone as an exemplar of mean-tone tuning; the literature argues that this form became the standard for Renaissance/Baroque keyboard tunings in many German/Italian contexts.
• 17th-century German organ practice shows a strong orientation toward ¼-comma mean-tone as a robust standard for organs; Werckmeister’s later discussions propose a circulating temperament with regular ¼-comma mean-tone as a core reference, but with accommodations for practical organ design (split pipes, etc.).
• The 18th-century glow of ¼-comma mean-tone is tied to the rise of tierce stops in organ stops for brighter, more brilliant textures; the scheme supported the harmonic brilliance sought in organ literature of that era.
• Descriptions of multiple shades of mean-tone in the 17th–18th centuries: ⅕-comma mean-tone, ¼-comma mean-tone, -comma mean-tone, and ⅓-comma mean-tone (with major 3rds either slightly larger or exactly pure depending on the shade); the literature shows extensive discussion of these shades and their practical effects.
• Notable figures and statements:
  • Cerone (1613) attributes ¼-comma mean-tone to master organ builders.
  • Lemme Rossi (1666) describes ⅕-comma mean-tone and references -comma mean-tone.
  • Sauveur (1701) matches the 53-part division to ⅕-comma mean-tone; Zaragoza (1674) and other scholars connect the 43-part and 55-part divisions to ⅕- and ⅙-comma mean-tone respectively.
  • The literature traces correlations between microtonal divisions (53-, 43-, 55-part divisions) and the mean-tone shades; this shows a continuous theoretical exploration of approximations to mean-tone across 16th–18th centuries.
• The 19-note and other microtonal harpsichords were discussed by Zarlino and Salinas (1571–1577); 19-note scales appear in the late 16th–early 17th centuries as experimental tonal systems; the history shows interest in various microtonal schemes without full standardization.
• In practice, the 18th-century German and Italian experiments illustrate a spectrum of mean-tone shades, with regular mean-tone and irregular temperaments vying for musical expression across keys.
• Summary: regular mean-tone temperaments from 1600 onward offered a robust set of tools for coloring harmony and cadences in Baroque keyboard music; they saturated the tonal palette with purer 3rds in some contexts while leaving a controlled amount of 5th/4th tempering to maintain keyboard practicality.

7. Irregular temperaments from 1680

• The 18th-century keyboard tuning landscape favors irregular temperaments that balance the desire for purer internal triads with the need for broader key modulation.
• Core idea: temper the 3rds lightly in some keys (often closer to the “modern” sense of purer 3rds in certain centers) while adjusting 5ths unevenly so that diatonic vs chromatic semitone sizes vary by key, creating distinct key colors.
• Circle of fifths shading: as one moves around the circle of fifths, the amount of tempering applied to 3rds changes; keys closer to the front tend to have more tempered 3rds, while those further back have different shading; the exact pattern varies by instrument and taste.
• Semitone sizes varied widely: largest—E–F and B–C; smallest—C–D♭ and possibly F–G♭ (or E♯–F♯). This unequal semitone distribution contributed to the unique character of each key.
• Effects on major/minor triads and leading notes: keys with fewer sharps/flats tended to have more resonant, smoother triads but cruder leading notes; minor keys exhibited more complex and darker tonal inflections.
• The E major triad often received special attention to balance, but not uniformly across all irregular temperaments.
• Visual/tonal portrait: 18th-century circle-of-fifths diagrams (fig. 2) show how 3rds could be tempered differently according to position in the circle, yielding a spectrum of key colors rather than a single universal color.
• Notable musical impact: irregular temperaments contributed to the distinctive timbral and tonal variety in late Baroque French harpsichord music (d’Anglebert, Couperin), Handel’s recitatives, and Bach’s keyboard works where key color is part of expression.
• The broader point: irregular temperaments provided a flexible framework that allowed composers to choose keys for their emotional character rather than to insist on a single uniform tuning across all keys.

