Help with even-order harmonics please

Hi Tuning members

Can members please help me define "even order harmonics"? I've been
studying the harmonic series and I'm confused by the definition
of "even order harmonics" and "odd order harmonics". I've seen
information on the web
http://www.ecmweb.com/ar/electric_fundamentals_harmonics/

which says, "The even multiples of the fundamental frequency are
known as even-order harmonics while the odd multiples are known as
the odd-order harmonics." -end-

Ok, therefore, (if I assume the fundamental frequency is _1_)
any "even" multiple of the fundamental(1), such as harmonics
2,4,6,8,10,12,14,16,18,20,22 are better known as even-order harmonics
and any odd multiple of the fundamental such as
3,5,7,9,11,13,15,17,19,21 are better known as odd-order harmonics?

I understand 2,4,6,8,10 etc. are all even numbers "but" what about
10? 10 is an even number but it is also an octave above harmonic 5
which is an odd number. Therefore isn't 10 basically an odd harmonic?
This also goes for any odd harmonic. If an odd harmonic is doubled,
it becomes an even number.

I've been all over the web and can't seem to find a straight answer
on this.

Thanks so much

Walter Lepore New Jersey

electricwally77 wrote:

> I understand 2,4,6,8,10 etc. are all even numbers "but" what about > 10? 10 is an even number but it is also an octave above harmonic 5 > which is an odd number. Therefore isn't 10 basically an odd harmonic?
> This also goes for any odd harmonic. If an odd harmonic is doubled, > it becomes an even number.

Yes, so 5 is on odd harmonic and 10 is an even harmonic. The only ambiguity is if you have a timbre composed entirely of even harmonics, in which case you numbered them wrong. An even harmonic will always be some number of octaves above either an odd harmonic or the fundamental.

Graham

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> electricwally77 wrote:
>
>> I understand 2,4,6,8,10 etc. are all even numbers "but" what
>>about 10? 10 is an even number but it is also an octave above
>>harmonic 5 which is an odd number. Therefore isn't 10 basically an
>>odd harmonic? This also goes for any odd harmonic. If an odd
>>harmonic is doubled, it becomes an even number.

>Yes, so 5 is on odd harmonic and 10 is an even harmonic. The only
>ambiguity is if you have a timbre composed entirely of even
>harmonics, in which case you numbered them wrong. An even harmonic
>will always be some number of octaves above either an odd harmonic
>or the fundamental.
>
>Graham

Hi Graham

you said...
>The only ambiguity is if you have a timbre composed entirely of even
>harmonics, in which case you numbered them wrong.

Could you expand on this. I'm still not getting it.

you said...
>An even harmonic will always be some number of octaves above either
>an odd harmonic or the fundamental.

Ok let's see if I got this right please. Starting with the
fundamental whose frequency is "1", I have the following series of
harmonics (keeping within a range of between 1-30) such as....
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,2
7,28,29,30

If I list the even-order harmonics which are defined as the number of
octaves above the fundamental they are: 2,4,8,and 16. Is this
correct? Eventhough "6" is an even number it is not an octave or
octaves above the fundamental "1"?
And if I list the the odd-order harmonics above the fundamental which
are defined as "none octave harmonics" they are:
3,5,6,7,9,10,11,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,30.
Is this correct?

Thanks
Walter

Hi Graham

you said...
>The only ambiguity is if you have a timbre composed entirely of even
>harmonics, in which case you numbered them wrong.

Could you expand on this. I'm still not getting it.

you said...
>An even harmonic will always be some number of octaves above either
>an odd harmonic or the fundamental.

Ok let's see if I got this right please. Starting with the
fundamental whose frequency is "1", I have the following series of
harmonics (keeping within a range of between 1-30) such as....
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,2
7,28,29,30

If I list the even-order harmonics which are defined as the number of
octaves above the fundamental they are: 2,4,8,and 16. Is this
correct?

Eventhough "6" is an even number it is not an octave or
octaves above the fundamental "1"? Is this correct please?

And if I list the odd-order harmonics above the fundamental which
are defined as "none octave harmonics" they are:
3,5,6,7,9,10,11,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,30.
The reason there are even numbers listed in the odd-order harmonics
(6,10,14,18 etc.)is because those even numbers are not octave(s) of
the fundamental. Is this correct?

Thanks
Walter

Hi Members

Maybe this will help understand my question better.

I heard that even-order harmonics are more pleasing to the ear than
odd-order harmonics. I'm learning to build "tube" guitar amplifiers
and depending on the type of construction I could construct the tube
and circuit so when a note on the guitar is struck, the harmonics of
the fundamental note coming out of the amplifier would produce even-
order harmonics or odd-order harmonics.

This is why I needed to understand more precisely what even-order and
odd order harmonics actually are in relation to the fundamental tone.

