Short answer: within 5.9 Hz of your target frequency.
The long answer, though, is a little more involved…
According to Jake Mandell, at 500 Hz, a normal person can reliably differentiate two tones 6 Hz apart. For harpists, this little tidbit of information is potentially revolutionary. It means that …
if I tune my harp string incorrectly by 5.9 Hz, or less, the average person won’t be hear that it’s out of tune.
This may sound unimportant, but…
I spend around 13 hours a year tuning my harp!
It takes me 3 minutes, around 250 times a year. If I can shorten my time by one minute every time I tune my harp, and still achieve a sound acceptable to most humans, I’ll save 4 hours of my life every year… and I’d gladly watch a few films rather than tune my harp, thank you very much!! 🙂
Quick caveat: 500 Hz is around the B natural above middle C. Does the normal person’s sensitivity to pitch change according to the register of the two tones; e.g. if we pluck 2 of the high strings on the harp, or 2 the lowest strings? Probably. Also, perhaps a normal person’s sensitivity to pitch changes according to whether two notes are sounded simultaneously OR one after the other. I’ve emailed an expert asking these questions and am currently waiting on their answer. But while I’m waiting, for a fun experiment, let’s presume that the average person’s differentiation of two tones is >6Hz at all registers of the harp, and for tones plucked simultaneously and consecutively. So…
For the average person to hear something as ‘in tune’ we have to make sure that the interval between 2 imprecisely-tuned strings is less than 6 Hz.
That’s easy, right? I’ll just tune each string within 5.9 Hz of its correct pitch! Bingo!
But … we hear strings in relation to one another. If one string is 5.9 Hz sharp, and the next string played is 5.9 Hz flat, the interval between both pitches will be bigger than normal by 11.8 Hz, and therefore the average listener will hear the interval as ‘wrong’.
So my second idea is: let’s make sure that each string is tuned to within 2.95 Hz of its intended correct pitch. Now, if one string is 2.95 Hz sharp, and the one played after it is 2.95 Hz flat, the difference will be 5.9 Hz, which is less than 6 Hz, and therefore (in theory!) our listener will think it sounds beautiful, even though in theory it’s out of tune!!!
Next step: the unit of measurement we’ve been using so far is “Hertz”. However, the majority of harp tuners use a unit of measurement called ‘cents’. So we have to translate the 2.95 Hz into cents.
I did this in an Excel file, which I include below; the column marked ‘2.95 Hz in cents’ is the hypothetical margin of error for a harpist so that they sound in tune… even when, precisely speaking, they’re not!! I’m a bit sceptical, myself… according to these calculations, the lowest C on my harp can be 77 cents out of tune and the average human won’t notice. I have more faith in humanity – I think they’ll notice the harp sounds a bit dodgy. Help me out here… try tuning your harp with my crazy experiment and tell me how it goes!
Note: Hertz are logarithmic (they multiply from one tone to the next), cents are linear. So the margin of error for each string is different.
| LEVER HARP 8VE | Note | 2.95 Hz in cents |
|---|---|---|
| LEVER HARP 8VE 5 | C 2 | 76 |
| LEVER HARP 8VE 5 | C # 2 /D b 2 | 72 |
| LEVER HARP 8VE 5 | D 2 | 68 |
| LEVER HARP 8VE 5 | D # 2 /E b 2 | 64 |
| LEVER HARP 8VE 5 | E 2 | 60 |
| LEVER HARP 8VE 4 | F 2 | 57 |
| LEVER HARP 8VE 4 | F # 2 /G b 2 | 54 |
| LEVER HARP 8VE 4 | G 2 | 51 |
| LEVER HARP 8VE 4 | G # 2 /A b 2 | 48 |
| LEVER HARP 8VE 4 | A 2 | 45 |
| LEVER HARP 8VE 4 | A # 2 /B b 2 | 43 |
| LEVER HARP 8VE 4 | B 2 | 40 |
| LEVER HARP 8VE 4 | C 3 | 38 |
| LEVER HARP 8VE 4 | C # 3 /D b 3 | 36 |
| LEVER HARP 8VE 4 | D 3 | 34 |
| LEVER HARP 8VE 4 | D # 3 /E b 3 | 32 |
| LEVER HARP 8VE 4 | E 3 | 30 |
| LEVER HARP 8VE 3 | F 3 | 28 |
| LEVER HARP 8VE 3 | F # 3 /G b 3 | 27 |
| LEVER HARP 8VE 3 | G 3 | 25 |
| LEVER HARP 8VE 3 | G # 3 /A b 3 | 24 |
| LEVER HARP 8VE 3 | A 3 | 23 |
| LEVER HARP 8VE 3 | A # 3 /B b 3 | 21 |
| LEVER HARP 8VE 3 | B 3 | 20 |
| middle C | C 4 | 19 |
| LEVER HARP 8VE 3 | C # 4 /D b 4 | 18 |
| LEVER HARP 8VE 3 | D 4 | 17 |
| LEVER HARP 8VE 3 | D # 4 /E b 4 | 16 |
| LEVER HARP 8VE 3 | E 4 | 15 |
| LEVER HARP 8VE 2 | F 4 | 14 |
| LEVER HARP 8VE 2 | F # 4 /G b 4 | 13 |
| LEVER HARP 8VE 2 | G 4 | 13 |
| LEVER HARP 8VE 2 | G # 4 /A b 4 | 12 |
| LEVER HARP 8VE 2 | A 4 | 11 |
| LEVER HARP 8VE 2 | A # 4 /B b 4 | 11 |
| LEVER HARP 8VE 2 | B 4 | 10 |
| LEVER HARP 8VE 2 | C 5 | 9 |
| LEVER HARP 8VE 2 | C # 5 /D b 5 | 9 |
| LEVER HARP 8VE 2 | D 5 | 8 |
| LEVER HARP 8VE 2 | D # 5 /E b 5 | 8 |
| LEVER HARP 8VE 2 | E 5 | 8 |
| LEVER HARP 8VE 1 | F 5 | 7 |
| LEVER HARP 8VE 1 | F # 5 /G b 5 | 7 |
| LEVER HARP 8VE 1 | G 5 | 6 |
| LEVER HARP 8VE 1 | G # 5 /A b 5 | 6 |
| LEVER HARP 8VE 1 | A 5 | 6 |
| LEVER HARP 8VE 1 | A # 5 /B b 5 | 5 |
| LEVER HARP 8VE 1 | B 5 | 5 |
| LEVER HARP 8VE 1 | C 6 | 5 |
| LEVER HARP 8VE 1 | C # 6 /D b 6 | 4 |
| LEVER HARP 8VE 1 | D 6 | 4 |
| LEVER HARP 8VE 1 | D # 6 /E b 6 | 4 |
| LEVER HARP 8VE 1 | E 6 | 4 |
| LEVER HARP 8VE 0 | F 6 | 4 |
| LEVER HARP 8VE 0 | F # 6 /G b 6 | 3 |
| LEVER HARP 8VE 0 | G 6 | 3 |
| LEVER HARP 8VE 0 | G # 6 /A b 6 | 3 |
| LEVER HARP 8VE 0 | A 6 | 3 |
| LEVER HARP 8VE 0 | A # 6 /B b 6 | 2 |