8. Equal temperament (to 1735 and beyond)

• The equal-temperament idea divides the octave into 12 equal parts, allowing all 12 keys to be used with the same interval structure.
• Early historical context: although equal temperament was not widely adopted by Renaissance keyboardists, there were long-standing discussions of equal divisions and their theoretical viability (18:17 semitone; 99 cents in practice; 100 cents in ET).
• Philosophical and practical arguments:
  • The ability to “tour through all the notes” with uniform intonation for performers and ensembles across keys made equal temperament attractive for modular transpositions and multimodal works.
  • Some claimed the equal division reduced the musical expressivity of keys, while others argued that variety should come from modulation and color rather than incorrect intonation.
• Key historical advocates and milestones:
  • Vicentino (1555) and Vincenzo Galilei (1581) discussed aiming for a consistent semitone ratio (18:17) for equal temperament on fretted instruments; this ratio yields a semitone close to the modern 100 cents.
  • Frescobaldi (early 17th c.) and Mersenne (1640s–1660s) discuss equal temperament in keyboard context, often facing resistance from those who favored mean-tone or irregular temperaments for tonal color.
  • By the 17th–18th centuries, Paris and Berlin/Iberian contexts see increased experimentation with equal temperament—Mersenne’s and Werckmeister’s discussions show a nuanced acceptance rather than outright rejection.
  • 1603 Artusi and later writers debated tempered intonation, sometimes mistakenly thinking irrational ratios could be dismissed; the development of equal temperament persisted despite debates.
  • 18th-century German theorists (Werckmeister, Meckenheuser, Neidhardt, Mattheson) exhibit growing interest in equal temperament, while many traditional organ builders remained skeptical.
• 18th-century popularization and the Bach era: the Bach circle (Well-Tempered Clavier and related discussions) becomes a focal point for arguments about equal temperament, the feasibility of transposing in all keys, and the expressive potential of keys under unequal temperaments. Marpurg (1776) champions equal temperament and argues for its ensemble benefits; he also critiques Kirnberger and other schemes that overreach in unequal temperaments.
• The broader cultural backdrop: the late 18th–early 19th centuries saw a shift toward equal temperament as the standard, facilitating large-scale modulation in Romantic-period works and enabling consistent keyboard design (pianos, organs) suited to equalization of interval sizes.

9. Fretted instruments

• Relationship between performance practice and temperaments on fretted instruments (lute, vihuela, guitar, viol) differs from keyboard instruments due to fretting location and string tension behavior.
• Key observational points:
  • Frets are fixed on the neck; players press strings against frets, rather than relying on a tuner to adjust pure intonation for every note.
  • Gut strings (historical) vary in tension and thickness; this variability affects the perceived pitch and complicates applying strict mathematical temperaments.
  • Practical adjustments by ear were common; players relaxed or sharpened notes slightly to obtain better consonances in context.
• Four major types of late Renaissance fretting prescriptions (as described in the article):
  • (i) Those with exclusively Pythagorean ratios (e.g., Finé, 1530).
  • (ii) Those embodying a precise mathematical model of equal temperament but too elaborate to be practically useful (e.g., Zarlino, 1588).
  • (iii) Those that appear erudite but fail to embody a feasible intonation model (e.g., Dowland, 1610).
  • (iv) Those simplified for practicality; Galilei’s rule (18:17) for fret placement down the neck is the best represented among these (Mersenne cites this rule as commonly used by instrument makers).
• Practical example set:
  • Vincenzo Galilei’s rule (18:17) for fret placement: dividing successive frets according to 18:17 ratios—highly practical for fretted instruments.
  • Ganassi’s Renaissance lute prescriptions would lead to fret positions derived from rational ratios with refinements by ear; he notes tempering 5ths to distinguish major and minor semitones.
  • Milán (1536) uses complex fret arrangements on vihuela to realize specific chromatic/diatonic semitones; he sometimes shifts a fret up or down by its width to strengthen certain notes; at times uses equal or near-equal semi-tone thinking in practice.
• General observation: equal temperament was often the theoretical norm for fretted instruments, whereas keyboard practice favored irregular/multi-shade temperaments; yet fretted instruments could reflect a broader tolerance for microtonal distinctions when needed for ensemble tuning and color.