Thank you
Walter

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
>
> Hi Members
>
> Maybe this will help understand my question better.
>
> I heard that even-order harmonics are more pleasing to the ear than
> odd-order harmonics. I'm learning to build "tube" guitar amplifiers
> and depending on the type of construction I could construct the
tube
> and circuit so when a note on the guitar is struck, the harmonics
of
> the fundamental note coming out of the amplifier would produce even-
> order harmonics or odd-order harmonics.
>
> This is why I needed to understand more precisely what even-order
and
> odd order harmonics actually are in relation to the fundamental
tone.
>
>
> Thank you
> Walter

it's very simple -- even-order harmonics = even-numbered harmonics;
odd-order harmonics = odd-numbered harmonics.

i think the reason even-order harmonics are considered pleasing is
that primitive guitar amp distortion clips the waveform
symmetrically, an operation which can only produce odd-order
harmonics. the sound is considerably more natural-sounding, and hence
more pleasing, if even-order harmonics are produced as well,
requiring an asymmetric response. at least that's my rudimentary
understanding of the issue.

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
> Hi Graham
>
> you said...
> >The only ambiguity is if you have a timbre composed entirely of
even
> >harmonics, in which case you numbered them wrong.
>
> Could you expand on this. I'm still not getting it.

because #2 would really be #1, #4 would be #2, #6 would be #3,
etc. . . . the note is simply an octave higher than you originally
thought it was, and has all harmonics.

> you said...
> >An even harmonic will always be some number of octaves above
either
> >an odd harmonic or the fundamental.
>
> Ok let's see if I got this right please. Starting with the
> fundamental whose frequency is "1", I have the following series of
> harmonics (keeping within a range of between 1-30) such as....
>
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,2
> 7,28,29,30
>
> If I list the even-order harmonics which are defined as the number
of
> octaves above the fundamental they are: 2,4,8,and 16. Is this
> correct?
>
> Eventhough "6" is an even number it is not an octave or
> octaves above the fundamental "1"? Is this correct please?

yup.

> And if I list the odd-order harmonics above the fundamental which
> are defined as "none octave harmonics" they are:
> 3,5,6,7,9,10,11,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,30.
> The reason there are even numbers listed in the odd-order harmonics
> (6,10,14,18 etc.)is because those even numbers are not octave(s) of
> the fundamental. Is this correct?

i don't understand why you list 3 and 6 but not 12, for example . . .

> it's very simple -- even-order harmonics = even-numbered harmonics;
> odd-order harmonics = odd-numbered harmonics.

Hi Wallyesterpaulrus

Can you please see my post number 42999 titled "amended" and tell me
where I went wrong?

Thanks
Walter

Hi Wallyesterpaulrus

Our messages crossed. Please ignore my amended request. You already
replied.

I said....
> > And if I list the odd-order harmonics above the fundamental which
> > are defined as "none octave harmonics" they are:
> >
3,5,6,7,9,10,11,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,30.
> > The reason there are even numbers listed in the odd-order
harmonics
> > (6,10,14,18 etc.)is because those even numbers are not octave(s)
of
> > the fundamental. Is this correct?

> i don't understand why you list 3 and 6 but not 12, for
example . . .

3 and 6 are not octave multiples of the fundamental "1".Therefore
they are odd-order harmonics. Correct?

Yes you are right, I forgot to list 12. 12 is not an octave multiple
of the fundamental "1".

Walter

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:

> > i don't understand why you list 3 and 6 but not 12, for
> example . . .
>
> 3 and 6 are not octave multiples of the fundamental "1".Therefore
> they are odd-order harmonics. Correct?

3 is odd, 6 is even.

> 3 is odd, 6 is even.

Now I'm confused.
6 is an even number. This I agree with but in context with the
fundamental "1" it is an odd-order harmonic because by definition, an
even-order harmonic is that which is an octave or octaves above the
fundamental "1". In this case the only "even" harmonics within the
harmonic range between 1 and 30 are 2,4,8,16,. Correct?

Thanks
Walter

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
>
> > 3 is odd, 6 is even.
>
> Now I'm confused.
> 6 is an even number. This I agree with but in context with the
> fundamental "1" it is an odd-order harmonic because by definition,
an
> even-order harmonic is that which is an octave or octaves above the
> fundamental "1". In this case the only "even" harmonics within the
> harmonic range between 1 and 30 are 2,4,8,16,. Correct?
>
> Thanks
> Walter

you have the wrong definition. even-order harmonics are even-numbered
harmonics. they are an octave or octaves above odd-numbered
harmonics, not necessarily the fundamental.

> you have the wrong definition. even-order harmonics are even-
numbered
> harmonics. they are an octave or octaves above odd-numbered
> harmonics, not necessarily the fundamental.

Is the following a valid question?

What are the even-order harmonics (not going higher than the 30th
harmonic) above the fundamental tone which has a frequency of "1".

Answer: 2,4,8,16 only

Walter

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
>
> > you have the wrong definition. even-order harmonics are even-
> numbered
> > harmonics. they are an octave or octaves above odd-numbered
> > harmonics, not necessarily the fundamental.
>
> Is the following a valid question?
>
> What are the even-order harmonics (not going higher than the 30th
> harmonic) above the fundamental tone which has a frequency of "1".

yes, that's a valid question.

