| A | B | F♯ | C♯ | ||||||||
| F | C | G | D | A | E | B | F♯ | ||||
| E♭ | B♭ | F | C | G | D | A | E | B | |||
| A♭ | E♭ | B♭ | F | C | G | D | |||||
| A♭ | E♭ |
See also Manipulating Time I.
The following version is in a twelve tone unequal temperament, so you can compare.
(Werckmeister 1691 tuning)
Trouble grasping the difference with just intonation? The fragments on the home page may help.
You can download zipped .csv files containing the exact intervals used in this piece. Explanation of column headers.
In the first attempt of this piece, measure 28 sounded as follows:
(measure 28, unmarked)
The top bar of this measure is:
The two G's indicated by an X have different pitches: they differ by an indirect syntonic comma and that is clearly audible. And unacceptable, in my opinion. This can be cured by guiding the optimizer. One does that by marking the second of these G's in the score with a special sign (only meant for the optimizer - not to be printed). This informs the optimizer that it is extremely desirable that the marked G has exactly the same pitch (disregarding octaves) as the last G before that point. In this case the first G indicated by an X above. The optimizer takes the marks into account when calculating a new combined pitch assignment. Measure 28 then becomes:
(measure 28, marked)
And indeed, the syntonic comma has gone. But when I listen to this new measure 28 through small PC speakers, I can hear a new bad interval in the fast notes of the third beat. The strange thing is: when I listen using quality headphones, it all sounds all right to me; interesting, but right!
Analysis shows that in this last 'corrected' version, on the second G marked X above, there is a harmonic minor third G-B♭ with pitch ratio 32/27. The earlier version (with the uncorrected indirect syntonic comma) had G-B♭ with pitch ratio 6/5. The 6/5 minor third can be restored by again guiding the optimizer. This time by marking the second of these G's in the score with a different sign, which informs the optimizer that on that particular G a harmonic minor third must have pitch ratio 6/5, not 32/27.
Alas, as was to be expected, the waterbed effect takes over: now there is a new annoying interval at 1 minute 5 seconds (not reproduced here). We could chase that and generate the next version, etc. For the time being, we remove the minor third mark from the second G and will be satisfied with the chord characterized as interesting, but right! above.
| C | G | D | A | E | B | F♯ | C♯ | |||
| E♭ | B♭ | F | C | G | D | A | E | B | ||
| A♭ | E♭ | B♭ | F | C | G | D | ||||
| A♭ | E♭ | B♭ |
Comparison:
| Measurement | Main version | Experiment 1 |
| Syntonic commas (direct & mid-air) | 3 | 7 |
| Pitches/octave | 30 | 27 |
| Harmonic minor 3rds 6/5 | 370 | 388 |
| Harmonic minor 3rds 32/27 | 70 | 52 |
You can download zipped .csv files containing the exact intervals used in this experiment. Explanation of column headers.