TEMPERAMENTS XXII: Neidhardt’s “für eine große Stadt” 1724Entire Contents Copyright © 2016 CBH
Some musicians despise Vallotti’s popularity as a generic all-purpose temperament. Its nice symmetry, with six narrow fifths together on the sharp side of F and the rest of the fifths perfect, gives an even gradation of key color. If you have been following these pages, though, you will know that a Major third derived from any chain of four perfect fifths in a row is worse than Equal Temperament. If it were possible to somehow split the long chain of six perfect fifths in Vallotti, perhaps something less grating could be obtained for the extreme keys. Like all temperaments, though, this must come at a cost.
I tend to add temperaments to this section of the Technical Library on demand, and this particular one, Neidhardt’s “für eine große Stadt”, was specified by visiting Swedish conductor Olaf Boman directing Berlin early music ensemble Solistenensemble Kaleidoskop in a punk fashion take on Handel’s Semele for the 2013 Sydney Festival. Mahan Esfahani also requested this temperament for his 2016 Brisbane Baroque solo harpsichord recital at QPAC of Frescobaldi, Pachelbel and Benda along with JS, CPE & JC Bach— and his first Sydney appearance at the Sydney Opera House in May 2017.
It would appear that Johann Georg Neidhardt (1680–1739) was somewhat of a temperament freak. In his Sectio Canonis Harmonici zur völligen Richtigkeit der Generum Modulandi of 1724 he went so far as to recommend which of his temperaments might be suitable for a village, a small city, a large city, or for making music in the Court itself—the latter in fact our familiar Equal Temperament. In the next decade, he provided monochord string lengths and discussed twenty-one temperaments in his Gäntzlich erschöpfte, Mathematische Abtheilungen des Diatonisch-Chromatischen, temperirten Canonis Monochordi etc. (The digitized original of a 1734 printing can be viewed online at the Bayerischen Staatsbibliothek website.) In that work, he revized his nomenclature and his original 1724 scheme “für eine große Stadt” does not appear.
Here we will contemplate his 1724 suggestion for the now Big City of Sydney.
Observe that this temperament has two sizes of narrowed fifths: Three familiar sixth-comma fifths of Vallotti, and six twelfth-comma (equal-tempered) fifths. Three remaining fifths are perfect to complete the circle, although none of them are located together. Our job is to place each fifth correctly around the circle.
If you have just tuned Vallotti, half the notes (C, G, D, A, E♭ and A♭) do not need to be moved: Only the other six pitches need to be retuned to change your temperament to Große Stadt:
1. Lower your f until it makes a perfect 5th with middle c', and tune up an octave to f'. (Instead of the determined three beats a second for f–a in Vallotti, this increases the beat speed of that interval to five times a second.)
2. Lower your b♭ to place it between e♭ and f' so that both fifths are slightly narrow. The equal-tempered fifth e♭–b♭ will only beat about once every two seconds, the fifth b♭–f' perhaps three times every four seconds. (If you have a metronome, you can confirm this latter beat rate at A440 by setting it to 45.)
3. The interval a–e' must have a slow beat, half what it was in Vallotti and comparable to b♭–f' a semitone higher. Temporarily raise your e' until it is pure with a the fifth below, then lower it slightly to make a–e' another equal-tempered fifth.
4. Tune an octave down to e, and raise your existing b to make a beatless fifth above that.
5. Two notes remain to be sharpened: Raise your d♭' until it makes a slow beat of about once per second with a♭', and raise your f♯ until it has a very slow beat with c♯' (the d♭' you’ve just tuned.) Tune up an octave to f♯', compare the same-sized fifths b–f♯' to d♭'– a♭' and you are done.
Now you can try the musical effect of this temperament. There is still a variety of key color available throughout the simple keys, although we’ve lost some of their Vallottian slightly gentler qualities. The remote keys are certainly more palatable, but perhaps Neidhardt’s real cleverness with this temperament is his distribution of fifths to give almost identical key color in the five sharp keys from A around to C♯, making them all eminently usable. The worst key is A♭, but no worse in fact than Vallotti.
To begin tuning Neidhardt’s Große Stadt from scratch, we will setup a chain of six narrow fifths like Vallotti, although some of these pitches will only be temporary. Follow these instructions:
1. Tune your a' (that’s the one in the octave above middle c' of course, but you knew that!) to your pitch source, and tune a in absolute perfect tune an octave below it.
2. Find the f a third below that a: Tune it perfect first of all, but then widen the interval by flattening the f until you hear three distinct beats per second.
3. Temporarily tune three fifths from F around the flat side of the circle of keys absolutely perfect, continuing until you hit A♭, tuning up and down octaves as necessary to stay within your bearings area in the middle of the keyboard.
4. Tune e' pure to a, then lower it a little, squeezing the interval so it has a perceptible but not overly blatant wave—every three-quarters of a second. Drop down the octave now to e, which of course must be tuned perfect.
5. Middle c' comes next. Tune it a perfect fifth above your f, and then squeeze it a little so the resulting fifth f–c' beats a little slower than the already tuned a–e' interval.
6. Find your g' above middle c', again tuning the fifth perfect and then squeezing it. (You’re a good deal higher up the keyboard now, so bear in mind that if this interval is the same theoretical size as the last, it’s going to beat just that little faster and trick you into thinking you have actually tuned it rougher: Every octave you go up doubles the beat speed.) Drop down the octave to the perfect g.
7. Jam your d' in between the g and the a', if that is the right phrase to use. First of all, tune it perfect to g, and then commence your juggling act, lowering your d' little by little until your fifth g–d' sounds about the same as your f–c'.
8. Tune two temporary perfect fifths: b pure to e, and f♯'above b. Drop down the octave to f#.
9. Locate your c♯' between f♯ and g♯' (really a♭'), making two slightly narrow fifths.
10. Now retune e' to its final position to form a slightly narrow fifth above a. Drop down an octave to e and confirm e–b is also slightly narrow, although only temporarily.
11. Raise your b to its final position a perfect fifth above e, and check you now have the slow-beating equal-tempered fifth required from b–f♯.
12. Two more pitches to retune, and you are done: Lower your f until it makes a perfect fifth with middle c', and tune up an octave to f'. Finally, lower your b♭ to place it between e♭ and f' so that both fifths are slightly narrow. The fifth e♭–b♭ will only beat about once every two seconds.
Check and determine you have everything in the right place: For practical purposes in these types of temperaments, the proportion and location of the various-sized fifths is more important than absolute mathematical precision. There are three sixth-comma narrow fifths g–d', c'–g' & d'–a' and six twelfth-comma narrow fifths e♭–b♭, f♯–c♯', a–e', b♭–f', b–f♯' & c♯'–g♯'. (I’ve listed these in ascending order within both groups to help you hear the increasing beat speeds as you rise in pitch.) And don’t forget the three pure fifths e–b, f–c', a♭–e♭'!
Anonymous [Kayano, Moxzan] Dodecagon — Chi-s akt temo Tokyo 2012, p98
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