Kent makes the point that people use ad hoc stretching of a Pure Octave ET anyway, and Kent suggests it is typically 1 cent. The Pure12th ET makes that 1.2347 cents. It is a minor difference. Here I just did the math:
Conclusions: for every interval, including all the 3rds, 6ths, major and minor, as well as the 4ths, 5ths, etc., one can only conclude that
% 1:1 (unison)
% 2:1 (octave)
% 5:3 (major sixth)
% 3:2 (perfect fifth)
% 4:3 (perfect fourth)
% 5:4 (major third)
% 6:5 (minor third)
%%
% Let's say you stretch every 1/2 step by 0.1 cent over a pure octave ET
OneCentStretchOctave = (2.^((0:.1:1.2)/1200)).*(2.^((0:12)/12));
OneCentStretchOctave'
%%
% 1.000000000000000
% 1.059524293114831
% 1.122591727700483
% 1.189413206748411 -- m3
% 1.260212187101555 -- M3
% 1.335225426713470 -- 4th
% 1.414703776387538
% 1.498913018643888 -- 5th
% 1.588134756519284 -- m6
% 1.682667355272188 -- M6
% 1.782826940142168
% 1.888948453500208
% 2.001386774925161 -- Oct
%
log_m3_1c=1200*log2(1.189413206748411);
log_M3_1c=1200*log2(1.260212187101555);
log_4th_1c=1200*log2(1.335225426713470);
log_5th_1c=1200*log2(1.498913018643888);
log_m6_1c=1200*log2(1.588134756519284);
log_M6_1c=1200*log2(1.682667355272188);
log_oct_1c=1200*log2(2.001386774925161);
%%
% Now use the Pure 12th formula instead
Pure12thOctaves = 3.^((0:12)/19)';
%
% 1.000000000000000
% 1.059526064738275
% 1.122595481859776
% 1.189419173187856 -- m3
% 1.260220615891982 -- M3
% 1.335236589858077 -- 4th
% 1.414717969546883
% 1.498930562988532 -- 5th
% 1.588156000719167 -- m6
% 1.682692677632456 -- M6
% 1.782856750895827
% 1.888983197268723
% 2.001426933358855 -- Oct
log_m3_P12=1200*log2(1.189419173187856);
log_M3_P12=1200*log2(1.260220615891982);
log_4th_P12=1200*log2(1.335236589858077);
log_5th_P12=1200*log2(1.498930562988532);
log_m6_P12=1200*log2(1.588156000719167);
log_M6_P12=1200*log2(1.682692677632456);
log_oct_P12=1200*log2(2.001426933358855);
%% Now compare the differences
%
Diffs = [log_m3_1c-log_m3_P12 log_M3_1c-log_M3_P12...
log_4th_1c-log_4th_P12...
log_5th_1c-log_5th_P12...
log_m6_1c-log_m6_P12...
log_M6_1c-log_M6_P12...
log_oct_1c-log_oct_P12];
Diffs'
% -0.008684347166593
% -0.011579129554491
% -0.014473911943526
% -0.020263476722107
% -0.023158259110346
% -0.026053041499722
% -0.034737388665917
% The differences are within a hundredth of a cent, so why bother? Just use
% the Pure 12th ET to do the stretch you would do anyway!
Original Message:
Sent: 03-30-2024 23:12
From: Bill Ballard
Subject: Piano Temperament Pure 12th
Peter Grey went: "However, you cannot have both...perfectly ascending and descending RBI's AND perfectly smooth ascending SBI's no matter how ideal the piano is."
For my first 10 years, I had a similar attitude, "I'll provide the 4ths and 5ths, and the manufacturer will provide the 3ds and 6ths." In my the 3ds&6ths temperament, the first RBI series ("runs brought in"?) I use to check out a finished temperament is the M6th series: it contains a M3d and a P4th on top of that, and thus tests both SBIs and RBIs simultaneously.
