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PTG Exam Temperament
In the process of choosing a temperament sequence, I had practiced the ones outlined in Reblitz, 2nd Ed., which included the Defebaugh and Potter temperaments, then the Sanderson one on the AccuTuner web site, and then I went through the PACE lessons and the Source Book. By then I'd gotten a feel for what worked well for me, and settled on Jim Coleman's "Baldassin Sanderson Kimbell Tremper" temperament (BSKT):
This is a modification of the original Baldassin-Sanderson temperament, along with the Sumrell-Stebbins concepts from "Let The Piano Tell You" in the 12/94 Journal. It seems the primary difference is the use of the "locate B3 between contiguous 4ths" technique. I had become confident enough with these techniques to sign up for the tuning exam, but then in a reckless move (or it seemed so at the time) decided to change my sequence a month before the exam because of a couple of insights. The first was that I was better at judging the slower beating M3rds in the A2-A3 range than I was the ones in the A3-A4 range. The second was that I saw I could use the concepts from the BSKT sequence, but have my intermediate single-octave temperament be C#3-C#4 rather than F3-F4, putting me in the slower region.
Where I made the mental leap was at a step very early in the classic sequence: after setting A3, the next step is usually to step down from A3 to an estimate for F3-A3, then immediately go back up to F4, setting up the "Let the Piano Tell You" technique on the intermediate temperament octave of F3-F4. But instead of stepping down, I hypothesized I could "step-up" and make the first estimate at A3-C#4, then tune down an octave to C#3-C#4, and likewise complete the rest of the temperament sequence "in the other direction." From that step on, it effectively *mirrors* the rest of the BSKT steps around the pivot point of A3, putting me in the area of the piano where I am able to judge the beat rates most effectively.
Now, I realize the old-hats out there are nodding, "oh yeah, that could work" (one of my mentors did), and maybe "that's an exercise left to the student," but I tell you, the moment of the insight was like a lightning bolt through my living room window. I was sitting quietly in front of my Chickering, contemplating whether my level of "tuning-fu" was going to be sufficient for the exam, then BOOM, instant inspiration for the rest of the day to figure out how it might work. Within two more practice sessions that week, I was already seeing better temperament results, smoother progressions of intervals, and better correlation to what Tunelab was telling me. So then all I had to do was in three weeks internalize it and get my practice sessions well under the 45 minute limit!
And since I haven't seen this combination of techniques arranged in this particular order, here it is below -- Yet Another Exam Sequence! At least, how I got through it, and hopefully in enough detail to be of use to people who are at the point I was, having the concepts understood, but still trying to figure out how to really put it all together.
The strategy is to use the slower beating area of the piano for more accuracy, during the early portion of the temperament when I have the fewest pre-existing bearings. Here are the tactics, and then right to the details:
- Divide A2-A4 in half, via Baldassin-Sanderson
- Divide C#3-C#4 into M3rds, via Sumrell-Stebbins
- Expand ladder of M3rds to A2-A4
- Divide middle M3rd of C#3-C#4 into whole-steps, via Kimbell-Tremper
- Finish A2-A3 whole step M3rds by bracketing from that middle note
- Divide middle two whole steps in half with contiguous 4ths
- Finish A2-A3 half step M3rds by bracketing from those two notes
- Expand to the upper octave
- A3, 4:2 from A4
- A2, 6:3 from A3
I used the Seiko SQ-50 third partial technique Jim Coleman published in the December 2008 PTG Journal. It's fast. Save the extra minute or two for the midrange work. Of course, the fork/F2 method is still indispensable as backup. On my exam, I didn't have to do anything after setting the two octaves, the double octave was sitting just under 1 bps. But this is where you would check and compromise your octave and double octave widths, and see below regarding its effect on C#3. Note I end up tuning A2 and B2, "extra" notes not required for the aural midrange section, and use them later to give me additional confidence in setting C2, which is required.
- C#4, 7bps above A3, temporary
This is the "initial estimate up, not down from A3" step. The initial approximation of A3-C#4 is around 7bps, and it'll get faster soon enough, but paying attention to Sumrell-Stebbins, the final value of F3-A3 will be more accurate when it is bracketed by contiguous M3rds that are both close to the same beat rate. I've never been very good at the toe-tapping, metronome counting stuff. Mike Swendsen told me early on, "about as fast as you can wah-wah-wah," and that's about as much as I've paid attention to absolute beat rates.