> Answer: 2,4,8,16 only

incorrect.

> Walter

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
>
> > you have the wrong definition. even-order harmonics are even-
> numbered
> > harmonics. they are an octave or octaves above odd-numbered
> > harmonics, not necessarily the fundamental.
>
> Is the following a valid question?
>
> What are the even-order harmonics (not going higher than the 30th
> harmonic) above the fundamental tone which has a frequency of "1".
>
> Answer: 2,4,8,16 only
>
> Walter

if you insert the words "an octave or several octaves" before the
word "above" above, then everything's correct.

Walter,

Perhaps you are confusing the even-order harmonics with the octave multiples harmonics. The two are different things.
If your tube amplifier produces even order harmonics, all *multiples* of two are entitled to be produced (amplified?), not only all *powers* of two. Paul's definition:
even-order harmonics = even-numbered harmonics;
odd-order harmonics = odd-numbered harmonics
is enough.
In other words, your tube amplifier, for electrical reasons, is only interested in the harmonics having 2 as a prime factor, independently from the other factors; that's all.

Regards
Leonardo

--- In tuning@yahoogroups.com, Leonardo Perretti <dombedos@t...>
wrote:
> Walter,
>
> Perhaps you are confusing the even-order harmonics with the octave
> multiples harmonics. The two are different things.
> If your tube amplifier produces even order harmonics, all
*multiples*
> of two are entitled to be produced (amplified?),

it's _produced_ -- tube amplifiers are prized for the sound of their
nonlinear distortion, which produces harmonics and combinational
tones based on all the pure frequency components fed into it. a
single sine wave (a pure fundamental, no harmonics) fed into a lousy
tube ampifier will come out with odd-order harmonics added, while a
single sine wave fed into a musically pleasing tube amplifier will
come out with both odd- and even-order harmonics added.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> it's _produced_ -- tube amplifiers are prized for the sound of their
> nonlinear distortion, which produces harmonics and combinational
> tones based on all the pure frequency components fed into it.

I've heard some non-tube amps have as an option the introduction of
this sort of distortion, to make for a warmer sound.

Hi Leonardo

You said...

>If your tube amplifier produces even order harmonics, all
>*multiples* of two are entitled to be produced (amplified?), not
>only all *powers* of two.

So what you are saying is that even order harmonics actual include
BOTH multiples of 2 such as 2,4,6,8,10,12,14,16 etc. as well as
powers of 2 such as
2 to the 1st power = 2
2 to the 2nd power = 4
2 to the 3rd power = 8
2 to the 4th power = 16

Thanks
Walter

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
> Hi Leonardo
>
> You said...
>
> >If your tube amplifier produces even order harmonics, all
> >*multiples* of two are entitled to be produced (amplified?), not
> >only all *powers* of two.
>
> So what you are saying is that even order harmonics actual include
> BOTH multiples of 2 such as 2,4,6,8,10,12,14,16 etc. as well as
> powers of 2 such as
> 2 to the 1st power = 2
> 2 to the 2nd power = 4
> 2 to the 3rd power = 8
> 2 to the 4th power = 16
>
> Thanks
> Walter

there is no need to separately consider 2, 4, 8, 16 . . . since they
are already in your first list above!

Hi, Walter

electricwally77 wrote:

>Hi Leonardo
>
>You said...
>
>>If your tube amplifier produces even order harmonics, all
>>*multiples* of two are entitled to be produced (amplified?), not
>>only all *powers* of two.
>
>So what you are saying is that even order harmonics actual include
>BOTH multiples of 2 such as 2,4,6,8,10,12,14,16 etc. as well as
>powers of 2 such as
>2 to the 1st power = 2
>2 to the 2nd power = 4
>2 to the 3rd power = 8
>2 to the 4th power = 16
>

Yes.
Powers of two are a special case where all prime factors of the number are =2. As almost one of the factors is 2, they are even numbers as well.

Regards
Leonardo

I figured out what I was doing wrong. I was using the 5th harmonic as
the reference point when I should have been using the fundamental (1)
as my reference point when identifying even harmonics and octave
equivalence.

The 5th harmonic from fundamental "1" (1,2,3,4,5)is an odd number and
I became confused and thought the 10th harmonic which is an even
number was actually an octave above the 5th harmonic. Therefore I
could not help thinking that the 10th harmonic was indeed an odd
harmonic because it was related to the 5th harmonic.

But I was wrong. I was using the 5th harmonic as the point of
reference when in fact the point of reference inmy question should
have been the fundamental which is "1".

The harmonics above the fundamental ("1") namely 5 and 10 are
actually different frequencies since what we are talking about is
essentially the harmonic series where the only notes that repeat
themselves above the fundamental (1) are the octaves. Harmonics 5 and
10 are not octaves of the fundamental (1) therefore they are
different frequencies. They are what they are. 5 is 5 Hz and 10 is 10
Hz. They are different sounds! The 5th and 10th harmonics are not
powers of 2 when using the fundamental (1) as a reference point.
i.e. 2,4 8, 16, 32 etc. therefore they are not octaves of each other.