10 Difficulties in interpreting theoretical writings

• Modern readers must be cautious: many Renaissance/Baroque writers discussed music in terms of a spectrum of theoretical models, not as a single historical practice.
• Problems to watch for:
  • The comma debate: syntonic comma vs. Pythagorean comma vs. miscellaneous “commas” used in mean-tone/regular schemes; misinterpretations arise when terms like “comma” are treated as a single universal quantity.
  • The use of irrational ratios in early monochord accounts; some writers used such numbers as a theoretical construct rather than reflecting practical usage.
  • Heuristic descriptions vs. mathematical rigor: writers like Aaron, Bermudo, Gaffurius sometimes conflated or confused enharmonic twin notes (D♯ vs E♭) in Pythagorean intonation with mean-tone practice.
• Specific cautionary notes:
  • Some early statements can be misread as absolute endorsements of equal temperament; the authors often presented both mathematical schemes and practical recommendations, leaving room for interpretive differences.
  • The Bach-era debates about “the well-tempered clavier” and Bach’s own tuning show the risk of overinterpreting a treatise’s mathematical precision; practical factors often trump theoretical neatness in performance.
• The use of historical sources: the article emphasizes cross-referencing a broad spectrum of treatises and instruments (organs, clavichords, harpsichords, lutes, viols) to avoid overgeneralizing from a single school or region.
• Modern scholarship warns against searching for a single “correct” temperament; instead, researchers often approach temperaments as a toolbox of approaches that were chosen situationally by composers, instrument builders, tuners, and performers.

11. Temperament and harmony

• Central problem: tempered tuning aims to achieve two kinds of equations that cannot both be perfectly achieved in a single system:
  1) A chain of fifths in tune should yield a consonant 3rd in tune (i.e., a harmonious major or minor triad).
  2) A chain of consonant major/minor 3rds in tune should yield a pure unison.
• Mean-tone temperaments satisfy the first (circuits of pure or near-pure 3rds) but not the second (not all 3rds, when moved chromatically, align to a single pure unison across all keys).
• Consequences of mean-tone limitation:
  • They cannot provide true enharmonic equivalence between, e.g., E♭ and D♯ across all keys; instead, those pitch classes drift apart as you modulate.
  • This makes modulation to certain keys crude or less flexible in enharmonic contexts, and it explains why mean-tone temperaments are ill-suited for perfect enharmonic modulation in the late Baroque/later repertoire.
• By contrast, equal temperament solves the problem by equalizing all semitones and making all major/minor 3rds be functionally in tune across all keys, enabling enharmonic modulation and transposition with consistent sonority.
• The historical progression:
  • Early Baroque and late Renaissance: irregular temperaments and mean-tone families offer richer triadic color but restrict some keys’ usability.
  • Late 18th century onward: equal temperament emerges as a practical standard for ensembles and keyboard instruments; the goal of perfect enharmonic modulation becomes less critical than the ability to play in all keys in a uniform way.
• The article’s interpretation of tonal color and key character:
  • Different keys are not merely transpositions with unchanged intervallic content; the timbre and distinguishable “color” of keys arise from their specific tempering profiles (e.g., in 18th-century irregular temperaments, E major vs C major have subtly different leading-tone qualities and diatonic/mode coloration).
  • The literature documents widely observed perceptual effects in performances (e.g., Bach’s Well-Tempered Clavier; Schubert in D-flat major’s nuanced warmth; Beethoven’s late piano works showing uneven but expressive contrasts).
• Notable historical reflections and conclusions:
  • Barpurg’s (1776) argument that equal temperament enables uniformity across keys for ensemble music; he emphasizes that a key’s perceived affect cannot be fully separated from the tuning it uses.
  • The development of “well-tempered” vs “equal-tempered” distinctions, and Bach’s and his contemporaries’ debates about how far one could go in transposition without compromising musical meaning.
  • The modern perspective: equal temperament underpins not only classical piano technique but also later musical styles (Romantic, impressionist, twentieth-century, and jazz harmony) where uniform semitone spacing supports complex chord progressions and advanced modulation.
• Practical implications across genres:
  • The ability to modulate across many keys with equal consonance across the keyboard is essential for ensemble playing, orchestration, and keyboard repertoire that requires broad key changes.
  • The historical preference for irregular temperaments (for tone color) gradually yields to the modern preference for uniformity and transpositional ease, especially in the piano and orchestra.
• In sum: temperaments sculpt harmony and timbre; they are not merely technical devices but carriers of expressive character across musical history. The transition from mean-tone and irregular schemes toward equal temperament reflects changing aesthetic priorities: from coloristic key prototypes to universal, instrument-wide tonal coherence and flexible modulation.

