When I wrote that Peter Serkin might like "another of the more elegant 20th century coloristic temperaments," that got me thinking about what exactly I might mean by that, and I got the notion of explaining how you might set up an "elegant" temperament. The Vallotti and Young concepts are certainly the most elegant from the point of view of providing an evenly graded range of M3 sizes (commonly referred to as "key color" as that is the main component - keeping in mind that the M3 affects the M6, etc.). The difference between Vallotti and Young is simply that Vallotti has narrow fifths from F to B, while Young has them from C to F# (upward around the circle of fifths). Vallotti makes more sense to me in his choice, and is far more commonly used.
The way the Vallotti design works is ingenious: you narrow six 5ths, those on the natural keys (FC, CG, GD, DA, AE, EB), and you widen those that involve an accidental (FBb, BbEb, EbAb, AbDb, DbGb, GbB or F#B). The interplay between these changes creates an even gradation of M3 sizes from CE to C#E# (DbF). The degree of widening and narrowing used in Vallotti leads to a somewhat edgy sound, objectionable to most modern pianists. Specifically the widest M3s are too wide, and the fifths on the natural keys are too narrow.
But a "half Vallotti" would probably be fine for almost anyone. How would you go about designing that, creating a cents offset table? I'll try to explain it in a way that can be grasped pretty easily. We need to start with the notion that the "norm" (ie, offsets of zero) means that all fifths are narrow by 2¢ (actually between 1.95 and 2¢, but let's round up). So if, for example you want to create a just fifth, you need to widen it by a total of 2¢, which you can do by raising the upper note +2¢, lowering the lower note -2¢, or some combination. Looking at it from the other perspective, if you see a table of offsets, you can compare the offsets of two notes a fifth apart and figure out the size of the fifth. +6 for the upper note and +6 for the lower means it is an ET fifth. +6 for upper and +4 for lower means it has been widened 2¢ from ET, hence it is a just fifth. +6 upper and +8 lower means it has been narrowed 2¢ from ET, so it is a total of 4¢ narrow. That is essentially a 1/6 comma fifth (rounding). Vallotti uses 1/6 comma fifths and just fifths.
To make a "half Vallotti," we could make half as much of a change, 1¢ in either direction rather than 2¢. And we can do that by simply following the circle of 5ths. Starting on A as zero, we will want to narrow all the natural key fifths by 1¢, and widen all the fifths with accidentals by 1¢. Moving downward around the circle of fifths, D will need to be raised 1¢, hence D - +1¢. G will need to be 1¢ narrow of ET as well, so it will need to be raised 2¢, hence G - +2¢. C - +3¢, F +4¢. We need to catch the natural fifths upward from A as well, and they will change in the opposite direction, so E will be -1¢, B -2¢.
The remaining fifths will need to be widened. We can do them all in the same direction, starting from F. So Bb will be +3, Eb +2, Ab +1, Db 0.0, Gb -1, and that completes the circle. Hence, putting them in order by half steps:
C +3¢
C# 0
D +1
D# +2
E -1
F +4
F# -1
G +2
G# +1
A 0
A# +3
B -2
You could also do "1/3 Vallotti" if you wished, by making the difference 0.7¢ rather than 1¢, etc., etc. in the same additive way moving around the circle of fifths.
Aurally, you can do the same kind of thing: simply make your fifths on the natural keys about 1 1/2 times narrower than ET (about 1 bps, and 4ths 1.5 bps), and make the fifths that involve accidentals a bit closer to pure but not quite.
Another interesting thing to do is to analyze the numbers in the chart to see how wide other intervals are, like M3s. For instance, CE: C is +3¢, E is -1¢, hence CE is 4¢ narrow of ET. So are FA and GB. C#F, F#A#, and G#C are wider than ET by 4¢. The other M3s are between those two extremes in size.
I hope this explanation makes some sense. It isn't all that hard to understand once you have managed to grasp the basics. I know that most people's eyes seem to glaze over when I try to explain these things, but I keep trying <G>.
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Fred Sturm
University of New Mexico
fssturm@unm.edu http://fredsturm.net "When I smell a flower, I don't think about how it was cultivated. I like to listen to music the same way." -Federico Mompou
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Original Message:
Sent: 04-15-2014 14:00
From: Fred Sturm
Subject: Peter Serkin Temperament
Hi Dennis,
I had no motivation other than to try to clarify what most people find hopelessly baffling. I would point out, though, that 1/7 and 1/8 comma mean tone (and, for that matter, most of the other fractions besides 1/4) were essentially mathematical constructs, not aural realities (at least "precise" aural realities) - until the late 20th century. They were set out in mathematical terms, in some detail, but the only way to tune them was by using a monochord: measure distances, pluck the string, transfer the pitch to your keyboard instrument. How precise is that? Not very.
It wasn't until Owen Jorgensen created aural methods of tuning these, based on the cents calculations provided by Murray Barbour, that these temperaments obtained an aural reality - to the extent people were able to follow those instructions. And the "equal beating" instructions are actually distortions of the original theoretical designs, to make it easier to tune what is, in essence "impossible" to do aurally with any precision. So I'm not sure that you can really argue that an aural refinement is in any way superior to an electronically generated tuning, done precisely.
Specifically with respect to these sizes of comma, they derive historically from the writings of a French theoretician by the name of Romieu (from one particular paper he presented), and Barbour says, "Romieu mentioned temperaments of 1/7, 1/8, 1/9, and 1/10 commas, but did not consider them sufficiently important to discuss." Romieu did discuss at length such fractions as 2/9, 3/10, 3/11, etc. He was purely a theoretician, writing in an academic way. There is no other source for the fractions of 1/7 and 1/8 comma that I know of, so I really think they need to be treated as a sort of late 20th century invention. With respect to the historical precedent, we are talking about a dry theoretical paper given to the French Royal Academy of Sciences in 1758, talking about a number of dry, mathematical/theoretical constructs. This is not how anyone tuned at the time.