If I was using the frequncy 5 as my fundamental then 10 would be an
octave above and the series would go as follows: 5, 10, 20,40,80 etc.

I hope I explained this correctly.

Regards
Walter

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:

> The harmonics above the fundamental ("1") namely 5 and 10 are
> actually different frequencies since what we are talking about is
> essentially the harmonic series where the only notes that repeat
> themselves above the fundamental (1) are the octaves. Harmonics 5
and
> 10 are not octaves of the fundamental (1) therefore they are
> different frequencies. They are what they are. 5 is 5 Hz and 10 is
10
> Hz. They are different sounds! The 5th and 10th harmonics are not
> powers of 2 when using the fundamental (1) as a reference point.
> i.e. 2,4 8, 16, 32 etc. therefore they are not octaves of each
other.

they are indeed octaves of each other -- 10 is an octave higher than
5.

> they are indeed octaves of each other -- 10 is an octave higher
than 5.

Only if 5 is my fundamental. Correct?

If I am using "1" as my fundamental i.e 1,2,3,4,5,6,7,8,9,10,11,12
etc. then 5 and 10 are not octaves of 1 or of each other. Correct?

Walter,

perhaps you are still wrong:

>The harmonics above the fundamental ("1") namely 5 and 10 are
>actually different frequencies since what we are talking about is
>essentially the harmonic series where the only notes that repeat
>themselves above the fundamental (1) are the octaves.

if we speak of even-order harmonics, then we are not speaking of "the harmonic series where the only notes that repeat themselves above the fundamental (1) are the *octaves*"; at least, not only of them.

Leonardo

hello Walter,

i'm not sure if you have cleared up your confusion
about this question yet, because i myself became
very confused trying to follow the discussion.

(it didn't help that the order in which i received
the messages in my mailbox, and the time stamped
on them, isn't the same as the order in which they
were posted!)

anyway, i'm just posting a little something here to
try and help.

it's not clear to me that you understand what we
mean by the word "octave". it comes from the Latin
word for "8th", since in the ordinary diatonic scale
(do re mi fa so la ti do) notes which are 8 steps
apart sound mysteriously "the same".

i say "mysteriously" because it's obvious that these
are two different notes, but there's something about
their sound that causes us generally to identify them
as being "the same". thus, they end up getting the
same letter-name, since the letters were associated
with pitches with reference to the above-mentioned
diatonic scale. thus, we have A B C D E F G for the
7 different notes of the scales, then the 8th note,
an "octave" above the first, gets the same letter, A.

well, there's a mathematical reason why that note sounds
"the same" as the starting note. it's because the
frequency ratio of those two notes is or approximates 2:1.

as has been pointed out in this thread, all even-numbered
harmonics are multiples of lower odd-numbered harmonics,
one or more octaves higher. thus, 6 is an octave above 3,
10 is an octave above 5, etc.

so the business about "powers of 2" simply concerns "octaves".

thus:

1 is the fundamental, 2 is an octave above it, 4 is
2 octaves above it, 8 is 3 octaves above it, and 16 is
4 octaves above it:

2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16

3 is the first harmonic which produces a note having a
different letter-name, 6 is the octave above that,
12 is 2 octaves above it, and 24 is 3 octaves above it:

3 * 2^0 = 3
3 * 2^1 = 6
3 * 2^2 = 12
3 * 2^3 = 24

the next "new note" comes from the 5th harmonic, and
10 is an octave above it, and 20 is 2 octaves above it:

5 * 2^0 = 5
5 * 2^1 = 10
5 * 2^2 = 20

the next "new note" comes from the 7th harmonic,
and 14 is an octave above it, and 28 is 2 octaves
above it:

7 * 2^0 = 7
7 * 2^1 = 14
7 * 2^2 = 28

the next "new note" comes from the 9th harmonic,
and 18 is the octave above it.

9 * 2^0 = 9
9 * 2^1 = 18

... and so on.

you will see from this that because 2 is by definition
the ratio of the octave, and 2 is also by definition
the factor which causes numbers to be classified as
"even" rather than as "odd", then ignoring octaves
means that you're concentrating only on odd-numbered
harmonics. tuning theorists refer to this as "odd-limit".

i deliberately included the trivial cases in the
calculations above, to show how all the harmonic numbers
could be notated as x * 2^y, where x is some odd-number
and y is a power of 2. if you're ignoring powers of 2,
the odd-number component is all that matters.

in fact, it's a generally-held tenet among tuning
theorists that odd-limit is important because each
odd-number ratio (or an approximation of it) is a
separate "identity" in a chord or other harmonic structure.

thus (assuming "octave equivalence") :

