Reinhard Kopiez

Hanover University of Music and Drama (Germany)/

Hochschule für Musik und Theater, Hannover (Germany)

(Study published in Music Perception, 2003, 20(4), 383-410)


Intonation of harmonic intervals:
Adaptability of expert musicians to equal temperament and just intonation


"Auf Blechblasinstrumenten ... ist die gleichschwebende Temperatur nicht erreichbar"

["On brass instruments ... equal tempered intonation is unattainable"]

(Vogel, 1961, p. 97)



Description of the study

Experiments in interval perception often reveal an astonishingly wide range of acceptability as far as mistuning is concerned. Mistunings of 20–40 cents seem to be acceptable in adjustment or evaluation tasks, yet this wide range of tolerance towards intonation deviation contrasts with the demands made on the intonation skills of expert musicians. Within the framework of a top-down approach, it is hypothesized that expert musicians use all acoustical cues available, such as timbre or musical context, in order to achieve optimal task adaptation and therefore are able to adapt to different tuning systems to a significant degree. On this hypothesis, it was decided to carry out an experiment which includes a "realistic" musical setting. The experimental conditions of this study used the paradigm of controlled varied condition with a 2 (tuning systems) * 2 (players) * 5 (renditions) * 5 (interval categories) design. Two professional musicians (trumpet players) were chosen as subjects to play the missing upper voice of a 4-part sound-example whilst listening via headphones to the remaining 3 parts in adaptive just intonation (JI) and equal temperament (ET). An analysis of variance showed a non-significant difference between the mean deviation of intonation in the two systems used. The results revealed a mean deviation of 4.9 cents in the ET condition (sd = 6.5) and 6.7 cents in the JI condition (sd = 8.1) and no significant differences were found between players. Results are based on the assumption of an unconscious "always the same" strategy, which means that the same intonation is employed for ET and JI versions. This overall tendency is interpreted as a "burn in"-effect and is the consequence of long term intonation practice in ET. Additionally, a factor analysis revealed four components which determine intonation patterns: these are the "major third factor", the "minor third and partials factor", the "instrumental tuning factor" and the "octave-minor seventh factor". To summarize, even in expert musicians, intonation is not determined by abstract tuning systems but is the result of an interaction among compositional features, the acoustic of the particular musical instrument and deviation patterns in specific intervals.

  • Download of publication in Music Perception, 2003, 20(4), 383-410 (pdf)



(a) test composition (score as eps-file ,104 k)

4-part ET-version as MIDI-file


4-part ET-version (mp3) (800k)

4-part ET-version (wav) (9 M)

4-part JI-version (5 limit) (mp3)
(800 k)

4-part JI-version (5 limit) (wav) (9 M)


3-part version in ET (mp3) (800 k)

3-part version in ET (wav) (9 M)


3-part version in JI (mp3) (800 k)

3-part version in JI (wav) (9 M)

(b) aural test

pythagorean (mp3)
pythagorean (wav)

equal tempered (mp3)
equal tempered (wav)

just (mp3)
just (wav)

(mp3, each approx. 260k; wav, each approx. 2.7 M)


(c) single performances
(solo voice and accompaniment are on separate channels):

best version in ET (mp3)
best version in ET (wav)
Player "K", mean deviation = +2 c, sd = 4.2 c (mp3, 800 k; wav, 8.4 M)

best version in JI (mp3)
best version in JI (wav)
Player "H", mean deviation = +4.6 c, sd = 3.6 c (mp3, 800 k; wav, 8.4 M)

worst version in ET (mp3)
worst version in ET (wav)
Player "K", mean deviation = +7.1 c, sd = 3.8 c (mp3, 800 k; wav, 8.4 M)

worst version in JI (mp3)
worst version in JI (wav)
Player "K", mean deviation = + 8.9 c, sd = 4.0 c (mp3, 800 k; wav, 8.4 M)




EXCEL-file of all frequency analyses

Data in SPSS format


Comments and discussions are welcome. Please contact: Reinhard Kopiez , [reinhard.kopiez(at)]