In this context, what do "aural refinements" mean? At least until the time that the tuning has been "given life," which it has in the persons of Tim McFarley and Peter Serkin. But note that it has evolved over the years from a pure 1/7 comma mean tone (originally) to a modified 1/8 comma mean tone. So how particular is Peter Serkin and about what? If it has evolved in this direction more or less at his request, all we can surmise from that is that the 1/7 comma wolf was probably troublesome, and so was the 1/8 comma wolf. But he does want some kind of difference of size of M3. It is very possible that another of the more elegant 20th century coloristic temperaments would please him more.
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Fred Sturm
University of New Mexico
fssturm@unm.edu
http://fredsturm.net
"When I smell a flower, I don't think about how it was cultivated. I like to listen to music the same way." -Federico Mompou
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Original Message:
Sent: 04-15-2014 11:18
From: Dennis Johnson
Subject: Peter Serkin Temperament
Hi all-
...and thanks for the detail Fred. If I may, this offers an opportunity to express one point regarding using numbers and tables for reproducing temperaments that always makes me uncomfortable. On one hand, if the concert went well and everyone is happy, great. On the other, as professional tuners we owe it to ourselves to understand exactly what is the goal of any given temperament. When it comes to ET we have very specific aural tests to rate the success or failure of exactly what was produced, independent of whether the player made a complaint or how exactly we reproduced numbers from someplace else. It's no different with this temperament. The goal here is to make 11 fifths exactly the same size and 8 major thirds exactly the same size, and do that at the place where the wolf (G#-Eb) beats in a 2:1 ratio with the diminished 4th (Ab-C), naturally all within a clean octave. Since the fifths are faster and major thirds slower than in ET it's relatively more noticeable when they aren't equal size. To do this accurately is not easy, but can be fun. And, in the end it's either spot on, or something else. I recommend using the table or other numbers to get it close, as a template, then aurally polish it to perfection. Same as you would do tuning ET. If that was the intent and I misunderstood, then all the better.
that's my 2 cents, now have a great day~!
dennis.
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Dennis Johnson, R.P.T.
St. Olaf College
Music Dept.
Northfield, MN 55337
sta2ned@stolaf.edu
(507) 786-3587
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Original Message:
Sent: 04-14-2014 17:47
From: Fred Sturm
Subject: Peter Serkin Temperament
For anyone who might be interested in trying to understand what is behind the numbers and the nomenclature involved in describing the "Serkin Temperament," I will make a stab at de-mystification.
First, there is the notion of a "fractional comma," in this case "1/8 comma." The comma referred to is the syntonic comma, the difference between a stack of four just fifths and a just major third: tune the just 5ths CG, GD, DA, AE and you end up with a CE "Pythagorean third," a good bit wider than an ET third. Tune E directly to C to make a just third (beatless) and E will be a good bit lower. The difference between the two placements of E is the syntonic comma: very close to 21.5 cents, or the arithmetical ratio of 81:80.
When we talk about 1/8 comma, we mean each fifth is narrowed by 1/8 of 21.5 cents, or 2.7 cents. An ET fifth is narrowed by just under 2 cents, so a 1/8 comma fifth is about 1/3 narrower than an ET fifth, not too noticeable by itself. It beats about like an ET 4th.
If you tune a standard mean tone using 1/8 comma fifths, you would tune 11 of them, leaving the Eb/G# "wolf" fifth to take up the difference. Add up the total of those eleven narrow fifths: 11 x 2.7 = 29.7 cents. The total of 12 fifths in the circle has to add up to the Pythagorean comma of 23.5 cents. Hence, the Eb/G# fifth will have to be 29.7 - 23.5 cents WIDE, or 6.2 cents wide. That is pretty noticeable, and that fifth will sound out of tune when it is exposed, and when you play a chord involving it (A flat triad). Furthermore, it will make the four major thirds involving it quite wide. The other eight major thirds will be precisely the same size, a fair bit narrower than ET. The other four will be a good bit wider than ET.
If you want to have more of a progression of M3 sizes, you can accomplish that most easily by tuning only 8 of the fifths narrow by 1/8 comma, tuning the rest just, as in the Vallotti pattern (this is not precisely true, but close enough for most purposes).
The earlier "Serkin Temperament" was based on 1/7 comma, so the eleven 1/7 comma fifths would be 3.1 cents narrow, and the wolf would be 10.7 cents wide. That is getting quite edgy, wherever that wolf fifth shows up, and the four wide M3s will be that much wider. However, it would be quite difficult to perceive and control the difference between a 1/7 comma and a 1/8 comma fifth, and to do that consistently, tuning aurally. It is the difference between 1.3 times narrower than an ET fifth and 1.5 times narrower. With an ETD, this is no problem, of course. It is when you add them all up that you really hear a difference, and it is mostly apparent in the wolf.
If you are interested in fooling around with this kind of thing, including calculating beat rates of various intervals for yourself, you can find some spreadsheets for the purpose, together with other materials, here: http://my.ptg.org/caut/resources/libraryview/?LibraryKey=a8f8135d-d76f-43b9-9486-b72523520cd9
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Fred Sturm
University of New Mexico
fssturm@unm.edu
http://fredsturm.net
"When I smell a flower, I don't think about how it was cultivated. I like to listen to music the same way." -Federico Mompou
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