We should expect disagreements between the SBIs and RBIs series. It's a product of the piano's inharmonicity curve, and the more ragged it is, the more disagreements there will be. For an example of how this works, pick any note in the middle of the temperament compass and add a P4th in either direction (# or b) and add a M3d in the opposite direction. (Pick, say, D3-F#3 and F#3-B3 or its converse, C#3-F#3 and F#3-A#3. The outside width of both these pairs is a M6th, and both have the common note F#3.)
For each of these two pairs of intervals, the expanded width (beyond the mathematically pure) is determined by the inharmonicity of the common note and the added note. If there's a disagreement between the 4ths series and the 3rds series, when each of these hits the common note, a likely assumption would be that it's the inharmonicity of the common note that's the outlier and causing the disagreement. But it would seem just as likely that the disagreement results from the inharmonicity of one (or both) of the added notes. Remember, they're separated by 3/4 of an octave. Even in a relatively smooth inharmonicity scale, major differences can develop in that distance, especially of one of the added notes is down in the hockey-stick end of the long bridge.
Regardless of individual preferences in tuning styles in this Forum, I'm confident that everyone here is turning out wonderful tunings.
------------------------------
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
Original Message:
Sent: 03-29-2024 16:55
From: Peter Grey
Subject: Piano Temperament Pure 12th
IMO (and remember it is strictly an opinion),
One can tune "ET" by concentrating even tempering on the 4ths and 5ths, and letting the 3rds and 6ths "fall where they may" (which on a well scaled instrument the progessions will be pretty darn good overall) OR one can concentrate on evenly tempering the 3rds and 6ths, creating (hopefully) a "perfect" ascension ladder of these intervals and let the 4ths and 5ths "fall where they may" (which on a well scaled instrument...).
However, you cannot have both...perfectly ascending and descending RBI's AND perfectly smooth ascending SBI's no matter how ideal the piano is. When we start "splitting hairs" (which is exactly what we ALL do when evaluating other people's tuning), we can, and will, find at least slight "discrepancies" in some of the intervals. It's the nature of what is required to actually produce one of these beasts (they're not perfectly uniform).
Therefore we make choices as to how we will create an ET scheme that satisfies within reason the tenets of what we mean by ET at this time in history, i.e. "all like intervals tempered the same" (with the term "same" tempered to mean reasonably or pretty close). In the aggregate, when done well, we end up with a reasonably smooth progession of ALL intervals with a few compromises to accommodate the idiosyncrasies of the instrument.
Good aural/analog tuners can accomplish this as can good digital tuners do the same. The only difference (IMO) is that aural/analog tuners are listening to the "whole sound" (or largely that) throughout the process, whereas digital tuners are often focusing on selected partials of the intervals, with both methods being "colored" slightly in different ways but not enough to make a functional or musical difference.
Show me ANYBODY'S tuning, I don't care who it is, and I can find stuff in it that isn't "perfect" (if I want to...because I CAN BE an obsessive hair splitter if I want to). Generally though I just like to sit down and play music on it. If its well done, I'm very happy.
Peter Grey Piano Doctor
------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com
Original Message:
Sent: 03-29-2024 15:52
From: Steven Rosenthal
Subject: Piano Temperament Pure 12th
The P12 tuning is a style, not a temperament per se, P8 or P12 we are still talking about an equal temperament. I have been using the P12 tuning since it became available for Verituner, 6 years or so to great effect. Any ET will yield a smooth progression of beats of all intervals, accelerating as you ascend and decelerating as you descend. This is our objective aural test to see if our tuning is adhering to the ET scheme. As you say, Steve, it's just physics. This even progression is what gives ET its utility in Western chromaticism. (Adherents to the use of colored temperaments have their points too)
On the recording I hear the M3's at the top of each chord speed up and slow down as it ascends, this indicates a problem somewhere along the line. Perhaps the beats are obscured by the 5 notes over 3 octaves, perhaps some notes drifted from the original tuning, perhaps there was a problem with the sampling, I can't hazard a guess. If you listen from 0:30 to 1:05 this is quite evident.
