undefined

what i am trying to find out for myself is whether urs and others
statement that the CPS is comparable to a diamond is justified

u say that both have similar chord counts - sure, but linear
temperaments aren't far off either. a chord is a periodic waveform
and the faster it is the more consonant - so simply comparing the
amount of chords is not a very accurate way of suggesting they have a
similar amount of harmonicity

the diamond suggests a unique and simple way to separate all the tones
into linear scales, if u were to hold down all the keys on a keyboard
for an otonality for an octave or to infinity the period of the
waveform is still the same. a CPS scale has no way to arrange the
tones in linear sets, so in terms of usability which is both
playability and harmonicity, i cannot see how it is comparable.

in your opinion what is the 'best' way to tune? how well it fits into
a chromatic keyboard is not really an issue worth considering i feel,
the chromatic keyboard itself is an artifact of the misappropriated
use of a concept foreign to musical tonality (the logarithm) and i
seem to get the same feeling from CPS..

i think the fact that so much great music has been made is really
amazing considering how bad 12tet sounds. it is obvious that
arranging the tones from lowest to highest is about the worst way u
can do it, and the chords that sound okay are only because they are
similr to more consonant chords in just intonation. like with the
cps, there are many chords and intervals that are only considered
acceptable because of their closeness numerically to the tones in the
diamond.

>what i am trying to find out for myself is whether urs and others
>statement that the CPS is comparable to a diamond is justified
>
>u say that both have similar chord counts - sure, but linear
>temperaments aren't far off either.

Both the diamond and CPS are things you could tune in a linear
temperament, using fewer notes than in just intonation.

Temperaments usually beat JI in terms of chords per note. That
is the main point of them.

>a chord is a periodic waveform
>and the faster it is the more consonant

The faster what is?

>so simply comparing the
>amount of chords is not a very accurate way of suggesting they have a
>similar amount of harmonicity

The idea of "9-limit" is that, to a rough degree, any chord containing
factors no greater than 9 is about as consonant as any other.

>the diamond suggests a unique and simple way to separate all the tones
>into linear scales, if u were to hold down all the keys on a keyboard
>for an otonality for an octave or to infinity the period of the
>waveform is still the same.

That's one way to say it.

>a CPS scale has no way to arrange the tones in linear sets,

Yes it does, as I've been trying to show you.

>so in terms of usability which is both playability and harmonicity,
>i cannot see how it is comparable.

Diamonds and CPS are complimentary objects in the same tonespace.

>in your opinion what is the 'best' way to tune?

I guess that depends.

>how well it fits into a chromatic keyboard is not really an issue
>worth considering i feel,

It is if you want to use instruments that have chromatic keyboards.
Of course this is a small part of the inquiry that I and others
have taken up on this list.

>the chromatic keyboard itself is an artifact of the misappropriated
>use of a concept foreign to musical tonality (the logarithm) and i
>seem to get the same feeling from CPS..

Hm. The chromatic keyboard is certainly a historical artifact.
But they happen to be cheap and I happen to have lots of practice
playing them.

>i think the fact that so much great music has been made is really
>amazing considering how bad 12tet sounds.

The vast majority of music is not tuned terribly accurately in
12-tET.

>it is obvious that arranging the tones from lowest to highest
>is about the worst way u can do it,

That may be true.

>and the chords that sound okay are only because they are
>similr to more consonant chords in just intonation. like with the
>cps, there are many chords and intervals that are only considered
>acceptable because of their closeness numerically to the tones in the
>diamond.

Not a valid comparison, I'm afraid.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >what i am trying to find out for myself is whether urs and others
> >statement that the CPS is comparable to a diamond is justified
> >
> >u say that both have similar chord counts - sure, but linear
> >temperaments aren't far off either.
>
> Both the diamond and CPS are things you could tune in a linear
> temperament, using fewer notes than in just intonation.
>
> Temperaments usually beat JI in terms of chords per note. That
> is the main point of them.
>
> >a chord is a periodic waveform
> >and the faster it is the more consonant
>
> The faster what is?
>

the faster the waveform of the chord is.

