Robert, you bring up interesting topics in your posts on tuning. Tuning a piano using perfect fifths used to present a conundrum for musicians due to what is called the Pythagorean Comma, which is the difference between twelve just perfect fifths and seven octaves. So if you start on a piano with C1 and tune in perfect fifths, C8 will be off from being 7 octaves above C1 by 23.5 cents. One-quarter of a semi-tone was always found to be unacceptable by the musical community according to my limited reading, and led to the development and use of what are now called "historic temperaments" such as versions of well temperament, meantone temperament, and just intonation - the types of temperaments that can be chosen today with digital pianos and with hybrid pianos. It was not until around 1917 when, of practical necessity, 12-Tone Equal Temperament (TET) based on the octave became the industry standard for tuning of pianos and hence for use by string instruments playing in accompaniment with a piano. When you get more experience with the pure fifth tuning perhaps you can give us some feedback on how you like it.
A comparison of numbers for various ET tunings that relate to octave stretch are the following:
(a) Those of us who tune using the industry ET standard (12-TET based on the octave) implicitly incorporate a multiplier per key of the 12th root of 2 = 1.059463. This gives a pure octave.
(b) The multiplier for an ET tuning based on perfect fifths (7-TET based on the fifth) is the 7th root of 1.5 = 1.059634.
(c) The multiplier with an ET based on perfect twelfths (19-TET based on the twelfth) is the 19th root of 3 = 1.059526.
(d) Finally, for an ET using 26ths (43-TET based on the twenty-sixth), if I have it right, the multiplier is the 43rd root of 12 = 1.059491.
My point (finally!) is that if you use the 26ths in the Verituner you are closer to the industry standard ET than using perfect twelfths, but you are still stretching the octave. The most stretch of the octave occurs when using an ET using perfect fifths.
As an analogy (or caution) to stretching the octave on a piano versus using pure/perfect octaves, think of a four-part choir. If all four parts were to be singing the same note, with a frequency spread from bass to soprano of that note being 20 cents off from pure octaves, anyone listening would notice the dissonance.
Kent, I just wanted to say that in my opinion in a discussion of piano temperaments, it tends to obfuscate the topic by bringing in "inharmonicity." It usually is not necessary to bring in the physical frequency since whether one tunes aurally or tunes visually with an ETD, the tuner (all of us) has as one of his/her primary objectives to match the musical harmonicity of the particular piano's strings. That is the reason that aural tuners (who use 12-TET based on the octave) have to choose to match a 4:2 or 8:4 or 6:3 etc octave, and why a visual tuner depends on the quality of the inharmonicity algorithm transparently built into their ETD to do the same. Similar to how a good violinist or cellist, when playing solo, will musically match the harmonicity of the strings in their musical instrument (and unavoidably ignore the inharmonicity). I always choose to use the industry standard ET that produces audibly pure octaves (with no beats), and I can hear the resultant impure fourths and fifths going up the scale; others who choose to use the temperament of 19-TET based on the twelfth like the different but uniquely pleasant sound that results.
Regards, Norman
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Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.nethttps://potomacpiano.com(301) 983.9321
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Original Message:
Sent: 12-03-2022 15:23
From: Robert Sluss
Subject: Swafford Pure 5th and Pure 26th styles
Thanks for your reply! I get what you're saying. Maybe I'll experiment with some "low-risk" pianos, e.g., the practice room instruments at the University of Florida's School of Music:)
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Robert Sluss
Lake City FL
(386) 752-1888
Original Message:
Sent: 12-03-2022 15:01
From: Kent Swafford
Subject: Swafford Pure 5th and Pure 26th styles
The example use case I always make for pure 5th equal temperament is a Bösendorfer Imperial that I used to tune regularly that had absolutely the lowest inharmonicity of any piano I ever tuned. Whereas on pianos with "normal" inharmonicity the cents value for C8 might commonly be 42 cents or so, because of the low inharmonicity this B'dorf' would commonly have only about 19 cents on C8! So, using pure 5th equal temperament would stretch the high treble enough for the piano to have a more normal melodic high treble, despite the low inharmonicity.
Pure 26th ET, on the other hand, is a very narrow tuning, just the bare minimum of stretch in the single octave.
All that said, the choice of the width of your equal temperament has more to do with your preferred beat rate relationships. That is, pure 12th ET has the beat rates of the fifths and octaves balanced. Pure octave ET would balance the beat rates of the 4ths and fifths. Pure 5th ET would yield clean perfect fifths and almost a vibrato in the octaves, not "balanced" at all. Your choice.
Kent
Sent from my iPad
Original Message:
Sent: 12/3/2022 2:25:00 PM
From: Robert Sluss
Subject: Swafford Pure 5th and Pure 26th styles
I have available in Vertuner, besides the Swafford Pure 12th style, the Pure 5th and the Pure 26th styles for tuning. Can anyone tell me what sizes of pianos/scales those two would be appropriate for?
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Robert Sluss
Lake City FL
(386) 752-1888
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