A standard test using single intervals would eliminate several variables. Then we can clearly hear the beat progression. Regardless of theory or style the proof is in the pudding, if you have a smooth progression of beats, your golden, if not then there is some work to be done. Steve N mentions letting M3s "fall where they may", this is in reference to the specific beats per second, not whether or not they progress as per standard.
There are samples of these progressions if you go here and click on Module 2; Hearing intervals.
Another note on "styles", this is where tuners can exercise some creative discretion in building a tuning as seen in the discussion above between Bill and Norman and it is where aural tuners can really shine.
------------------------------
Steven Rosenthal RPT
Honolulu HI
(808) 521-7129
Original Message:
Sent: 03-29-2024 14:38
From: Steven Norsworthy
Subject: Piano Temperament Pure 12th
Here's the math for both octave ET and pure-12 ET
>> 2.^((0:12)/12)
1.0000 1.0595 1.1225 1.1892 1.2599 1.3348 1.4142 1.4983 1.5874 1.6818 1.7818 1.8877 2.0000
>> 3.^((0:19)/19)
1.0000 1.0595 1.1226 1.1894 1.2602 1.3352 1.4147 1.4989 1.5882 1.6827 1.7829 1.8890 2.0014 2.1206 2.2468
2.3805 2.5222 2.6724 2.8315 3.0000
Now compute the difference at the M3
1200*log2(1.2602/1.2599)
0.4122 cents sharper for pure-12 ET over octave ET
Now the question: how well did I line up with actual measurements I just made from the recordings I posted at the M3? Here it is
Theoretically
1200*log2(1.2602/1.25) = 14.0695 cents sharp for pure-12 ET at the M3
and in pure octave it is
1200*log2(1.2599/1.25) = 13.6574 cents sharp
In actual measurement from my pure-12 ET, I got several examples at
13.90 - 14.01 range of variances, which means I am indeed getting consistent progressive beat rates
Kent suggests that there is a natural 'masking' that one may not be hearing which is due to the coherent of the slow beat rates of the 5ths and octaves, i.e., lower stronger partials. That is his answer as to why you may not have been hearing what is actually a very consistent M3, M10, M17 progression that you were expecting.
Nearly ideal!
Any questions?
Steve
------------------------------
Steven Norsworthy
Cardiff By The Sea CA
(619) 964-0101
Original Message:
Sent: 03-29-2024 14:09
From: Norman Brickman
Subject: Piano Temperament Pure 12th
Steven R and all -- I think we can understand now this tuning that Steve N posted. To reiterate, he is chromatically playing five notes: the fundamental and partials 2,3,4,5. So a heavy emphasis is on the octave and double octave, which can easily dominate, particularly when working lower in the scale. The equal beating that we are clearly hearing in the lower two octave (but elsewhere as well) primarily originates from the octave and double-octave coincident partials. (Steve sent me a cleaner recording for the first octave of the chromatic progression that agrees with his audio that is posted above.)
Conclusion: the ETD tuning is labeled as a P12 ET, but (in my opinion) it is not. In the lower part of the scale it stretches the octave much more than the specified 1.23 cents impurity, but I estimate that the stretch elsewhere in the scale also exceeds spec. PLUS (a second problem) it actually continually increases the octave stretch as one goes lower in the scale (!), hence the reason for the audible coincident partial beat rate remaining uniform for, say, most noticeably the lowest two octaves but elsewhere as well.
It should be easy enough for the ETD developer to reprogram to correct the problem. (Or is the extra octave stretch controlled by ETD user controls?) It is no secret that I prefer pure octaves via use of our piano industry ET tuning standard. But with 5 experienced tuners liking what I consider an "Enhanced Octave Impurity Tuning," perhaps this new tuning temperament will be preserved within the ETD.
Steve N -- it would be great to hear the same chromatic progression for a proper P12 ET tuning when you get it, with its 1.23 cents impure octaves. Regards, Norman.
