58 good intervals

According to my Interval Calculator program the following intervals between 1/1 and 5/1 are all good. The number on the right is the frequency of the higher note if the frequency of the lower note is 12. I write intervals so that the numerator (left hand side) is less than the denominator (right hand side). This will annoy some, just flip them around in your mind.

1/1 12
8/9 13.5
7/8 13.7143
6/7 14
5/6 14.4
4/5 15
7/9 15.4286
3/4 16
8/11 16.5
5/7 16.8
7/10 17.1429
2/3 18
7/11 18.8571
5/8 19.2
3/5 20
7/12 20.5714
4/7 21
5/9 21.6
6/11 22
7/13 22.2857
1/2 24
6/13 26
5/11 26.4
4/9 27
7/16 27.4286
3/7 28
5/12 28.8
2/5 30
5/13 31.2
3/8 32
4/11 33
5/14 33.6
6/17 34
1/3 36
6/19 38
5/16 38.4
4/13 39
3/10 40
5/17 40.8
2/7 42
5/18 43.2
3/11 44
4/15 45
5/19 45.6
6/23 46
1/4 48
6/25 50
5/21 50.4
4/17 51
3/13 52
5/22 52.8
2/9 54
5/23 55.2
3/14 56
4/19 57
5/24 57.6
6/29 58
1/5 60

In my system if all the possible dyads/intervals in a chord (lowest and highest notes are not more than 5/1 apart) are within 6.775877 cents of any of the 58 intervals listed above then the chord is good. Can anyone identify an interval in this list that they think is dissonant?

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> According to my Interval Calculator program the following intervals between 1/1 and 5/1 are all good. The number on the right is the frequency of the higher note if the frequency of the lower note is 12. I write intervals so that the numerator (left hand side) is less than the denominator (right hand side). This will annoy some, just flip them around in your mind.

It would have annoyed your elementary school math teacher, who obviously didn't hit you with a ruler hard or often enough. For the record, you are not writing the numbers you claim to be writing, you are writing their reciprocals.

Sorry Gene,

I have been working in isolation from other microtonalists since 1995 and I did not discover until January of this year that the conventional way to write an interval has the numerator greater than or equal to the denominator. For fifteen years I used the opposite approach. When I posted my list of good intervals I was going to rewrite each interval but this was going to take some time so I just posted what I had.

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > According to my Interval Calculator program the following intervals between 1/1 and 5/1 are all good. The number on the right is the frequency of the higher note if the frequency of the lower note is 12. I write intervals so that the numerator (left hand side) is less than the denominator (right hand side). This will annoy some, just flip them around in your mind.
>
> It would have annoyed your elementary school math teacher, who obviously didn't hit you with a ruler hard or often enough. For the record, you are not writing the numbers you claim to be writing, you are writing their reciprocals.
>

genewardsmith wrote:
> > --- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>> According to my Interval Calculator program the following intervals
>> between 1/1 and 5/1 are all good. The number on the right is the
>> frequency of the higher note if the frequency of the lower note is
>> 12. I write intervals so that the numerator (left hand side) is
>> less than the denominator (right hand side). This will annoy some,
>> just flip them around in your mind.
> > It would have annoyed your elementary school math teacher, who
> obviously didn't hit you with a ruler hard or often enough. For the
> record, you are not writing the numbers you claim to be writing, you
> are writing their reciprocals.

Well, if you're talking about intervals used in harmony (dyads), it's the same interval; you're just looking at it from the other direction. It's only a convention that an octave is written as 2/1 (up from the bottom note) rather than 1/2 (down from the top note). But you can write intervals like these as proportions (1:2, 4:5, 5:8, etc.) in which case it doesn't matter what order you have them in.

Thanks Herman,

that's exactly what I thought,

John.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> genewardsmith wrote:
> >
> > --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >> According to my Interval Calculator program the following intervals
> >> between 1/1 and 5/1 are all good. The number on the right is the
> >> frequency of the higher note if the frequency of the lower note is
> >> 12. I write intervals so that the numerator (left hand side) is
> >> less than the denominator (right hand side). This will annoy some,
> >> just flip them around in your mind.
> >
> > It would have annoyed your elementary school math teacher, who
> > obviously didn't hit you with a ruler hard or often enough. For the
> > record, you are not writing the numbers you claim to be writing, you
> > are writing their reciprocals.
>
> Well, if you're talking about intervals used in harmony (dyads), it's
> the same interval; you're just looking at it from the other direction.
> It's only a convention that an octave is written as 2/1 (up from the
> bottom note) rather than 1/2 (down from the top note). But you can write
> intervals like these as proportions (1:2, 4:5, 5:8, etc.) in which case
> it doesn't matter what order you have them in.
>

John,

Out of the intervals you listed that (also) fit within the octave, written in larger/smaller number form...I think the following are somewhat dissonant:
11/8
10/7
13/7

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Sorry Gene,
>
> I have been working in isolation from other microtonalists since 1995 and I did not discover until January of this year that the conventional way to write an interval has the numerator greater than or equal to the denominator. For fifteen years I used the opposite approach. When I posted my list of good intervals I was going to rewrite each interval but this was going to take some time so I just posted what I had.

I don't object to you using intervals all less than 1. What I object to is doing so, and then claiming they are greater than 1. That's like spelling things backwards and then claiming it's correct spelling.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Well, if you're talking about intervals used in harmony (dyads), it's
> the same interval; you're just looking at it from the other direction.

If you try to use rational numbers for undirected intervals, it makes complete hash of the mathematics unless you decide at the outset that only rationals >= 1 (or <= 1, like John apparently uses) are going to be used to represent undirected intervals. Which leaves open the question, what do you propose to use for directed intervals? In any case, if you write p/q you ARE writing a rational number, and moreover, it's considered correct when writing ratios not to equate p:q and q:p, and to consider the ratios to be equal if and only if the corresponding rational numbers are equal. This isn't very helpful, since it identifies the two, but there it is. The real reason for the distinction is historical rather than mathematical, I'm afraid.

> It's only a convention that an octave is written as 2/1 (up from the
> bottom note) rather than 1/2 (down from the top note). But you can write
> intervals like these as proportions (1:2, 4:5, 5:8, etc.) in which case
> it doesn't matter what order you have them in.

If only it were true. But it isn't. That's not how the thing is supposed to work, and not how it ever did work, going back to Euclid.

At 10:58 PM 5/31/10 -0000, genewardsmith wrote:

>--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>>
>> According to my Interval Calculator program the following intervals
between 1/1 and 5/1 are all good. The number on the right is the frequency
of the higher note if the frequency of the lower note is 12. I write
intervals so that the numerator (left hand side) is less than the
denominator (right hand side). This will annoy some, just flip them around
in your mind.
>
>It would have annoyed your elementary school math teacher, who obviously
didn't hit you with a ruler hard or often enough. For the record, you are
not writing the numbers you claim to be writing, you are writing their
reciprocals.

Wouldn't that depend on whether one chooses to measure frequencies or
wavelengths? As they are related according to the speed of sound, which
number stays consistent from submarine to airplane?

-- Beco dos Gatinhos <http://www.rev.net/~aloe/music/tuning.html>

--- In tuning@yahoogroups.com, ALOE@... wrote:

> >It would have annoyed your elementary school math teacher, who obviously
> didn't hit you with a ruler hard or often enough. For the record, you are
> not writing the numbers you claim to be writing, you are writing their
> reciprocals.
>
> Wouldn't that depend on whether one chooses to measure frequencies or
> wavelengths?

No. Numbers are not defined in terms of wavelengths, wavelengths are defined in terms of numbers.