I'm glad someone other than me is asking this. I've been thinking hard about this for some time now.
i've never used a 5 string for any extended length of time (few hours max) but both tuning in 5ths and a five string are better than an extension in my opinion (which people are more than welcome to, and often do, disagree with).
the only advantages that I really see in a c-extension is that you don't need to buy a new bass. you can stay with the same bass you know and love and get a few extra notes without changing much. the notes speak clearly due to the length of the string and small mass in comparison to a low B/C string. Also, if you have a run of low Ds, you can "set it and forget it".
on the down side, I've never met a C-extension that's in tune (i've tried and listened to quite a few. like I said, I've been thinking about this a while. usually the low C must be Sharp for the E to be in tune). The old "typewriter" style system was noisy and heavy, and not particularly articulate, and while the new "capo" style is lighter, it is difficult to finger due to extended string length (most players use 1-4 for a half tone), which means that it is also slow.
5-string basses are a good option if you can find one that sounds good. This, unfortunately, is the rub. some people here (Paul Warburton, for example), found a 5er that sounds like warm, chocolatey goodness and will swear by them. others here have found nothing but tubby, heavy, thick-topped monsters with A strings that have no volume and speak slower than the rest of the strings. The moral of the story being that if you want a fiver, you have to search around and be willing to shell out a lot of hard cash to get a good one. Some people find the string spacing, etc hard to deal with because they've played 4 string basses for so long (the same problem comes up in 4 and 5 string electrics, surprisingly enough), others can get used to it.
So Pros? you can find a good one, and learn to play it well. Cons? you need to find one, and you can't use the old bass you had, and you need to learn to play it as well as you did your old one.
5ths? well, I'm a bit controversial in opinion here. i think that it not only makes a lot of sense, but also works really well in practice. I will admit that tuning in this way is a big pain for some things, but the basic turn of phrase around the people who play this way seems to be "if it was hard on 4ths bass, it's easier in 5ths. if it was ridiculously simple in 4ths, it's probably hard in 5ths. I feel like I can speak a bit here because I regularly do both. I use a 4ths bass at school to learn with my teacher, and i use a 5ths bass whenever i got to any kind of jam, whether folk, jazz, whatever. I also practice classical stuff at home in 5ths for my own amusement. There's also a learning curve. you need to relearn how to play the bass. Basic technique transfers, and some basic shifting concepts, but all your scales and arpeggios are different, and all your excerpts need to be relearned. added benefit? you've never heard your bass sound so good.
There are arguments that the low C string just can't put out the power of a c-extension string, and not only is that untrue in every incidence of my experience, but 5ths basses tend to be louder instead. They are also more articulate , though much of that aspect has to do with the fact that all the 5ths guys here cut the wings off their bridges. The other argument that isn't entirely true is that of shifting. This depends heavily what music you're playing. all those insane mozart runs are suddenly much easier because you're tuned like a cello, but some of those easier under the fingers, meaty Brahms 4ths bass parts become more difficult because you're not playing the exact instrument they were written for. I've been told Ein heldenlieben is way easier in 5ths, but I can't confirm. I have the privilege if seeing the National Arts Center Orchestra perform (cheap student tickets are 15 bucks. sweet deals). There are two 5ths players (principal and asst. principal), and the other three bassists are in 4ths. I would argue that, having seen them play the same part side by side, that neither does more shifting than the other, they just shift in different places (I will add the caveat of the 5ths "hand position" of a minor third here, although it is very rare that you ever hold that position, generally a very closed hand is used).
so, long winded, slightly inflammatory explanation over? it's up to you. got a bass you just love to pieces? c-extension unless you want to make a big commitment to 5ths. looking to upgrade your bass and think that extra string won't be a bother? go for a 5er and don't let anyone stop you.
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