- C#3, from C#4, 6:3 maybe slightly narrow, temporary
- F3, "Let The Piano Tell You" on intervals C#3-F3-A3-C#4
Once C#3 is set, then F3 becomes the knob you turn to make the three contiguous M3rds either all the same beat rate, "slow, medium, fast," or "fast, medium, slow" according to Sumrell-Stebbins, and giving a good location for the first solid M3rd in the ladder, F3-A3. I find that I can make better judgements in this range about whether the middle M3rd is a little lower or a little higher than halfway between the outer M3rds. Now fix the temporary notes, and finish the ladder of M3rds:
- C#3, bracket the C#3-F3 M3rd between the A2-C#3 and F3-A3 M3rds
- C#4, from C#3, 6:3 maybe slightly narrow, check A3-C#4 progression
- F4, from F3, 4:2 maybe slightly wide, check A3-C#4-F4-A4 progression
Since F3-A3 was set previously, and A2 even earlier, the C#3 step is another opportunity to adjust a single note with the goal of getting its M3rd between the two surrounding it. Same as the Sumrell-Stebbins technique, but you don't move the outer M3rds around. In PACE #24, the discussion of the Sumrell-Stebbins technique indicates it does not take A2 into account, and therefore may not be as effective as the Baldassin- Sanderson double-octave technique as far as determining the octave compromises. But I think this is because the step-by-step instructions in that article limit the consideration of C#3 to only a 6:3 comparison with C#4. I think with only the single modification I've made above you gain that advantage as well: instead of C#4, use C#3 as the initial correction of the temporary notes in the outer two M3rds, then you have the opportunity to compromise it against the A2 M3rd.
In the three steps above, those last two notes tuned in the two-octave ladder, C#4 and F4, really highlight what makes this sequence work. Since I have a harder time tuning the faster beating M3rds in the A3-A4 range, I now have backup checks for when I'm setting them. Not only do I have the progression of M3rd beat rate among A3-C#4-F4-A4, I can also check and possibly compromise them using the C#3 and F3 an octave below. And in the case I still don't like what I hear, I haven't gone so far yet that it feels difficult to re-visit the A2-C#3-F3-A3 ladder. I just check them again and make adjustments in case I hadn't quite gotten my "ears on" when I started. It's also useful later when I'm setting the rest of the notes above C#4; I'll have more notes where I can make use of other interval tests.
When I started down this path, I hesitated because with the classic intermediate octave of F3-F4 the contiguous M3rds are double checked in a ladder up to and including A4, the known reference point. My intermediate C#3-C#4 octave is at least two notes removed from that starting point, depending on where I measure it. I can't say for sure why I get better results this way; what I've described in the previous paragraph is why I *think* it works. Maybe I do something I haven't articulated or hasn't even broached my conscious mind, which compensates for other problems in the sequence. Maybe it is simply another path to arrive at some reasonable intermediate conditions which then can be refined, I don't know.
But I press on anyway. The next two paragraphs are not important steps for understanding the midrange sequence I'm describing in this article, but it may be important to the *exam* sequence. I've seen it written many times: get through the entire midrange once, quickly, to make sure you get a reasonable shot at each note, and to deal with the exam detuning.
So at this point in the PTG exam, do a rough fill-in of the other notes in the intermediate temperament octave. The one I like for quickly filling in the notes in the gaps between the M3rds ladder is presented by Bill Bremmer in his "Midrange Piano Tuning" document:
Specifically, starting at step 20 in the section sub-titled "The Marpurg Shortcut." I like the pattern of systematically setting notes relative to each of the M3rd ladder notes, because it's optimized for the situation I have, which is that I already have a ladder of M3rds in the octave. I go for rough, fast results by setting pure 4ths and 5ths, possibly trading off some risk of not having an exam-passing first pass. I'd rather have more time left at the end. Having completed the C#3-C#4 octave, I tune C3 from C4, then quickly tune 4:2 octaves above C#4 up to B4. Again, for speed, I don't do separate M10th checks on those octaves, I'm just listening for the 4:2 note to get pretty close to pure. This cleans up the exam detuning in the midrange and makes it harder to inadvertently get a M3rd on the "wrong side" during the fine- tuning that follows.
