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In applied statistics we have a very important and useful theorem called the Central Limit Theorem (CLT). This basically states that if you have several different random distributions at play in some process that with a large enough sample size two things will emerge no matter how different in shape the individual distributions are and even if these distributions are highly non-symmetric. These are:
1. If you add up the random values from the various and different distributions the resulting distribution will be the normal distribution (the common and symmetric Bell Curve). This will happen even if none of the underlying distribution are Bell Curves.
2. If you average all of the averages of the individual distributions, the result will be the average of the above Bell Curve.
I've always been fascinated by this; it's bigger than humans. It’s as if nature or God (or both) seek symmetry no matter what the process may be; from manufacturing ball bearings, to various processes in biology, chemistry, astronomy, etc.
The only example that comes to my mind in the music world is a large choir. Let’s say you have a 300 member moderately or poorly talented choir and everyone tries to sing the same note. Since we have 300 different random distributions, some will sing it flat and some will sing it sharp and maybe a few or none will hit it spot on. Yet, the CLT will tell you that the ensemble average (what the listeners hear) will be close to the target note and at least be recognizable if not pleasing.
I understand that the above will happen at the individual note-level and at the song-level.
My question is, has anyone seen this principle play out in music?
https://soundcloud.com/user-646279677BiaB 2025 Windows For me there’s no better place in the band than to have one leg in the harmony world and the other in the percussive. Thank you Paul Tutmarc and Leo Fender.
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The only example that comes to my mind in the music world is a large choir. Let’s say you have a 300 member moderately or poorly talented choir and everyone tries to sing the same note. Since we have 300 different random distributions, some will sing it flat and some will sing it sharp and maybe a few or none will hit it spot on. Yet, the CLT will tell you that the ensemble average (what the listeners hear) will be close to the target note and at least be recognizable if not pleasing. In actual practice, the size of your choir will always be smaller than the number of people you need to average out the people off-pitch. Plus, a choir isn't static. Each choir member more or less believes that they are on pitch. The people who are on pitch will be trying to blend, so they will be relatively quiet. Meanwhile, the people who are woefully off pitch will sing even louder, because they're trying to show everyone else what the right note is. In contrast, when you've got a group of string instruments starting on a pitch, because they aren't fretted, they'll all likely be slightly off pitch. But because they're listening to each other, the instruments will quickly home in on the proper pitch, where "proper pitch" is defined as whatever the first chair is playing. Except for Ed, who lost his place several bars back, and is miming his part hoping that no one notices.
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Absolutely not. In fact, the opposite is normal.
Symphony orchestras play sharper as the concert progresses. The Haynes Flute Co. has documentation on this effect. When you order one of their $20K–$80,000 (and up) flutes, you specify how sharp you expect it to play and then adjust down at the beginning of the evening. Chamber ensembles are far less likely to do this as each player can hear all the others and the keyboard if present.
I’ve been directing choirs since 1969. Some are more in tune than others. If I had to generalize, the bigger they are, the flatter they sing. Hall acoustics have as much to do with this as anything else.
16-20 well trained choristers is about as large an ensemble as I have ever heard sing acapella with near-perfect intonation.
In my first orchestra holding a bass chair, I learned about ensemble intonation: If I hear the other basses but not myself, I’m in tune but the second I hear myself clearly, I’m not anymore. I discovered that to be true as a choral singer as well and train my singers accordingly.
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I'm no expert, but I understand that pianos are tuned slightly sharper than normal pitch at the higher octaves (with each note interval per octave representing the 12th root of two - the semitone interval), and this is deliberate, because the human ear hears 'flatter' at higher frequencies.
Certainly my Yamaha Concert Grand is tuned this way. In this case, it relates to the human ear perception of 'what is in tune',
I think I read somewhere that the 88th note on a piano is approximately a semitone sharper.