> >so simply comparing the
> >amount of chords is not a very accurate way of suggesting they have a
> >similar amount of harmonicity
>
> The idea of "9-limit" is that, to a rough degree, any chord containing
> factors no greater than 9 is about as consonant as any other.
>

this is not true. try measuring the waveforms of different chords,
the more consonant ones are very fast, the less consonant ones are
slow. chord with strange ratios may sound like a faster chord that is
numerically close but infact they are much much slower.

> >the diamond suggests a unique and simple way to separate all the tones
> >into linear scales, if u were to hold down all the keys on a keyboard
> >for an otonality for an octave or to infinity the period of the
> >waveform is still the same.
>
> That's one way to say it.
>
> >a CPS scale has no way to arrange the tones in linear sets,
>
> Yes it does, as I've been trying to show you.
>

u have said there are modes to the CPS - what are they? how would u
arrange them?

> >it is obvious that arranging the tones from lowest to highest
> >is about the worst way u can do it,
>
> That may be true.
>
> >and the chords that sound okay are only because they are
> >similr to more consonant chords in just intonation. like with the
> >cps, there are many chords and intervals that are only considered
> >acceptable because of their closeness numerically to the tones in the
> >diamond.
>
> Not a valid comparison, I'm afraid.
>
> -Carl
>

what i am trying to say is having a modulatory advantage is
meaningless if the intervals cannot be arranged in linear scales. u
say this can be done with a cps, can u plz give me an example? thanks

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >what i am trying to find out for myself is whether urs and others
> >statement that the CPS is comparable to a diamond is justified
> >
> >u say that both have similar chord counts - sure, but linear
> >temperaments aren't far off either.
>
> Both the diamond and CPS are things you could tune in a linear
> temperament, using fewer notes than in just intonation.
>
> Temperaments usually beat JI in terms of chords per note. That
> is the main point of them.
>

yes thats what i am trying to say, that the number of chords per note
is not a good basis for comparison

> >a chord is a periodic waveform
> >and the faster it is the more consonant
>
> The faster what is?
>
> >so simply comparing the
> >amount of chords is not a very accurate way of suggesting they have a
> >similar amount of harmonicity
>
> The idea of "9-limit" is that, to a rough degree, any chord containing
> factors no greater than 9 is about as consonant as any other.
>

well i guess u could say that depending on how broad u mean by
'about'. eleven is a lot further from seven than seven is from five
god that sound sstupid
hah

>> >what i am trying to find out for myself is whether urs and others
>> >statement that the CPS is comparable to a diamond is justified
>> >
>> >u say that both have similar chord counts - sure, but linear
>> >temperaments aren't far off either.
>>
>> Both the diamond and CPS are things you could tune in a linear
>> temperament, using fewer notes than in just intonation.
>>
>> Temperaments usually beat JI in terms of chords per note. That
>> is the main point of them.
>>
>> >a chord is a periodic waveform
>> >and the faster it is the more consonant
>>
>> The faster what is?
>
>the faster the waveform of the chord is.

You mean the frequency of the waveform?

>> >so simply comparing the
>> >amount of chords is not a very accurate way of suggesting they have a
>> >similar amount of harmonicity
>>
>> The idea of "9-limit" is that, to a rough degree, any chord containing
>> factors no greater than 9 is about as consonant as any other.
>
>this is not true. try measuring the waveforms of different chords,
>the more consonant ones are very fast, the less consonant ones are
>slow. chord with strange ratios may sound like a faster chord that is
>numerically close but infact they are much much slower.

Can you give an example?

>what i am trying to say is having a modulatory advantage is
>meaningless if the intervals cannot be arranged in linear scales. u
>say this can be done with a cps, can u plz give me an example? thanks

Sure.

Find the thing that looks like an upside-down pentagram :) here

http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=2&m=f&o=0

If you look at the points labeled B-C, C-D, A-C, and C-E, you will
notice they form the outline of the bottom of a pentagon. They
represent the scale 1/1 9/8 3/2 7/4. You can see these tones in
here

!
2|5 [1 3 5 7 9] dekany.
10
!
9/8
7/6
5/4
21/16
35/24
3/2
5/3
7/4
15/8
2/1
!