------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321
Original Message:
Sent: 03-28-2024 22:43
From: Norman Brickman
Subject: Piano Temperament Pure 12th
Bill, yes indeed. Right on. And I think that Bill Bremmer also brings in the 8:4 octave in some of his discussions. I'm really not much of a minor-sixth guy for using the 8:4 octave, but I'm a regular for use of the others. Hopefully we select the proper one to use at the proper time. And agreed, you only mentioned the single octaves. Regards, Norman.
------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321
Original Message:
Sent: 03-28-2024 21:22
From: Bill Ballard
Subject: Piano Temperament Pure 12th
Norman went: "Concerning the discussions in this thread on the M3 in a P12 ET tuning versus an octave-based ET tuning, on the same PTG Journal page referenced above you say that "the M3rds will end up additionally widened by less than 0.4 cent over pure-octave equal temperament."
Again, which octave relationship are we talking about, the 2:1, the 4:2, or the 6:3? And that's just the single octaves.
------------------------------
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
Original Message:
Sent: 03-28-2024 19:59
From: Norman Brickman
Subject: Piano Temperament Pure 12th
Kent, thanks for the update. The author you mentioned is Claudio Di Veroli in your article in the PTG Journal of October 2017 on page 33, for those interested. There you refer to "traditional pure-octave equal temperament", and then in the next paragraph you bring in Di Veroli's opinion where you paraphrase him ("in usual octave stretching") and explain that he refers to the mid-range of the keyboard. The piano industry tuning standard is what you call the "traditional pure-octave equal temperament" – with no octave stretch. When a visual tuner sets an ETD for a "traditional pure-octave ET," I presume that the ETD does not then stretch the octave!
We know that piano tuners enjoy their independence. In your and my case -- I follow the piano industry standard and tune with perfect octaves and imperfect twelfths; while you, in turn, tune with perfect twelfths and imperfect octaves. My tests / demonstrations upon completion include the simultaneous playing of, say, C2+C3+C5+C6, and on a better piano having it sound like a single note being played. I'm sure that you have different types of tests and demonstrations of a quality tuning with perfect 12ths.
Concerning the discussions in this thread on the M3 in a P12 ET tuning versus an octave-based ET tuning, on the same PTG Journal page referenced above you say that "the M3rds will end up additionally widened by less than 0.4 cent over pure-octave equal temperament." This is a part of what David has been referring to.
Regards, Norman.
------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321
Original Message:
Sent: 03-28-2024 18:20
From: Bill Ballard
Subject: Piano Temperament Pure 12th
I'm replying to no one in particular, but I have been using the P12 aurally for the top half of the piano, after stumbling on it 35 years ago (and writing about in the PTJ a few years later). For the temperament and the bottom half, I use a 6:3 octave. I might have used the P12 for the temperament, except that it didn't lend itself to the 3ds&6ths scheme of dividing a compass octave into smoothly ascending ladders of thirds and fitting 6ths in between them. (Yes, the late Bernard Stopper did have a scheme for building ladders of 3rds, but with its dependence on occasional 4ths and 5ths which had to be tempered subjectively, working with ladders inside a P12 compass really seemed more kluge than scheme.)
To use Dan Levitan's Expansion Units, the P12 (8 EUs) seemed to be the next right step the the 6:3's 27 EUs and a better tempering than the 4:2's 12 EUs. The 6:3 had to be abandoned pretty quickly by the top of the 4th octave, and on smaller pianos, it was too wild even in the temperament.
Hearing all this talk of tunings based on pure octaves, I'm reminded on Dan's appearance on a Piano Technicians Masterclass. Someone asked him what he thought of P12. He replied," Which one, the 3:1, the 6:2, or the 9:3?"
------------------------------
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
Original Message:
Sent: 03-28-2024 12:19
From: David Pinnegar
Subject: Piano Temperament Pure 12th
Dear Kent
Haha - yes I appreciate what you say . . . but many musicians really do consider the piano to be discordant and strained but in the industry it's the sound that people are familiar with.
In my opinion it's one of the reasons that many who play the piano aren't listening to the sound, the music that they're making, reducing their performance to the technicality of the fingers at the keys. There was indeed a famous interview with Lang-Lang when he actually talked about this and that he was now starting to listen to the sound rather than to listening to the sound as he thought it should be merely in his head.
