# Is there a word for a group of numbers which have equal mantissa in some base?

In music, the letter A denotes multiple pitches: not only 440Hz, but also 220Hz, 110Hz, 55Hz, etc. and 880Hz, 1760Hz, 3520Hz, etc. (these are approximations). The common feature of these is they are all powers of two times each other - i.e.:

I would like to say:

The note A in music is an example of a(n) _____, since it represents all numbers of the form  for  and .

As in the title, mantissa means roughly what I want (though its meaning varies from context to context). Is there any word or common phrase that describes this situation?

• @Chenmunka I was deliberating on whether to post here or there; I settled with here since a word-request was more on-topic here. Sep 19, 2017 at 8:59

The frequencies of note A in music are elements in the geometric progression.

This is imperfect though, not only because it doesn't fit well with your example sentence but also because a geometric progression has a first term.

In your equation n can be any member of Z, that is any integer, negative, zero or positive. In a geometric progression n starts at zero and goes up either to a final term or to infinity. It does not allow for negative n.

If you are thinking particularly of music then a frequency of 55/64 is too low to really be a note anyway, so you don't need negative indices, but this is presumably just an example you are using.

We cannot just say multiple because in the example a multiple of 55/64 would be 55/64 times any number 1, 2, 3, 4, 5, 6, 7, 8 etc; not just 1, 2, 4, 8 etc.

Another term for a geometric progression is a geometric sequence; but NOT a geometric series, which is something else.

I hope somebody comes up with a better suggestion.

• Thanks. Do you know whether 'infinite geometric sequence' would be mathematically correct? Sep 19, 2017 at 23:01
• My understanding is that an infinite geometric sequence is only infinite in a positive direction. So n in your equation could go from 0, 1, 2 etc up to infinity. A finite geometric sequence goes from 0 to some specified positive integer. I do not know of a name for a sequence that has n from minus infinity to plus infinity. Sorry. (I suspect there must be such a name, It seems an obvious mathematical extension - which is what mathematicians do. If you don't get a better answer here I suggest mathematicians SE will find someone who knows - unless @EdwinAshworth can help.. Sorry. Sep 19, 2017 at 23:12
• What about 'infinite bidirectional geometric sequence' (sorry, fishing)? Sep 19, 2017 at 23:13
• Well, it could be , I just don't know. I've never heard of such a expression. Sep 19, 2017 at 23:14

I would probably explain that by saying they are all multiples of each other. Multiple is a more widely known mathematical term, and its a consequence of the observation you make that for any two examples of such cases, one will be a multiple of the other.

• But they are not multiples. They are powers. The multiples of 2 are (2,4,6,8,10,12,14,16, ...) but the powers of 2 are (2, 4, 8, 16, 32, ...). Sep 19, 2017 at 15:57
• @Flater And that makes them multiples of each other. To take your example, all of the second sequence are multiples of each other; e.g. 16 is a multiple of 4. The first sequence are not all multiples of each other, 14 is not a multiple of 10. Sep 19, 2017 at 16:06
• It is true that they are all multiples of each other, but a group of numbers like 3, 6, 12, 36, 144, 2880 are still multiples/factors of each other. Sep 19, 2017 at 23:01
• Yes, and they are 3·2ⁿ, n∈ℤ. Sep 20, 2017 at 9:03
• @JonHanna 144 is 3 x 48 and 48 cannot be expressed as a power of 2. Sep 20, 2017 at 10:26