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No, but I know folks who would hear those sharp and flat notes within that 300 and be unhappy with the ensemble. 😀
Bud
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I think the big choirs are more likely populated with people who can actually sing in key and had to pass an audition to get on the choir... Choirs like the Mormon Tabernacle and a few other high profile ones come to mind. Your average church choir is looking for almost anyone who claims they can sing, to fill the chairs. The result is an averaging of pitches to get reasonably close to sounding halfway decent. Except for Sister Agatha. Agatha always sings sharp and louder than everyone else with a voice that is reminiscent of something halfway between a factory time clock whistle and a banshee from the pits of Hades.
In the past, I had actually joined the church choir. I was one of the few who had permission to do "special music" solo's without the requirement of being in the choir. I did acquiesce to the requests of the music minister to join. It was a nightmare for me. I can hear very small deviations in pitch and being in the midst of a group of people who were singing either sharper or flatter than the given key of the song, well, I would often find myself just listening rather than adding to the cacophony. After a couple of weeks of utter boredom and frustration, I had another discussion with the MM and told him it simply wasn't my thing and he could either use me where I could be helpful or opt to not use me.
In another time.....even earlier. I knew a couple of piano tuners through the years. While watching and observing the actual tuning process, I noticed that they would tune using the strobe tuner and then reference the note against another note lower on the piano. Then they would tweek that higher note based on the lower one rather than what the strobe tuner said. Asking about that, I was told that you trust the piano to tell you the truth rather than the strobe tuner. It had to do with the characteristics of a given piano. While it was generally true that certain notes on a piano were correctly to pitch, those notes were something to pay attention to and adjust on a per basis. Kinda like the same way many guitarists have tuning issues with the G string on their guitar. I know that on my guitars, I can tune the G string but then I need to adjust it slightly flat on my guitars for it to work well in chords. It's fascinating how that works.
You can find my music at: www.herbhartley.comAdd nothing that adds nothing to the music. You can make excuses or you can make progress but not both. The magic you are looking for is in the work you are avoiding.
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Herb,
James Taylor pointed out that very thing about guitars in a video I watched once. He said most guitars are notorious for this inherent flaw, that some of the strings have to be slightly out of tune at times to sound right.
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Agreed, I fight with that string quite often ..
Tune guitar via tuner
Play a few chords
Tweak G almost every single time by ear.
I do not work here, but the benefits are still awesome
Make your sound your own!
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I almost started thinking it was a Strat thing (all 3 did it) .. so glad it's not just me.
I do not work here, but the benefits are still awesome
Make your sound your own!
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Back in the band days we always tuned the G within a chord context. Never thought about the issue until now. Interesting y’all!
Bud
Our albums and singles are on Spotify, Apple Music, Amazon Music, YouTube Music, Pandora and more. If interested search on Janice Merritt. Thanks! Our Videos are here on our website.
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James Taylor pointed out that very thing about guitars in a video I watched once. He said most guitars are notorious for this inherent flaw, that some of the strings have to be slightly out of tune at times to sound right. I've seen that one too! I love the giant tuner he's got.  (Yes, I know it's a video screen)
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That's a cool JT video. Won't argue with HIM about guitar tuning...
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I'm no expert, but I understand that pianos are tuned slightly sharper than normal pitch at the higher octaves (with each note interval per octave representing the 12th root of two - the semitone interval), and this is deliberate, because the human ear hears 'flatter' at higher frequencies. This is called Stretch tuning or Octave Stretching. There's more of a stretch in the bass as one goes lower than in the treble. Some Peterson tuners have these curves built in and they also publish them for those who wish to adjust the temperment manually. The length of the soundboard is the key factor though pianists who play softer than average do not require as much stetch. Peterson 490-ST Manual with recommended stretch guides James Taylor pointed out that very thing about guitars in a video I watched once. He said most guitars are notorious for this inherent flaw, that some of the strings have to be slightly out of tune at times to sound right. JT is describing tempered tuning. This wasn't new when JS Bach wrote about it in 1722, either.
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