For instrumental musicians or barber shop quartet singers all familiar with bringing perfect intervals to the harmony between them there's always a listening dimension and a tuning modification to achieve best harmony between the instruments. Between such musicians movement towards perfect thirds are a specific target and it's possibly for this sort of musician that the standard uniform stretched thirds are less than musical. It's for this reason that universal adoption of particularly wide thirds throughout the scale isn't likely to endear the piano to people and isn't therefore in the best interests of the industry.
Best wishes
David P
-- - - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594
Original Message:
Sent: 3/27/2024 9:14:00 PM
From: Kent Swafford
Subject: RE: Piano Temperament Pure 12th
David wrote:
"With respect I find thirds in perfect 12th tuning too wide, harsh, and give the impression to other musicians that the instrument is discordant and unmusical."
Pure 12th equal temperament is simply a codified way of executing the (more or less) same stretch that has always been done in fine piano tuning. If you dislike equal temperament, then I suppose you'll dislike the M3rds in pure 12th ET, and that is fine.
However, to describe a pure 12th ET tuning as discordant and unmusical is probably just wrong, plain and simple, as evidenced by its wide adoption around the world over the past 35 years.
All the best,
Kent
Original Message:
Sent: 3/27/2024 8:57:00 PM
From: David Pinnegar
Subject: RE: Piano Temperament Pure 12th
With respect I find thirds in perfect 12th tuning too wide, harsh, and give the impression to other musicians that the instrument is discordant and unmusical.
Of course there will be a solidity and resonance to the sound with coincidence of 3rd harmonics with perfect 12ths, but this serves the stridency of the instrument and not the musical expressions of the music.
Mood is critically expressed by the thirds.
Best wishes
David P
-- - - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594
Original Message:
Sent: 3/27/2024 8:44:00 PM
From: Kent Swafford
Subject: RE: Piano Temperament Pure 12th
Steve wrote:
" 'aural tuners' are generally more concerned with major 3rds and all their combinations of the 10th and 17th octave versions. He and I agree that we should simply 'let the major 3rds fall where they wish to fall' because in the history of of Western European musical composition from the 18-20th century, the foundation of that kind of harmony is always the 5ths and octaves, so if we let them balance each other then the major and minor 3rds take 2nd place."
This is essentially correct, but I'd phrase it slightly differently. To calculate a pure 12th ET, ETD tuning calculators tend to use the 12th (3:1), the octave (2:1, 4:2, 6:3), and the fifth (3:2 and 6:4). If the tuning calculator keeps the 12ths pure, and at the same time balances the fifth and octave as near as it can get both to a temper of 1.23 cents, then the 3rds, 10ths, 17ths, are bound to be close to their target tempers as well. Because of inconsistent inharmonicity, _all_ of the tuning intervals, including the 5ths, octaves, and 12ths may possibly be drawn away from their theoretical amount of temper, but _all_ the tuning intervals will be very close to correct, within reasonable tolerances, with none needing special priority.
Kent
Original Message:
Sent: 3/27/2024 4:46:00 PM
From: Steven Norsworthy
Subject: RE: Piano Temperament Pure 12th
Hi Nathan,
Kent Swafford and I have been working on this now for the past month. We carefully examined the IH curves of my piano and determined that the 3:1 or Pure 12th tuning would be best. You are hearing that result when I play the chords using 1,2,3,4,5 together.
The Pure 12th ET was discovered by Bernhard Stopper several decades ago. Kent Swafford became the leading writer and proponent of it, having published many papers.
