7 ms·

There’s a bit more to this: a “perfect” fifth is not actually two notes differing by a factor of 2^(7/12), but rather by a factor of 1.5. It just so happens tha

There’s a bit more to this: a “perfect” fifth is not actually two notes differing by a factor of 2^(7/12), but rather by a factor of 1.5. It just so happens that 2^(7/12) is very close to 1.5, which makes it possible to tune a piano so that all fifths and all octaves sound approximately right. Theoretically, though, there will always be some error: any non-integer power of two is necessarily irrational. Thus, there is no perfect way to divide an octave into equally hz-spaced increments so that you hit the fifth along the way to doing the whole octave.

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VBprogrammer 5y ago
I believe it's also one of the reasons the G-string is hardest to tune by ear because in equal temperament it's actually the furthest away from the true note.
- hunter2_ 5y ago
  Yes, between most guitar strings is a perfect 4th interval (where equal temperament ratio [0] and the integer ratio it approximates [1] are quite close to each other) but between the G string and B string is a major 3rd interval (where equal temperament ratio [3] and the integer ratio it approximates [4] are a bit farther from each other). When using a tuner to tune a guitar, you get the equal temperament normally desired for it's ability to play equally in any key. When tuning by ear, if you are just minimizing the beating you hear (instead of targeting the beating that exists in equal temperament, like good piano technicians are trained to do) you'll end up with integer ratios. Accidentally ending up at the integer ratio of a major third is "farther off" than doing so for a perfect fourth. Typically this manifests as a difficult time tuning the B string though, not the G string, if you go low to high. Solution: tune low to high for the E,A,D,G and tune high to low for the high E,B (doing a perfect octave for the E to E where the equal temperament ratio [4] perfectly matches the integer ratio [5]). [0] 1.334840:1 or 2^(5/12):1 [1] 1.333333:1 or 4:3 [2] 1.259921:1 or 2^(4/12):1 [3] 1.25:1 or 5:4 [4] 2:1 or 2^(12/12):1 [5] 2:1
  - ghostpepper 5y ago
    Does this mean there's actually a scientific reason to tune by ear instead of with an electronic tuner? I feel like I naturally do some of these things, like tuning low to high but then (seemingly arbitrarily) tune the high e against the low E.
    - hunter2_ 5y ago
      Yes, integer ratios sound better than equal temperament ratios. The downside is that you end up with an instrument that has some integer ratios and some disgusting wolf intervals because it's mathematically impossible to have all integer ratios between every possible interval in every possible key you might decide to play in. That problem is what equal temperament solves for, and is a good reason to use an electronic tuner unless you know your songs will sound better without.
  - narag 5y ago
    When tuning by ear, if you are just minimizing the beating you hear... I assume you are talking about fretless instruments. With a guitar you just play a unison using the frets and the thicker string.
    - hunter2_ 5y ago
      No, my entire comment is about tuning a typical fretted guitar. Playing a unison as you say, and turning the peg until beating completely disappears, yields integer ratios instead of equal temperament ratios. The result of that can sound nicer than equal temperament for some keys/chords but terrible for other keys/chords.
      - narag 5y ago
        Unison means same sound. There's no ratio, unless you mean 1:1.
        hunter2_ 5y ago
- osigurdson 5y ago
  I've always found the B string hardest to tune by ear. Generally I need to fine tune it in the context of a few chords.
  - hunter2_ 5y ago
    Whether G is hard or B is hard depends on whether you're going high-to-low or low-to-high, as the issue is the interval between them, so it's just a matter of which one you're using as a known-good reference and which one you're trying to tune. There's an easier solution than taking multiple passes as you play different chords: see my other comment.
    - osigurdson 5y ago
      Thanks. I understand your approach but what method do you use to tune the high E? I assume you mean that the low E is used to tune the high E but perhaps you pick an E on the G string for this purpose. Is there any "best" E on the low side to use as a reference?
      - hunter2_ 5y ago
        You always need some point of reference to start with, or if you don't have an external reference, you can just go with anything at all. I would typically choose either of the E strings, just to be able to immediately match the other E string to it, and then go inward (up to G and down to B).
        osigurdson 5y ago
        I tried it and it works pretty well. One thing that I noticed is I subconsciously have been tuning GBE open because it’s a pain to reach around to the tuning pegs. How does this impact things?
        hunter2_ 4y ago
        When you play G and B open and tune them to eliminate beating, you're in pretty bad shape because you're making a "just" (integer ratio) major 3rd which is quite far from an equal temperament major 3rd. When you play B and E open and tune them to eliminate beating, you're in okay shape because you're making a "just" (integer ratio) perfect 4th which is not terribly far from an equal temperament perfect 4th. See my earlier comment [0] with all the numbers if you want to quantity "pretty bad" vs "okay". In that comment, my initial references to footnotes 3 and 4 should actually be 2 and 3, but it's too late to correct it. [0] https://news.ycombinator.com/item?id=31145233 https://news.ycombinator.com/item?id=31145233
- efortis 5y ago
  That's interesting. I start with "A" on the G-string, and then use its "B" to tune the B-string and so on.
ghostpepper 5y ago
This is super interesting and I feel like reading it helped a few concepts click for me, so thank you.
hunter2_ 5y ago
> all fifths and all octaves sound approximately right Octaves are actually 100% right regardless of equal temperament versus perfect/just/etc. The only real exception is when you get into octave stretching to compensate for string inharmonicity, but that's a whole separate matter.

