Temperament Design

Meantone and well-tempering gradually evolved into the modern Equal Temperament, and just as with the earlier system, there was a proliferation of methods and schools, with a very conservative rate of progress.

The Yamaha or Japanese style adopted by the Sydney Conservatorium was developed in a research setting, unconstricted by loyalties and traditions. The academics who developed it looked at as many traditions as they could and developed the method using the best practices as the basis.

The Design of the temperament has these goals: Accuracy, Stability and Speed. They are inter-related. As an example, if a piano can be completely tuned in less than an hour, starting with the temperament in the middle and tuning to the top and then from the temperament towards the low end, then it can be expected to be stable. This is because the iron-frame and cabinet take some time to react to the change in stress. The bass strings are positioned at an angle, with the lower part of the strings more towards the middle of the piano than the top part. Putting extra tension on the bass end can make the top end increase in tension by way of a see-saw effect centred aroung the middle.

  • A temperament obviously encompasses an octave. Any more than one octave introduces uncertainty, unnecessary retuning within the procedure,
  • On many pianos, the tenor strings change from plain-wire to copper-wound in the low tenor, and a large number make this change at F33, with E being the last copper-wound string. Copper-wound strings have a lower inharmonicity, and including a wound string in the temperament complicates tuning tht temperament.
  • For this reason the temperament is set between F33 and F45. This octave is in the most audible range, centred around middle C. The beat speeds are easy to count here.

    If the temperament area were shifted down a semitone or two, the beat speeds to be counted would be slower and would take more time to judge. Similarly, taking the temperament area up would mean that more notes would be in the uncountable area, and in the same area (above F45) the inharmonicity increases making for uncertainty.

  • In general, strings increase in inharmonicity from the lowest notes upwards. There is often a slight bump in the inharmonicity curve at the lowest notes of the bass, where the strings are thicker due to the compromise in string length, and there is a similar slight rise in the low tenor for the same reason. Tuning the temperament in this octave just above the low tenor allows for a smooth rise in pitch, and even if the lowest note or two do increase in inharmonicity, the pitch-curve of the higher partials will be smooth, making it possible to have a neat tuning curve across the piano.
  • By fortunate coincidence, when starting on A37 the first interval tuned is a fourth above, D42s, and the beat speed for A-D is 1 per second. The first third chord which is checked is F33-A37, and the beat speed here is exactly 7 beats per second. (This is all at A=440hz). Other tuning methods starting on C or F do not have such a neat reference so near the beginning of the process.
  • Historically discussion of Equal Temperament referred to a "flattening of the fifth" as the characteristic, the fifth interval being seen as important. In practice, the inversion — sharpening of the fourth, is faster to achieve. Hammer/pin technique requires each tuning event to involve taking the string over the intended pitch, and striking the key hard enough to force the string down to the intended position. This is known as setting the pins. So if you are intending to hear 1 beat per second pull the D up so that (for instance), there are 2bps. It will take no more than 1 second to hear the two beats, possibly less. Then after the hard blow on the key, it will take no more than a second to recognise 1 beat.

    Going the other way, dropping a fifth from, say, zero beats down to 1bps, might involve 2 to 3 seconds listening to ascertain that the interval is beatless, another 2 or 3 to hear that there is only, say, 1 beat every 2 seconds, and finally 1 second to hear the intended speed of 1bps. Obviously not the way to do it.

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