In that temperament, here is the math, which you can run this code in Matlab:
% Pure 12ths Tuning Concept
% 3.^((0:19)/19); % 20 half steps from the fundamental to the 3rd partial
% 1.0000 1.0595 1.1226 1.1894 1.2602 1.3352 1.4147
% 1.4989 1.5882 1.6827 1.7829 1.8890
% 2.0014 2.1206 2.2468 2.3805 2.5222 2.6724 2.8315 3.0000
% The first octave is stretched
% log2(3)*12; % = 2.001426
% 3^(12/19); % = 2.001426 Ideal Octave Ratio Stretched
% ans/2; % = 1.0007 Ideal Octave Ratio 'Stretch Factor'
% 1200*log2(ans); % = 1.2347 cents sharp
% The first 5th is compressed by the same amount as the stretch of the first octave
% 3^(7/19); % 1.4989 Ideal 5th Ratio Compressed
% 1200*log2(ans/1.5); % = -1.2347 cents flat
As Kent told me, 'aural tuners' are generally more concerned with major 3rds and all their combinations of the 10th and 17th octave versions. He and I agree that we should simply 'let the major 3rds fall where they wish to fall' because in the history of of Western European musical composition from the 18-20th century, the foundation of that kind of harmony is always the 5ths and octaves, so if we let them balance each other then the major and minor 3rds take 2nd place.
I had 4 concert tuners last week at my house, each with 30+ years of experience and all RPT's. Without exception all of them said it was "The best sounding tuning they have ever heard."
At least some like it!
Steve
Original Message:
Sent: 3/27/2024 4:34:00 PM
From: Nathan Monteleone
Subject: RE: Piano Temperament Pure 12th
> For a concert performance involving a duet of piano + violin or other string instruments, I am always amazed at the skill of the string player in adapting to the ET (12-TET based on the octave) of the piano, with its enharmonics and with every musical interval except the octave being (aurally) impure. I would assume that there are never concert performances where the piano is tuned in perfect 12ths – but I have a feeling that I will shortly be told how wrong I am!? Regards, Norman.
I don't really have enough experience tuning for violinists to say for sure, but the couple that I do work with don't seem to mind. Pure 12 ET's offsets vs. the more traditional 4:2 octave midrange are really pretty minor; I forget the exact number but it's something like 1.3 cents per octave?
------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com
Original Message:
Sent: 03-27-2024 16:21
From: Norman Brickman
Subject: Piano Temperament Pure 12th
Steve, thanks for the post. Is there a musical reason that you chose perfect twelfths to tune by? I personally think of tuning in twelfths in an ET as adding "character" to the music, as opposed to adding the "mood" and "color" that is associated with historic temperaments (which are not ETs). And with the above all being compared to our piano-industry-standard ET with its aurally-pure octaves.
(All reference in this post being to "musical"/aural frequency, NOT actual physical frequency via the Railsback Curve / inharmonicity.)
I presume that you used an Equal Temperament for the P12 tuning, 19-TET based on the 12th. Correct? In the audio that you provide there are obvious coincident partial beats. At first I assumed I was hearing the effects of the 1.23 cents non-pure octave, but then as you continue chromatically I was not sure – I might have a follow-up question after you respond.
For a concert performance involving a duet of piano + violin or other string instruments, I am always amazed at the skill of the string player in adapting to the ET (12-TET based on the octave) of the piano, with its enharmonics and with every musical interval except the octave being (aurally) impure. I would assume that there are never concert performances where the piano is tuned in perfect 12ths – but I have a feeling that I will shortly be told how wrong I am!? Regards, Norman.
------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321
Original Message:
Sent: 03-25-2024 18:31
From: Steven Norsworthy
Subject: Piano Temperament Pure 12th
We should legitimately be able to post anything related to piano temperament in PianoTech or CAUT.
During the tuning process, the 'Pure 12th' method was followed and the unisons were tuned to within 0.1 ¢.
The link is meant as an 'aural check,' of Pure 12th Tuning, playing parallel 5-partial chords: In the left hand, playing the octaves, and in the right hand playing partials 3,4,5 (a 2nd-inversion major triad). Playing them chromatically starting at A0 and ending up at C8.
https://youtu.be/8hmVoHHfxrk
Respectfully submitted,
Steve
------------------------------
Steven Norsworthy
Cardiff By The Sea CA
(619) 964-0101
------------------------------