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Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
microtemperament, microtempering
A temperament where each just interval approximates to a single tempered interval, and more than two intervals (including the octave) are required to define the tuning.
If a just interval approximates to different tempered intervals, you have a well temperament.
If only two intervals, usually an octave and a fifth, are required to define the tuning, you have a linear temperament. Examples are meantone and schismic scales.
Kudos to Dave Keenan for blazing this trail.
For an octave-invariant scale, the number of bridge intervals for a microtemperament is two less than the number of harmonic axes. For 7-prime limit, there are 3 harmonic axes: 3, 5 and 7. So only 1 bridge is required for a microtemperament. Two bridges therefore must give you a linear temperament, as indeed they do here.
[from Graham Breed, Onelist Tuning Digest # 531, message 3]
What if the number is three less, four less, etc.? Still could be a microtemperament, I would say. And if the number is one less? Many linear temperaments are examples of microtemperament, like Helmholtz's and Groven's schismatic temperaments.
[from Paul Erlich, Onelist Tuning Digest # 532, message 2]
Paul Erlich wrote:
For an octave-invariant scale, the number of bridge intervals for a microtemperament is two less than the number of harmonic axes.What if the number is three less, four less, etc.? Still could be a microtemperament, I would say.
Yes, my oversight.
And if the number is one less? Many linear temperaments are examples of microtemperament, like Helmholtz's and Groven's schismatic temperaments.
Well, that's why we need a definition.
[from Paul Erlich, Onelist Tuning Digest # 532, message 16]
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