On to the rest of "BSKT steps pivoted around A3." Ignore the fact that a fast temperament has just been set. Assume all you have is the two octave ladder of M3rds, and simply be happy that the upcoming notes will already be close to where you'll want them. The next thing is the location of that tricky whole step in the middle of the octave, and in this sequence it is G3. Note that G3 is down a whole step from A3, rather than up a whole step to B3 that is used in the conventional sequence, no matter which Baldassin-Sanderson variant you use. I set two temporary perfect fifths from known notes such that those temporary notes make up the contiguous 4ths with G3:
- D3, from A3, perfect 5th (equal beating M6th-M10th at F2)
- C4, from F3, perfect 5th (equal beating M6th-M10th at G#2)
- G3, equal beating between contiguous 4ths, D3-G3-C4
Compared to the traditional techniques of stacking 4ths or zig-zagging 4th/3rd/4th on either side of this middle note, I found I could get closer in less time with this technique. Using direct interval measurements in Tunelab over a number of practice sessions helped to confirm it. Once G3 is set, I can bracket whole-step M3rds in-between the M3rds of the ladder:
- D#3, from G3, in-between C#3-F3 and F3-A3
- B3, from G3, in-between F3-A3 and A3-C#4
That completes the whole-step ladder of M3rds in the intermediate octave. Next use contiguous 4ths to locate the two notes that will allow for half-step bracketing of the remaining M3rds. Also, at this point I have enough notes that I can begin to employ same-direction 4th/5th checks on the notes being set.
- F#3, between equal contiguous 4ths, C#3-F#3-B3, check F#3-C4 5th
- G#3, between equal contiguous 4ths, D#3-G#3-C#4, check C#3-G#3 5th
With those two notes set, I can finish the rest of the half-step M3rd bracketing to complete the C#3-C#4 octave. What follows is a bit of a departure from the strict mirroring of Jim Coleman's BSKT sequence. In theory, the sequence would find F#3 with contiguous 4ths, then tune the two neighboring M3rds, and then repeat with G#3. I changed the ordering because I find it more natural to do both of the contiguous 4ths operations together, and then follow it with the steps below. I like this ordering because I basically start at the bottom and move up by half steps, thinking about which notes I've already tuned, and filling in the "other" note when I notice it hasn't been tuned yet. So starting at C#3 and going up, here's my thought process:
- Skip C#3, ladder note
- Set D3, from F#3, between C#3-F3 and D#3-G3, check 4th/5th G3/A3
- Skip D#3, already have D#3-G3 after the whole step G3 technique
- Set E3, from G#3, between D#3-G3 and F3-A3, check 4th/5th A3/B3
- Skip F3, ladder note
- Set A#3, from F#3, between F3-A3 and G3-B3, check 4th/5th F3/D#3
- Skip G3, already have G3-B3 after the whole step G3 technique
- Set C4, from G#3, between G3-B3 and A4-C#4, check 4th/5th G3/F3
- Skip A3, ladder note, done
That finishes the one-octave temperament sequence, and when you tell your examiner the temperament octave you want to be scored on is C#3-C# 4, maybe they'll give you a quizzical look, then shrug their shoulders and go on.
Now what's left in the midrange is the faster beating area. These can be set with a whole host of checks, depending on how much time you have left. For the two remaining gaps in the M3rd ladder, C#4-F4, F4-A4, and the leftover notes C3, A#4, B4, I took the following strategy: set the note that splits the surrounding M3rds by either a whole step or a half step, check and maybe compromise the octave below, and with the same- direction 4th/5th checks, and the M3rd and M10th progressions (E4 and up). This is the same technique as in the BSKT sequence after the intermediate octave is completed. So here's the order:
- D#4, from B3, between A3-C#4, C#4-F4, check D#3, 4th/5th check A#3/G#3
- D4, from A#3, between A3-C#4, B3-D#4, check D3, 4th/5th check A3/G3
- E4, from C4, between B4-D#4, C#4-F4, check E4, 4th/5th check B3/A3
That completes the half-steps all the way up through F4, so you can apply the same "split whole, split half" technique to the next M3rd gap using the notes you just tuned above:
- G4, from D#4, etc.
- F#4, from D4
- G#4, from E4
And then there are a few leftover notes:
- A#4, from F4
- B4, from G4
- B2, from D#3, between A2-C#3 and C#3-F3
- C3, from E3, between B2-D#3 and C#3-F3
And that's it! Apply whatever refinement methods you like best. I start with a parallel progressions of 4/5/6ths, double-checking with same-direction 4th/5th tests, and go on from there with other checks. Apologies in advance if I mis-credit the original discoverers of techniques, the people I reference are simply the ones that wrote about the information wherever I first saw it.