1 = "root"
3 = "perfect 5th"
5 = "major 3rd"
7 = "minor 7th"
9 = "major 9th"
11 = "(harmonic) 11th"
13 = "(harmonic) 13th"
15 = "major 7th"
17 = "minor 9th"
19 = "augmented 9th" or "flat 10th"

etc.

i included the "(harmonic)" for the "11th" and "13th"
because they are so far off from the "11th" and "13th"
that we get in the regular 12-tone equal-tempered scale
(nearly midway between two 12edo notes) that they really
don't resemble either of the two 12edo approximations,
but rather have a very unique sound all their own. the
same is somewhat true of 7 representing the "minor 7th".

i can't speak on the mechanics of guitar amplifiers,
but i hope this helps you understand a little better.

i personally place more stock on the concept of
"prime-limit", with the idea that the lowest prime-numbers
each have some type of "affect" associated with them,
but i don't know how much this concerns your particular
question. i only say something about it because it
*might* be useful to your research.

-monz

> From: "electricwally77" <earth7@optonline.net>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, March 26, 2003 11:57 AM
> Subject: [tuning] Re: Help with even-order harmonics please-amended-
>
>
> Hi Leonardo
>
> You said...
>
> >If your tube amplifier produces even order harmonics, all
> >*multiples* of two are entitled to be produced (amplified?), not
> >only all *powers* of two.
>
> So what you are saying is that even order harmonics actual include
> BOTH multiples of 2 such as 2,4,6,8,10,12,14,16 etc. as well as
> powers of 2 such as
> 2 to the 1st power = 2
> 2 to the 2nd power = 4
> 2 to the 3rd power = 8
> 2 to the 4th power = 16
>
> Thanks
> Walter

----- Original Message -----
From: "electricwally77" <earth7@optonline.net>

> > they are indeed octaves of each other -- 10 is an octave higher
> than 5.
>
> Only if 5 is my fundamental. Correct?

No.

* David Beardsley
* microtonal guitar
* http://biink.com/db

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
> > they are indeed octaves of each other -- 10 is an octave higher
> than 5.
>
> Only if 5 is my fundamental. Correct?

it doesn't matter what the fundamental is, or if there even is a
fundamental at all.

> If I am using "1" as my fundamental i.e 1,2,3,4,5,6,7,8,9,10,11,12
> etc. then 5 and 10 are not octaves of 1 or of each other. Correct?

10 is always an octave above 5. the frequency ratio 2:1 is always an
octave.

----- Original Message -----
From: "David Beardsley" <davidbeardsley@biink.com>
> ----- Original Message -----
> From: "electricwally77" <earth7@optonline.net>
>
> > > they are indeed octaves of each other -- 10 is an octave higher
> > than 5.
> >
> > Only if 5 is my fundamental. Correct?
>
> No.

If 5 was your fundamental, it wouldn't be 5/4, it would be 1/1.

As for guitar amp harmonics, I have some nice tube amps
and I've never heard anything higher than 5/4. Anything else
is so faint that I'd have to blast the amp.

* David Beardsley
* microtonal guitar
* http://biink.com/db

--- In tuning@yahoogroups.com, David Beardsley <davidbeardsley@b...>
wrote:

> As for guitar amp harmonics, I have some nice tube amps
> and I've never heard anything higher than 5/4.

you mean 5/1?

> Anything else
> is so faint that I'd have to blast the amp.

what experimental setup are you referring to? are you feeding a sine
wave into the amp? if you're talking about just playing a note on the
guitar, the 5th harmonic is present already, and the amp need not
contribute any distortion for it to be present. either way, it'll be
hard to hear, since the ear interprets an entire harmonic series as a
single pitch with a particular timbre. on the other hand, i've often
been able to make out the 17th or 19th harmonic on a bass guitar note
with *no* distortion.

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

> --- In tuning@yahoogroups.com, David Beardsley <davidbeardsley@b...>
> wrote:
>
> > As for guitar amp harmonics, I have some nice tube amps
> > and I've never heard anything higher than 5/4.
>
> you mean 5/1?

I guess so. What I remember hearing is a very nice 4:5:6 chord
coming from one note. Nice and clear, very obvious.

> > Anything else
> > is so faint that I'd have to blast the amp.
>
> what experimental setup are you referring to? are you feeding a sine
> wave into the amp? if you're talking about just playing a note on the
> guitar, the 5th harmonic is present already, and the amp need not
> contribute any distortion for it to be present. either way, it'll be
> hard to hear, since the ear interprets an entire harmonic series as a
> single pitch with a particular timbre. on the other hand, i've often
> been able to make out the 17th or 19th harmonic on a bass guitar note
> with *no* distortion.

Looping a few low E notes on my G&L 63 tone JI (Not a drone)
into a THD Univalve. I think I was using a Valve Arts KT66
as a power tube and Electro Harmonix 12AX7 for pre-amp and driver
tubes. Or maybe a 6550 as a power tube? That's what I perform with.
Nothing experimental about this, it's cutting edge guitar technology,
you know: that new fangled vacuum tube stuff.

Did I answer your question?