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VBprogrammer 5y ago

I believe it's also one of the reasons the G-string is hardest to tune by ear because in equal temperament it's actually the furthest away from the true note.

hunter2_ 5y ago
Yes, between most guitar strings is a perfect 4th interval (where equal temperament ratio [0] and the integer ratio it approximates [1] are quite close to each other) but between the G string and B string is a major 3rd interval (where equal temperament ratio [3] and the integer ratio it approximates [4] are a bit farther from each other). When using a tuner to tune a guitar, you get the equal temperament normally desired for it's ability to play equally in any key. When tuning by ear, if you are just minimizing the beating you hear (instead of targeting the beating that exists in equal temperament, like good piano technicians are trained to do) you'll end up with integer ratios. Accidentally ending up at the integer ratio of a major third is "farther off" than doing so for a perfect fourth. Typically this manifests as a difficult time tuning the B string though, not the G string, if you go low to high. Solution: tune low to high for the E,A,D,G and tune high to low for the high E,B (doing a perfect octave for the E to E where the equal temperament ratio [4] perfectly matches the integer ratio [5]). [0] 1.334840:1 or 2^(5/12):1 [1] 1.333333:1 or 4:3 [2] 1.259921:1 or 2^(4/12):1 [3] 1.25:1 or 5:4 [4] 2:1 or 2^(12/12):1 [5] 2:1
- ghostpepper 5y ago
  Does this mean there's actually a scientific reason to tune by ear instead of with an electronic tuner? I feel like I naturally do some of these things, like tuning low to high but then (seemingly arbitrarily) tune the high e against the low E.
  - hunter2_ 5y ago
    Yes, integer ratios sound better than equal temperament ratios. The downside is that you end up with an instrument that has some integer ratios and some disgusting wolf intervals because it's mathematically impossible to have all integer ratios between every possible interval in every possible key you might decide to play in. That problem is what equal temperament solves for, and is a good reason to use an electronic tuner unless you know your songs will sound better without.
- narag 5y ago
  When tuning by ear, if you are just minimizing the beating you hear... I assume you are talking about fretless instruments. With a guitar you just play a unison using the frets and the thicker string.
  - hunter2_ 5y ago
    No, my entire comment is about tuning a typical fretted guitar. Playing a unison as you say, and turning the peg until beating completely disappears, yields integer ratios instead of equal temperament ratios. The result of that can sound nicer than equal temperament for some keys/chords but terrible for other keys/chords.
    - narag 5y ago
      Unison means same sound. There's no ratio, unless you mean 1:1.
      - hunter2_ 5y ago
osigurdson 5y ago
I've always found the B string hardest to tune by ear. Generally I need to fine tune it in the context of a few chords.
- hunter2_ 5y ago
  Whether G is hard or B is hard depends on whether you're going high-to-low or low-to-high, as the issue is the interval between them, so it's just a matter of which one you're using as a known-good reference and which one you're trying to tune. There's an easier solution than taking multiple passes as you play different chords: see my other comment.
  - osigurdson 5y ago
    Thanks. I understand your approach but what method do you use to tune the high E? I assume you mean that the low E is used to tune the high E but perhaps you pick an E on the G string for this purpose. Is there any "best" E on the low side to use as a reference?
    - hunter2_ 5y ago
      You always need some point of reference to start with, or if you don't have an external reference, you can just go with anything at all. I would typically choose either of the E strings, just to be able to immediately match the other E string to it, and then go inward (up to G and down to B).
      - osigurdson 5y ago
        I tried it and it works pretty well. One thing that I noticed is I subconsciously have been tuning GBE open because it’s a pain to reach around to the tuning pegs. How does this impact things?
        hunter2_ 4y ago
        When you play G and B open and tune them to eliminate beating, you're in pretty bad shape because you're making a "just" (integer ratio) major 3rd which is quite far from an equal temperament major 3rd. When you play B and E open and tune them to eliminate beating, you're in okay shape because you're making a "just" (integer ratio) perfect 4th which is not terribly far from an equal temperament perfect 4th. See my earlier comment [0] with all the numbers if you want to quantity "pretty bad" vs "okay". In that comment, my initial references to footnotes 3 and 4 should actually be 2 and 3, but it's too late to correct it. [0] https://news.ycombinator.com/item?id=31145233 https://news.ycombinator.com/item?id=31145233
efortis 5y ago
That's interesting. I start with "A" on the G-string, and then use its "B" to tune the B-string and so on.

ghostpepper 5y ago

This is super interesting and I feel like reading it helped a few concepts click for me, so thank you.

hunter2_ 5y ago

> all fifths and all octaves sound approximately right Octaves are actually 100% right regardless of equal temperament versus perfect/just/etc. The only real exception is when you get into octave stretching to compensate for string inharmonicity, but that's a whole separate matter.