Piano tone isn't just a matter of hammer voicing. Hammer voicing voices the note but not the instrument. It's for this reason that I place importance on how vibrations relate to each other, producing either a sweet or a harsh voice.
Those of us who have grown up playing instruments and in particular string or brass instruments will be familiar with pure intervals giving a sweet tone or in specialist choirs singing Pythagorean intonation, a strident sound. Those of us who might have grown up with Hammond organs and dialling in harmonic drawbars will be familiar with sounds being near in tone, but the "harmonics" being based on Hammond version of equal temperament tone generators, the sound not being quite "there" and nearly always sounding like a Hammond. Harmonics make the tone. In the piano we have the harmonics, and inharmonics of the strings, and the harmonics of the scale notes when played as chords and the harmonics brought together by the instrument. Like the Hammond, this is why, in my view, equal temperament distortions of the natural scale are not ideal.
As a matter of curiosity I wanted to see how Perfect 12th tuning can affect tone. We've seen in the spreadsheet analysis of P12 that it can result in very low beat rates as far as 5ths and 3rd harmonics are concerned, but what might it do overall to the tone? By use of sequences of perfect fifths, the thirds have to be stretched towards Pythagorean.
The spreadsheet I've posted before I realised was faulty in looking at accordance or not of the odd harmonics but failing to take into account the stretch in the 2nd and 4th partials, and so warranted a reworking, although with surprisingly little difference.
The spreadsheet had grown like topsy and as standards as against others could be measured, I kept ET, Meantone, Kellner data built in. I'm not entirely happy with the Meantone data and will have to revisit it but it might still be useful as a provisional comparator.
Sheet 4 of the spreadsheet extends the analysis, counting beats between scale notes and 3rd and 5th harmonics - rows 4 to 59.
In that range of notes, 486 beats are counted in Equal Temperament, 658 in Meantone, predominantly on account of dismal 5ths, 491 in Kellner so comparable to Equal as a candidate alternative, and 549 in P12 on account of the expanded 3rds.
Subject to checking of the data and some divisors, P12 comes out very well for near beatless coincidence of harmonics with scale notes and as Steven Norsworthy has reported
|
Equal Temperament |
Meantone |
Kellner |
Perfect 12th |
| Proportion same |
37.79 |
35.79 |
43.14 |
58.86 |
| Proportion 1-2 beats |
14.72 |
16.05 |
9.03 |
5.35 |
| Proportion 2-3 beats |
9.03 |
7.02 |
7.36 |
4.35 |
| Proportion 3-4 beats |
4.01 |
6.02 |
5.02 |
3.68 |
| Proportion 4-5 beats |
2.68 |
3.68 |
4.68 |
2.34 |
| Proportion 5-6 beats |
2.34 |
4.01 |
2.01 |
2.68 |
| Proportion 6-7 beats |
1.67 |
2.68 |
2.68 |
2.01 |
| Proportion 7-8 beats |
3.68 |
1.67 |
2.01 |
3.34 |
| Proportion 8-9 beats |
2.68 |
3.01 |
4.35 |
3.01 |
| Proportion 9-10 beats |
2.68 |
1.34 |
2.34 |
2.68 |
When we look at a measure of beats of thirds and fifths in different keys and accounting for increasing pitch we find, although I have reservations about the Meantone figures as D major for instance is anomalous
| Keys |
Equal Temperament |
Meantone |
Kellner |
Perfect 12th |
| C |
11 |
5 |
4 |
30 |
| C# |
11 |
10 |
16 |
30 |
| D |
11 |
31 |
7 |
30 |
| Eb |
11 |
5 |
13 |
30 |
| E |
11 |
19 |
14 |
30 |
| F |
11 |
3 |
6 |
28 |
| F# |
11 |
3 |
18 |
28 |
| G |
11 |
6 |
8 |
28 |
| Ab |
11 |
17 |
17 |
28 |
| A |
11 |
32 |
12 |
27 |
| Bb |
11 |
4 |
10 |
27 |
| B |
11 |
41 |
16 |
27 |
| C |
11 |
5 |
4 |
27 |
| TOTALS of all keys |
135 |
135 |
142 |
345 |
we see that due to the strong Pythagorean nature of the thirds in P12 we can expect it to be quite a strident tonality, whereas Meantone is predictably sweet but let down badly in B major at least (we expect B Ab F# and C# to be usable only for "special effect") but shows a balance overall to the perfectly balanced Equal Temperament of which Kellner competes closely with an excellent balance of very good and less good but still quite good keys.
Because I've found Kellner to work I haven't explored beyond Kirnberger III. The 6th comma temperaments, Vallotti, Young etc I cannot hear in practice as different from ET and have been caught out embarrassingly thinking that instruments tuned in those tunings were ET. However with 6 perfect fifths they will still resonate well although display less resonance of nearer perfect thirds. Stanhope, Kirnberger I and II are worthy of further exploration.
The essence of the Well Temperament series of tunings with perfect fifths is to transfer from the concept of sweet perfect thirds as in Meantone in the home keys, to the starkness and unlocking of vibrations expressed by wide thirds in the Pythagorean arena of perfect 5ths and perfect 12ths in the remote keys. We thus open up a whole new spectrum of tonalities as music modulates. Liszt used the Pythagorean dislocation of notes and harmonics in B major on which to skate about as on ice "Student of Adolfo Barabino masterclass demos B Major unequal temperament"
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| Student of Adolfo Barabino masterclass demos B Major unequal temperament |
| This is a student of Adolfo Barabino's masterclass playing Liszt and demonstrating the effects of unequal temperament. Proponents of the Bradley Lehman temperament, which I believe to be flawed, say that B major is an unusable key in the Kellner Bach temperament so here we have Liszt skating around in this etheric key. |
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(a recording 14 years ago with the instrument played impromptu and not tuned for the event) and Barabino Masterclass student Liszt Piano Sonata Unequal Temerament tuning
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| Barabino Masterclass student Liszt Piano Sonata Unequal Temerament tuning |
| Student of Adolfo Barabino plays the Liszt Sonata on a piano tuned to unequal temperament at Hammerwood Park |
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is the full piece with the instrument then tuned.
When we change to the home key of Bb major, the colour and tonality are chromatically very different.
"Bach in a good Unequal Temperament"
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| Bach in a good Unequal Temperament |
| Pianist Helen Yorke at Hammerwood Park, Sussex, UK in Unequal Temperament. The piano is a Bechstein from 1885 tuned by David Pinnegar to a High Definition version of the Kellner tuning https://www.youtube.com/watch?v=sb6pt3OvU_o. Keyboard Partita No.1 in B-flat Major, BWV 825 |
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In my opinion the tonality of the instrument has changed with the change of key and this brings very great interest to the listener and is capable of keeping the modern audience engaged.
Upon the basis of the results hinted at in the spreadsheet perhaps it's possible to see why the opinions about temperament I express are far from arbitrary and how some tunings might be candidates for good tonality and musical effect whilst others might be appropriately regarded as outliers.
Most analysis I've seen on temperaments has been around cents, but cents aren't what we hear. Beats and accordance, association of frequencies and disassociation and it's in this way that there's potentially more mileage to be found in continuing such analysis.
I'm not expert in manipulating Excel spreadsheets so am sure that some keen eyed members might find faults and look forward to any better solutions which might be found.
Best wishes
David P
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David Pinnegar BSc ARCS
Hammerwood Park, East Grinstead, Sussex, UK
+44 1342 850594
"High Definition" Tuning
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