I want to thank ALL who contributed to this thread. Many comments are very helpful. I will now wind the clock twice weekly; that may keep the time setting. At the moment it is consistently off by 10 seconds fast. I'll take that.
10 seconds in 24 hours is not bad for a clock this, but this clock has no seconds hand, so how did you determine it was off by 10 seconds? If you are going by when it strikes, that can be a little off. The first task is to verify that the clock responds to rate adjustments. and you seem to have done that. There is always a little slop or backlash in the adjusting mechanism, so turning the key one "click" in the opposite direction from the last adjustment probably isn't actually changing the rate at all. Try to keep the drive gears under tension. For example, your clock is a bit fast, so move the key one click toward slow and wait for the clock to respond, then another click etc. but always going toward slow. If you overshoot the target and the clock is now slow, one click toward fast will have no effect because of the slack in the adjustment mechanism
. Then turn the key a full turn toward fast, then back off one click less than a full turn toward slow. The adjuster will now be moving toward slow again with no backlash.
It can help if you give the pendulum a little tug downward after each adjustment to make sure the suspension spring is seated in the upper support (an adjustment toward "fast" can sometimes push the suspension spring upward).
I suggest that you make a graph showing the minutes (or seconds) fast/slow over a full 7 days without making any adjustments to the rate. You are chasing ghosts trying to make instantaneous adjustments throughout the week with the mainspring at various percent wound. Start the week fully wound (minus about one key turn) and set the clock. The goal is to have it end the week as close as possible to the actual time of day. You will likely find that it gains a few seconds during the first few days, settles down mid-weel, and begins losing a few seconds near the end of the week as the spring runs down. Then take a look at your graph and determine how much too fast it was at the fastest point on the graph. Divide that number by 2 and next week, set it that much slower than the actual time of day. Your clock will now be a bit slow early in the week, a bit fast mid-week, and a bit slow at the end of the week, but the actual error from the correct time of day will be about half as much than if you set the fully wound clock to the exact time of day to begin the week.
The time section of this clock is basically a Seth Thomas # 89 movement with modifications to link to the chimes unit. +/- one minute per week is considered pretty good for this movement. Some may even say +/- 5 minutes over a week is acceptable.
Regarding the flat bell tone, as I understand it, a bell does not have a true fundamental pitch but rather produces a complex set of harmonics that sounds like a specific musical note. Perhaps your true pitch listener is listening for something that isn't there. Bottom line is does the set of bells have a pleasant sound together, or does it sound like something is "wrong". Attempting to tune the bells is opening a can of worms. I agree with others, OK to change the position that the hammer hits the bell, and make sure the hammer tips are all in good condition.