* David Beardsley
* microtonal guitar
* http://biink.com/db

> > If I am using "1" as my fundamental i.e ,2,3,4,5,6,7,8,9,10,11,12
> > etc. then 5 and 10 are not octaves of 1 or of each other. Correct?

> 10 is always an octave above 5. the frequency ratio 2:1 is always
an
> octave.

I stand corrected. You are right. 10 is a double of 5 thus 10 is an
octave above 5.

However when using 1 (the fundamental) as the reference point, 5 and
10 are not octaves of 1. Correct(I hope)?

Walter

> If 5 was your fundamental, it wouldn't be 5/4, it would be 1/1.
> * David Beardsley

David, where did you get 5/4 ? Please expand on this

Walter

--- In tuning@yahoogroups.com, David Beardsley <davidbeardsley@b...>
wrote:
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> > --- In tuning@yahoogroups.com, David Beardsley
<davidbeardsley@b...>
> > wrote:
> >
> > > As for guitar amp harmonics, I have some nice tube amps
> > > and I've never heard anything higher than 5/4.
> >
> > you mean 5/1?
>
> I guess so. What I remember hearing is a very nice 4:5:6 chord
> coming from one note. Nice and clear, very obvious.

yup, just one of the many wonderful effects you can get with an
electric guitar and a nice amp, turned up. the higher harmonics are
exceedingly difficult to hear out as individual pitches, especially
as they get close to one another, but contribute to the timbre -- the
more of the higher harmonics there are, the more "buzzy" the sound.
and we've all heard a real buzzy guitar distortion sound; whether
jimi or kurt, tubes are the likely culprit. (other methods have
included slicing the speaker cone a la dave davies . . .)

> > > Anything else
> > > is so faint that I'd have to blast the amp.
> >
> > what experimental setup are you referring to? are you feeding a
sine
> > wave into the amp? if you're talking about just playing a note on
the
> > guitar, the 5th harmonic is present already, and the amp need not
> > contribute any distortion for it to be present. either way, it'll
be
> > hard to hear, since the ear interprets an entire harmonic series
as a
> > single pitch with a particular timbre. on the other hand, i've
often
> > been able to make out the 17th or 19th harmonic on a bass guitar
note
> > with *no* distortion.
>
> Looping a few low E notes on my G&L 63 tone JI (Not a drone)
> into a THD Univalve. I think I was using a Valve Arts KT66
> as a power tube and Electro Harmonix 12AX7 for pre-amp and driver
> tubes. Or maybe a 6550 as a power tube? That's what I perform with.
> Nothing experimental about this, it's cutting edge guitar
technology,
> you know: that new fangled vacuum tube stuff.
>
> Did I answer your question?

i meant "experimental" in the scientific sense of determining the
odd-order and even-order distortion products; not in the musical
sense of experimental music, but you probably knew that already.

anyway, i play gigs with one of my two tube amps nearly every night .
. . love 'em.

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
> > > If I am using "1" as my fundamental i.e
,2,3,4,5,6,7,8,9,10,11,12
> > > etc. then 5 and 10 are not octaves of 1 or of each other.
Correct?
>
>
> > 10 is always an octave above 5. the frequency ratio 2:1 is always
> an
> > octave.
>
> I stand corrected. You are right. 10 is a double of 5 thus 10 is an
> octave above 5.
>
> However when using 1 (the fundamental) as the reference point, 5
and
> 10 are not octaves of 1. Correct(I hope)?
>
> Walter

correct! they are octaves of a note a major third above the 1. 386
cents, to be exact.

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
>
> > If 5 was your fundamental, it wouldn't be 5/4, it would be 1/1.
> > * David Beardsley
>
> David, where did you get 5/4 ? Please expand on this
>
> Walter

sorry to interject here,

1/1 -- probably comes from partch. if we're talking partch here, then
an overtone series above the 5th harmonic of 1/1 would be known as a
"5/4 Otonality" (so it is possible, in partch's system, to use a
fundamental other than 1/1). 5/4 is the pitch you get when you
transpose the 5th harmonic it down two octaves to be within the
octave between 1 (1/1) and 2 (2/1) (again, it's 386 cents higher than
1). the pitches of this 5/4 Otonality include 15/8, 25/16, 35/32,
45/32, 55/32 . . . since these denominators are powers of two, they
just represent transposition by some number of octaves, so you can
ignore them and see that this is just 5, 15, 25, 35, 45, 55 . . . all
are harmonics above a fundamental of 5.

Gene Ward Smith wrote:

> I've heard some non-tube amps have as an option the introduction of
> this sort of distortion, to make for a warmer sound.

Anything that implements guitar distortion should favor the even harmonics as discussed. I built my own distortion patch in Kyma, so I can share my experience here.

The even-order harmonics are favored because when a tube amplifier gets into the nonlinear region it treats the positive and negative parts of the signal differently. If you start with a sine wave, symmetric clipping will give you only odd harmonics. That doesn't sound so bad, but it's only one sound and probably not the one you want. Any kind of asymmetry will add even harmonics, which leads to a warmer sound. One explanation for this is that the octave above the fundamental is stronger.