If you want pentatonics, you can either go up to the 11-limit
with the eikosany, or "stellate" the above scale. I've already
given the (30-tone) stalled 2|5 dekany scale in a previous
post. You can see it contains 1/1 9/8 3/2 7/4 plus the missing
5/4.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >what i am trying to find out for myself is whether urs and others
> >> >statement that the CPS is comparable to a diamond is justified
> >> >
> >> >u say that both have similar chord counts - sure, but linear
> >> >temperaments aren't far off either.
> >>
> >> Both the diamond and CPS are things you could tune in a linear
> >> temperament, using fewer notes than in just intonation.
> >>
> >> Temperaments usually beat JI in terms of chords per note. That
> >> is the main point of them.
> >>
> >> >a chord is a periodic waveform
> >> >and the faster it is the more consonant
> >>
> >> The faster what is?
> >
> >the faster the waveform of the chord is.
>
> You mean the frequency of the waveform?
>
> >> >so simply comparing the
> >> >amount of chords is not a very accurate way of suggesting they
have a
> >> >similar amount of harmonicity
> >>
> >> The idea of "9-limit" is that, to a rough degree, any chord
containing
> >> factors no greater than 9 is about as consonant as any other.
> >
> >this is not true. try measuring the waveforms of different chords,
> >the more consonant ones are very fast, the less consonant ones are
> >slow. chord with strange ratios may sound like a faster chord that is
> >numerically close but infact they are much much slower.
>
> Can you give an example?
>
> >what i am trying to say is having a modulatory advantage is
> >meaningless if the intervals cannot be arranged in linear scales. u
> >say this can be done with a cps, can u plz give me an example? thanks
>
> Sure.
>
> Find the thing that looks like an upside-down pentagram :) here
>
>
http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=2&m=f&o=0
>
> If you look at the points labeled B-C, C-D, A-C, and C-E, you will
> notice they form the outline of the bottom of a pentagon. They
> represent the scale 1/1 9/8 3/2 7/4. You can see these tones in
> here
>
> !
> 2|5 [1 3 5 7 9] dekany.
> 10
> !
> 9/8
> 7/6
> 5/4
> 21/16
> 35/24
> 3/2
> 5/3
> 7/4
> 15/8
> 2/1
> !
>
> If you want pentatonics, you can either go up to the 11-limit
> with the eikosany, or "stellate" the above scale. I've already
> given the (30-tone) stalled 2|5 dekany scale in a previous
> post. You can see it contains 1/1 9/8 3/2 7/4 plus the missing
> 5/4.
>
> -Carl
>

what about the other tones in the scale? does this same method work
for all the cps scales? is there any other rational way to arrange
the other tones

>> Find the thing that looks like an upside-down pentagram :) here
>>
>> http://tinyurl.com/zvwbr
>>
>> If you look at the points labeled B-C, C-D, A-C, and C-E, you will
>> notice they form the outline of the bottom of a pentagon. They
>> represent the scale 1/1 9/8 3/2 7/4. You can see these tones in
>> here
>>
>> !
>> 2|5 [1 3 5 7 9] dekany.
>> 10
>> !
>> 9/8
>> 7/6
>> 5/4
>> 21/16
>> 35/24
>> 3/2
>> 5/3
>> 7/4
>> 15/8
>> 2/1
>> !
>>
>> If you want pentatonics, you can either go up to the 11-limit
>> with the eikosany, or "stellate" the above scale. I've already
>> given the (30-tone) stalled 2|5 dekany scale in a previous
>> post. You can see it contains 1/1 9/8 3/2 7/4 plus the missing
>> 5/4.
>
>what about the other tones in the scale? does this same method work
>for all the cps scales? is there any other rational way to arrange
>the other tones

Look at the diagram. Do you see any point that isn't part of a
pentagon like this?