Key equations and numeric references (for quick study)

• Octave equivalence and cents: 1 ext{ octave} = 1200 ext{ cents}
• Equal temperament semitone (ET): 100 ext{ cents per semitone}
  ightarrow 12 imes 100 = 1200 ext{ cents}
• 18:17 semitone (historical equal-temperament approximation): ext{semitone}
  ightarrow ext{approximately } 99 ext{ cents} \ ( ext{approximately } 100 ext{ cents})
• Circle of fifths tempering (Pythagorean comma context): average fifth tempered by approximately ext{Pythagorean comma}/12 \ ext{(about } 2 ext{ cents)}
  • 5th: ext{5th} ext{ is smaller than pure by } rac{ ext{Pythagorean comma}}{12} \ ext{(about } -2 ext{ cents)}
  • 4th: ext{4th} ext{ is larger than pure by } rac{ ext{Pythagorean comma}}{12} \ ext{(about } +2 ext{ cents)}
• Major/minor 3rds in ET:
  • Major 3rd ET deviation from pure: +14 ext{ cents}
  • Minor 3rd ET deviation from pure: -16 ext{ cents}
  • Ratio implication: Major 3rd shifted about 7 imes 2 ext{ cents} = 14 ext{ cents}; Minor 3rd shifted about 8 imes 2 ext{ cents} = 16 ext{ cents}
• 2–3 relevant commas (for context): syntonic comma ext{≈ } 21.51 ext{ cents}; Pythagorean comma ext{≈ } 23.46 ext{ cents}
• 19-note equal division (historical experimentation): a reference point for non-12-tone mean-tone work; not a standard in practice but shows interest in microtonality in historical theory.
• 12-tone equal division of the octave in the 20th century: a reference for equality of all keys across the keyboard historically; the modern thinking aligns with this, but 18th-century practice varied widely.

Connections to broader themes

• Tempo and timbre: temperaments influence timing, rhythmic perception, and timbre because the quality of consonances is altered by small pitch deviations; this affects the perception of rhythm and phrasing.
• Performance practice: temperaments reflect the practical realities of performers (keyboardists, organists, lutenists, gambists) and the instruments’ construction, strings, and timbral quality.
• Harmony and modulation: temperaments shape how keys relate to one another; the degree of dissonance or consonance in different keys, and the possibility of enharmonic modulatory paths, are tied to the specific tuning used.
• Ethics/philosophy: temperaments reveal deeper debates about musical truth vs. musical convenience, and about whether the goal is to preserve purity of intervals or to maximize expressive, modulatory possibilities across a repertoire.
• Real-world relevance: the historical evolution from irregular to equal temperaments underpins modern piano, organ, and ensemble tuning practices; it also informs period-performance choices and scholarly reconstructions of historical instruments and performances.

Summary takeaways

• Temperament is a spectrum, not a single system: quasi-Pythagorean, regular mean-tone, irregular temperaments, and equal temperament each served different musical aims across centuries.
• The main technical driver is balancing two opposing goals: stable, consonant intervals (especially 3rds) and the ability to modulate across many keys with a uniform tonal palette.
• Historical practice shows a pragmatic approach: different instruments, repertoires, and locales used different temperaments; the modern standard (equal temperament) arose from a long historical process of negotiation, experimentation, and alignment with instrument design and ensemble needs.
• The art of tempering is inseparable from the art of performance and composition: composers improvised in many tonal landscapes, and tuners/makers crafted systems to realize those landscapes on real instruments.

References and examples mentioned in the transcript

• Ex.1 Du Fay, Mon chier amy
• Ex.2 Matteo da Perugia, Pres du soloil
• Ex.3 Paumann (D-Mbs Cim 325b)
• Ex.4 Bull (G.B. Och Mus. 1113)
• Ex.5 Gabrieli (I-Tn Giordano II)
• Ex.6 Valente (Naples, 1580)
• Ex.7 Cabezón (Madrid, 1578)
• Ex.8 Schlick (Tabulaturen etlicher Lobgesang, Mainz, 1512)
• Ex.9 Schlick (op. cit., p.51)
• Ex.10 L. Couperin (F-Pn Mus. Rés. Vm7 675)
• Ex.11 Chambonnières (Paris, 1670)
• Ex.12 Anon (transcription of Lully’s Bellérophon, 1679)
• Ex.13 Platti (Nuremberg, c.1746)
• Ex.14 Fischer (Ariadne musica neo-organoedum, 1702)
• Ex.15 Schubert, Piano Sonata in C (d 840), opening
• Ex.17 Schütz, Die so ihr den Herren fürchtet swv364
• Ex.18 S.L. Weiss, Stammbuch notation (Stoc widened enharmonic modulation)
• Ex.19 Du Fay, Belle, plaissant et graciese