With real signals, they won't look much like a sine wave, so the above explanation doesn't really hold. But it's still a good rule of thumb. Most guitar notes will end up like square waves if you clip them enough. And apparently this is still true with a tube amplifier -- the differences are only for moderate distortion. Transistors are different, see below.

The way I emulated this is:

1) split the signal into positive and negative components

2) apply different gains using an ideal clipping amplifier

3) put them back together again

4) somwhere in the mix you could apply a high pass filter to remove the asymmetry of the signal. I don't always do this.

You can build electronics to do the same kind of thing. There are circuit diagrams on the web, I don't have URLs to hand. You need diodes to separate out the positive and negative parts. This is where you have to think about transistors -- they introduce their own artefacts which are harsher than those of either tubes or ideal operational amplifiers. Apparently these involve inharmonic tones, which would be a particular problem with just intonation. So for best results use op-amps rather than straight transistors.

While I'm at it, tube amplifiers also give smoother clipping than op-amps. That is, the corner of the signal is curved rather than sharp. To emulate this, apply an arctan waveshaper (which will also amplify the signal a bit) before the gain, if that's relevant to whatever system you're using. These days, I plot my waveshaper curve algorithmically so that I can take into account both the smoothing and the asymmetry.

I don't have any tube amplifiers to compare to, so I don't know how close I'm getting. But it does sound good. One advantage is that all parameters are controllable. So you can have the asymmetric gain without any clipping, which is interesting.

Graham

----- Original Message -----
From: "electricwally77" <earth7@optonline.net>
To: <tuning@yahoogroups.com>
Sent: Wednesday, March 26, 2003 10:06 PM
Subject: [tuning] Re: Help with even-order harmonics please-amended-

>
> > If 5 was your fundamental, it wouldn't be 5/4, it would be 1/1.
> > * David Beardsley
>
> David, where did you get 5/4 ? Please expand on this

5/4 is the ratio that represents the 5th harmonic. It's the
distance between the 4th harmonic (an octave of 1/1)
and the 5th harmonic.

* David Beardsley
* microtonal guitar
* http://biink.com/db

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, March 27, 2003 1:56 AM
Subject: [tuning] Re: Help with even-order harmonics please-amended-

> --- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
> >
> > > If 5 was your fundamental, it wouldn't be 5/4, it would be 1/1.
> > > * David Beardsley
> >
> > David, where did you get 5/4 ? Please expand on this
> >
> > Walter
>
> sorry to interject here,
>
> 1/1 -- probably comes from partch. if we're talking partch here,

I thought we were discussing the harmonic series.

* David Beardsley
* microtonal guitar
* http://biink.com/db

Hi David, Graham, Leonardo, Wallyester,Monz, & Gene

Thanks for all your input and help thus far on my question
about "Even Order Harmonics". I'm amazed every day at the power of
the internet and how it brings people with the same interests
together. I've been struggling to learn tuning theory for quite some
time now between the everyday events of a job and raising a family.

I'm amazed at all the detailed and well presented responses.

I'm also amazed at the number of members who are well versed in the
field of tube amplifiers. I wasn't sure if I should bring up the
topic of tube amplifiers which is the reason I was asking about even
order harmonics. I guess all this stuff ties in closely with each
other.

I've noticed the following field of studies are all closely related:
Tuning, Music Composition, Tube Amp building, Instrument Building
Musicology, Music Theory , Music History, Mathematics, Physics and
many other areas.

David, (or other members) can you please expand on your reply to
message 43043 which basically is…

>>I stand corrected. You are right. 10 is a double of 5 thus 10 is an
>> octave above 5. However when using 1 (the fundamental) as the
>>reference point, 5 and 10 are not octaves of 1. Correct(I hope)?
>>
>> Walter

>correct! they are octaves of a note a major third above the 1. 386
>cents, to be exact.

Fellow Tuning Members,
I'm not understanding "a major third above 1". How can 5 and 10 be
octaves of 386? I'm confused because (I think) we are now mixing
cents (386) with frequencies (5Hz and 10Hz). Please explain as basic
as possible. Thanks all.

Walter

----- Original Message -----
From: "electricwally77" <earth7@optonline.net>

>Fellow Tuning Members,
>I'm not understanding "a major third above 1". How can 5 and 10 be
>octaves of 386? I'm confused because (I think) we are now mixing
>cents (386) with frequencies (5Hz and 10Hz). Please explain as basic
>as possible. Thanks all.

Let's look at the harmonic series on a guitar string.

The open string is the fundamental. 1/1

The harmonic over the 12th fret is 2/1, the 1st octave.

The harmonic over the 7th fret is 3/2, a new harmonic that is a Just perfect
5th.

The harmonic over the 5th fret is 4/2 or 4/1, another octave of 1/1.

You'll find the 5th harmonic a bit flat of the 4th fret. 5/4 is a Just major
3rd.