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> Find the thing that looks like an upside-down pentagram :) here
> >>
> >> http://tinyurl.com/zvwbr
> >>
> >> If you look at the points labeled B-C, C-D, A-C, and C-E, you will
> >> notice they form the outline of the bottom of a pentagon. They
> >> represent the scale 1/1 9/8 3/2 7/4. You can see these tones in
> >> here
> >>
> >> !
> >> 2|5 [1 3 5 7 9] dekany.
> >> 10
> >> !
> >> 9/8
> >> 7/6
> >> 5/4
> >> 21/16
> >> 35/24
> >> 3/2
> >> 5/3
> >> 7/4
> >> 15/8
> >> 2/1
> >> !
> >>
> >> If you want pentatonics, you can either go up to the 11-limit
> >> with the eikosany, or "stellate" the above scale. I've already
> >> given the (30-tone) stalled 2|5 dekany scale in a previous
> >> post. You can see it contains 1/1 9/8 3/2 7/4 plus the missing
> >> 5/4.
> >
> >what about the other tones in the scale? does this same method work
> >for all the cps scales? is there any other rational way to arrange
> >the other tones
>
> Look at the diagram. Do you see any point that isn't part of a
> pentagon like this?
>
> -C.
>

i meant is there a generalised procedure to arrange the tones in a cps
into linear sequences, and do the other tones in the scale form linear
sequences - it seems like they dont.

--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >
> > >> >what i am trying to find out for myself is whether urs and others
> > >> >statement that the CPS is comparable to a diamond is justified
> > >> >
> > >> >u say that both have similar chord counts - sure, but linear
> > >> >temperaments aren't far off either.
> > >>
> > >> Both the diamond and CPS are things you could tune in a linear
> > >> temperament, using fewer notes than in just intonation.
> > >>
> > >> Temperaments usually beat JI in terms of chords per note. That
> > >> is the main point of them.
> > >>
> > >> >a chord is a periodic waveform
> > >> >and the faster it is the more consonant
> > >>
> > >> The faster what is?
> > >
> > >the faster the waveform of the chord is.
> >
> > You mean the frequency of the waveform?
> >
> > >> >so simply comparing the
> > >> >amount of chords is not a very accurate way of suggesting they
> have a
> > >> >similar amount of harmonicity
> > >>
> > >> The idea of "9-limit" is that, to a rough degree, any chord
> containing
> > >> factors no greater than 9 is about as consonant as any other.
> > >
> > >this is not true. try measuring the waveforms of different chords,
> > >the more consonant ones are very fast, the less consonant ones are
> > >slow. chord with strange ratios may sound like a faster chord
that is
> > >numerically close but infact they are much much slower.
> >
> > Can you give an example?
> >

dont know how to give an example but this is from wikipedia

"If two notes are simultaneously played, with frequency ratios that
are simple fractions (e.g. 2/1, 3/2 or 5/4), then the composite wave
will still be periodic with a short period, and the combination will
sound consonant. For instance, a note vibrating at 200 Hz and a note
vibrating at 300 Hz (a perfect fifth, or 3/2 ratio, above 200 Hz) will
add together to make a wave that repeats at 100 Hz: every 1/100 of a
second, the 300 Hz wave will repeat thrice and the 200 Hz wave will
repeat twice. Note that the total wave repeats at 100 Hz, but there is
not actually a 100 Hz sinusoidal component present."

what i am trying to say -
think abot a major third chord in 12tet

the frequencies woould be

1.0 1.259921.. 1.5

if u think about the acoustics, this chord does not actually have a
definite period, because the major third is irrational, simply no
matter how many times the wave repeats are they all adding up, every
cycle it sounds slightly different. yet even though we can hear a
dissonance we, accept this as 'consonant' because it sounds very
similar to the 'real' chord 1.0 1.25 1.5 which has a definite period
and the waveform is exactly the same each time

similarly, if u had instead used the ratio 500/401 for a major third,
it sounds consonant, but the consonance u r hearing is not the
repeating pattern that the 500/401 makes, instead u just hear it as
5/4 cause it is close - so the interval is not making the chord, the
chord is 'made' in spite of the interval.

the more factors are involved, and the higher the primes, the chord
becomes slower, and eventually too slow for our brain to see pattern
and recognise it as a definite pitch. (our brain can count waves very
slowly though, if u listen to binaural beats with a very marginal
difference - so that it takes maybe 15 seconds for the waves to add
up, if u close ur eyes and listen u can hear it.)

i posted a week ago or something the periods for the linear scales in
the 9lim.

