What we're talking about here is the "compass" of the temperament octave, the width of the octave that the 12 chromatic steps have to fit within. There's direct acoustic proof that a 3:1 octave is wider than a 4:2 octave, but no similar proof that it's narrower than the 6:3, only empirical observation. But it's safe to say that a temperament based on a "3:1 compass" would have a stretch somewhere between these two.
But what has left me less than impressed about miracle-of-the-21st-century P12 temperament is that it's not backwards compatible, i.e.., it can't be done aurally. It is simple enough for an ETD to do: 1.) calculate the complete tuning based on a 2:1 temperament, 2.) note the difference in off-sets between the 2:1 octaves and the 3:1 P12s, and 3.) widen the temperament semitones to reflect this. In fact any self-respecting ETD allows the the choice of 3:1 as the basic "octave" interval.
But as was quickly pointed out, even the ETD has to know at each step in the temperament, what the off-set is at that specific step. Kind of like arriving at the proper tempering of a 5th on a given piano, tuning a circle of 5ths/4ths aurally. You don't know the correct answer until you've been around the mulberry bush.
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William Ballard RPT
WBPS
Saxtons River VT
802-869-9107
"Our lives contain a thousand springs
and dies if one be gone
Strange that a harp of a thousand strings
should keep in tune so long."
...........Dr. Watts, "The Continental Harmony,1774
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Original Message:
Sent: 08-03-2019 12:55
From: Scott Kerns
Subject: An Exploration of the Mathematics of Pure 12th Equal Temperament
You're certainly correct David. However, Roshan clearly states: "The purpose of this post is to explore the mathematics of Pure 12th Equal Temperament." Doesn't there have to be some mathematics involved before you start tuning? It's all way over my head but I appreciate the exploration.
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"That Tuning Guy"
Scott Kerns
www.thattuningguy.com
Tunic OnlyPure, TuneLab & PianoMeter user
Original Message:
Sent: 08-03-2019 11:29
From: David Pinnegar
Subject: An Exploration of the Mathematics of Pure 12th Equal Temperament
Dear Roshan
Whilst I admire your mathematics and juggling with which I'd have been familiar with facility four decades ago there's a significant trouble. That occurs because instruments behave differently. Earlier this year I organised a seminar on tuning and the experience from which organ builder Martin Renshaw comes is relevant
https://youtu.be/k61eHv9piMc?t=983
in instruments and their buildings reacting in different ways. In the same way is inharmonicity to the piano. As a result whilst one can specify in mathematics to one's heart's content, it's not actually till you're actually tuning and hearing those things with which one's bringing alignment that real tuning happens.
Sometimes these things bring difficulties of expression. Jason Kanter has been kindly trying to bring my description of what I do to the mathematics of graphing accessible to him and what I do has such significant deviation from standard practice that we can't graph it yet with existing software. It's only when you can apply the mathematics to the real measured world of a real piano that any tuning system can really come to life.
Best wishes
David P
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David Pinnegar BSc ARCS
Curator and House Tuner - Hammerwood Park, East Grinstead, Sussex UK
antespam@gmail.com
Seminar 6th May 2019 - http://hammerwood.mistral.co.uk/tuning-seminar.pdf "The Importance of Tuning for Better Performance"
Original Message:
Sent: 08-02-2019 23:01
From: Roshan Kakiya
Subject: An Exploration of the Mathematics of Pure 12th Equal Temperament
The purpose of this post is to explore the mathematics of Pure 12th Equal Temperament.
I will be using the following source as a reference throughout this post.
http://www.piano-stopper.de/dl/PTG2008_StopperTemperament.pdf
The Pythagorean comma is the discrepancy between 12 Pure Fifths and 7 Pure Octaves.
Pythagorean comma (1st Definition) = (3/2)12 / (2/1)7 = 531441/4096 / 128/1 = 531441/4096 × 1/128 = 531441/524288.
The frequency ratio of a larger interval can be calculated by multiplying the frequency ratios of the intervals that are smaller than it.
For example, Pure Major Third (5/4) × Pure Minor Third (6/5) = Pure Fifth (3/2).
The formula of the Pythagorean comma can be rearranged as follows.
(3/2)12 = (2/1)7 × Pythagorean comma.
312/212 = 27 × Pythagorean comma.
312 = 212 × 27 × Pythagorean comma.
312 = 219 × Pythagorean comma.
This formula suggests that 12 Pure Twelfths are larger than 19 Pure Octaves. The discrepancy between them is the Pythagorean comma. Therefore, the Pythagorean comma can also be defined as follows.
Pythagorean comma (2nd Definition) = 312/219 = 531441/524288.
The 2nd Definition of the Pythagorean comma directly involves the 2nd harmonic (Pure Octave) and the 3rd harmonic (Pure Twelfth) of the 1st harmonic (Unison).
Pure 12th Equal Temperament can be constructed by sharpening each of the 19 Pure Octaves by the 19th root of the Pythagorean comma.
312 = (2/1 × (531441/524288)1/19)19.
531441 = 531441.
"The answer to the question why not take any other equal temperamant between pure octaves and pure fifths is given by the recent discovery of the inherent beat symmetries that only occur when the duodecimes are in tune, eliminating beats and therefore producing improved clarity and resonance, as with pure tuned intervals".
An analysis of the beat rates of the Tempered Fifths and the Tempered Octaves of Pure 12th Equal Temperament must be performed in order to verify this discovery.
Frequency ratios:
Pure Twelfth = 3/1 = 3.
Tempered Fifth = (3/1)7/19 = 37/19.
Tempered Octave = (3/1)12/19 = 312/19.
Frequencies:
A4 (Unison) = 440.00 Hz.
E5 (Tempered Fifth) = 440.00 Hz × 37/19 = 659.53 Hz.
A5 (Tempered Octave) = 440.00 Hz × 312/19 = 880.63 Hz.
E6 (Pure Twelfth) = 440.00 Hz × 3 = 1320.00 Hz.
Beat rates:
Tempered Fifth (A4-E5) = Difference between 3rd partial of A4 and 2nd partial of E5 = 440.00 Hz × 3 − 440.00 Hz × 37/19 × 2 = 0.94 Hz.
Tempered Octave (E5-E6) = Difference between 1st partial of E6 and 2nd partial of E5 = 440.00 Hz × 3 × 1 − 440.00 Hz × 37/19 × 2 = 0.94 Hz.
Beat ratio:
A4-E5 / E5-E6 = 0.94 Hz / 0.94 Hz = 1.00.
A beat ratio of 1.00 indicates that the Tempered Fifth (A4-E5) and the Tempered Octave (E5-E6) will beat at the same rate. Therefore, this result verifies the discovery of the beat symmetry that occurs if Pure Twelfths are used.
Interestingly, Pure 12th Equal Temperament also causes the Pure Fifths and the Pure Octaves to be tempered by the same amount in opposite directions.
Pure Fifth = 1200 × log2(3/2) = 701.955 cents.
Tempered Fifth = 1200 × 7/19 × log2(3/1) = 700.720 cents.
Pure Octave = 1200 × log2(2/1) = 1200.000 cents.
Tempered Octave = 1200 × 12/19 × log2(3/1) = 1201.235 cents.
Difference between Tempered Fifth and Pure Fifth in cents = 700.720 cents − 701.955 cents = −1.235 cents.
Difference between Tempered Octave and Pure Octave in cents = 1201.235 cents − 1200.000 cents = +1.235 cents.
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Roshan Kakiya
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