>> > >try measuring the waveforms of different chords,
>> > >the more consonant ones are very fast, the less consonant ones are
>> > >slow. chord with strange ratios may sound like a faster chord
>> > >that is numerically close but infact they are much much slower.
>> >
>> > Can you give an example?
>> >
>
>dont know how to give an example but this is from wikipedia
>
>"If two notes are simultaneously played, with frequency ratios that
>are simple fractions (e.g. 2/1, 3/2 or 5/4), then the composite wave
>will still be periodic with a short period, and the combination will
>sound consonant. For instance, a note vibrating at 200 Hz and a note
>vibrating at 300 Hz (a perfect fifth, or 3/2 ratio, above 200 Hz) will
>add together to make a wave that repeats at 100 Hz: every 1/100 of a
>second, the 300 Hz wave will repeat thrice and the 200 Hz wave will
>repeat twice. Note that the total wave repeats at 100 Hz, but there is
>not actually a 100 Hz sinusoidal component present."
>
>what i am trying to say -
>think abot a major third chord in 12tet
>
>the frequencies woould be
>
>1.0 1.259921.. 1.5
>
>if u think about the acoustics, this chord does not actually have a
>definite period, because the major third is irrational, simply no
>matter how many times the wave repeats are they all adding up, every
>cycle it sounds slightly different. yet even though we can hear a
>dissonance we, accept this as 'consonant' because it sounds very
>similar to the 'real' chord 1.0 1.25 1.5 which has a definite period
>and the waveform is exactly the same each time

Yes, shorter periods tend to be associated with consonance. One
of my favorite measures of consonance for dyads is

numerator * denominator

for a ratio in lowest terms. This isn't an octave-equivalant
measure, however. When you assume octave-equivalence, odd limit
seems to be about as good as they come.

-C.

>>if u think about the acoustics, this chord does not actually have a
>>definite period, because the major third is irrational, simply no
>>matter how many times the wave repeats are they all adding up, every
>>cycle it sounds slightly different. yet even though we can hear a
>>dissonance we, accept this as 'consonant' because it sounds very
>>similar to the 'real' chord 1.0 1.25 1.5 which has a definite period
>>and the waveform is exactly the same each time
>
>Yes, shorter periods tend to be associated with consonance. One
>of my favorite measures of consonance for dyads is
>
>numerator * denominator
>
>for a ratio in lowest terms.

I forgot to say that this gives the length of the period of the
dyad.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >>if u think about the acoustics, this chord does not actually have a
> >>definite period, because the major third is irrational, simply no
> >>matter how many times the wave repeats are they all adding up, every
> >>cycle it sounds slightly different. yet even though we can hear a
> >>dissonance we, accept this as 'consonant' because it sounds very
> >>similar to the 'real' chord 1.0 1.25 1.5 which has a definite period
> >>and the waveform is exactly the same each time
> >
> >Yes, shorter periods tend to be associated with consonance. One
> >of my favorite measures of consonance for dyads is
> >
> >numerator * denominator
> >
> >for a ratio in lowest terms.
>
> I forgot to say that this gives the length of the period of the
> dyad.
>
> -Carl
>

what is a dyad?

At 06:09 PM 9/24/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>>
>> >>if u think about the acoustics, this chord does not actually have a
>> >>definite period, because the major third is irrational, simply no
>> >>matter how many times the wave repeats are they all adding up, every
>> >>cycle it sounds slightly different. yet even though we can hear a
>> >>dissonance we, accept this as 'consonant' because it sounds very
>> >>similar to the 'real' chord 1.0 1.25 1.5 which has a definite period
>> >>and the waveform is exactly the same each time
>> >
>> >Yes, shorter periods tend to be associated with consonance. One
>> >of my favorite measures of consonance for dyads is
>> >
>> >numerator * denominator
>> >
>> >for a ratio in lowest terms.
>>
>> I forgot to say that this gives the length of the period of the
>> dyad.
>>
>> -Carl
>>
>
>what is a dyad?

The interval between a pair of pitches. -C.