This gives us a Just major chord = 1/1, 5/4 3/2.

The fundamental, 1/1 with it's octaves, 2/1 & 4/2 are the same note.

And so on....follow?

Time for a walk and a hair cut. Later.

* David Beardsley
* microtonal guitar
* http://biink.com/db

Hi Members,

Near the bottom in message # 43028 I said .....

>The 5th and 10th harmonics are not
> powers of 2 when using the fundamental (1) as a reference point.
> i.e. 2,4 8, 16, 32 etc. therefore they are not octaves of each
other.

I used the wrong choice of words here which caused me to be wrong. My
last sentence above should have read:

Therefore harmonics 5 and 10 are not octaves of the fundamental (1).
However they are octaves of each other.

Thanks David for pointing that out to me.

Regards
Walter

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...> wrote:
> I figured out what I was doing wrong. I was using the 5th harmonic
as
> the reference point when I should have been using the fundamental
(1)
> as my reference point when identifying even harmonics and octave
> equivalence.
>
> The 5th harmonic from fundamental "1" (1,2,3,4,5)is an odd number
and
> I became confused and thought the 10th harmonic which is an even
> number was actually an octave above the 5th harmonic. Therefore I
> could not help thinking that the 10th harmonic was indeed an odd
> harmonic because it was related to the 5th harmonic.
>
> But I was wrong. I was using the 5th harmonic as the point of
> reference when in fact the point of reference inmy question should
> have been the fundamental which is "1".
>
> The harmonics above the fundamental ("1") namely 5 and 10 are
> actually different frequencies since what we are talking about is
> essentially the harmonic series where the only notes that repeat
> themselves above the fundamental (1) are the octaves. Harmonics 5
and
> 10 are not octaves of the fundamental (1) therefore they are
> different frequencies. They are what they are. 5 is 5 Hz and 10 is
10
> Hz. They are different sounds! The 5th and 10th harmonics are not
> powers of 2 when using the fundamental (1) as a reference point.
> i.e. 2,4 8, 16, 32 etc. therefore they are not octaves of each
other.
>
> If I was using the frequncy 5 as my fundamental then 10 would be an
> octave above and the series would go as follows: 5, 10, 20,40,80
etc.
>
> I hope I explained this correctly.
>
> Regards
> Walter

Hi Leonardo

Yes you were correct, I was in fact confusing even-order harmonics
with the octave multiples harmonics. The two are indeed different
things.

I was confusing "multiples of two" with "powers of two".

I can't believe I fell into that trap!

Thanks
Walter

--- In tuning@yahoogroups.com, Leonardo Perretti <dombedos@t...>
wrote:
> Walter,
>
> Perhaps you are confusing the even-order harmonics with the octave
> multiples harmonics. The two are different things.
> If your tube amplifier produces even order harmonics, all
*multiples*
> of two are entitled to be produced (amplified?), not only all
> *powers* of two. Paul's definition:
> even-order harmonics = even-numbered harmonics;
> odd-order harmonics = odd-numbered harmonics
> is enough.
> In other words, your tube amplifier, for electrical reasons, is
only
> interested in the harmonics having 2 as a prime factor,
independently
> from the other factors; that's all.
>
> Regards
> Leonardo

Wallyesterpaulrus said.....

> you have the wrong definition. even-order harmonics are even-
numbered
> harmonics. they are an octave or octaves above odd-numbered
> harmonics, not necessarily the fundamental.

Got it!

Thanks
Walter

Hi David

> If 5 was your fundamental, it wouldn't be 5/4, it would be 1/1.

Your right! That was my problem to begin with.

Besides what Leonardo pointed out earlier in which I was
confusing "power of two " with "multiples of two".

And as Wallyesterpaulrus pointed out, even-order harmonics are even-
numbered harmonics. they are an octave or octaves above odd-numbered
harmonics, not necessarily the fundamental.

I was using two reference points instead of one to identify the
relationship between even an odd harmonics(refering to the 5th
harmonic as 1/1 when in fact it should have been referred to as 5/1
or 5/4).

Thank you all for the education.

Regards
Walter

--- In tuning@yahoogroups.com, David Beardsley <davidbeardsley@b...>
wrote:
> ----- Original Message -----
> From: "David Beardsley" <davidbeardsley@b...>
> > ----- Original Message -----
> > From: "electricwally77" <earth7@o...>
> >
> > > > they are indeed octaves of each other -- 10 is an octave
higher
> > > than 5.
> > >
> > > Only if 5 is my fundamental. Correct?
> >
> > No.
>
> If 5 was your fundamental, it wouldn't be 5/4, it would be 1/1.
>
> As for guitar amp harmonics, I have some nice tube amps
> and I've never heard anything higher than 5/4. Anything else
> is so faint that I'd have to blast the amp.
>
> * David Beardsley
> * microtonal guitar
> * http://biink.com/db

Hi David

You said......

> Let's look at the harmonic series on a guitar string.
> The open string is the fundamental. 1/1
> The harmonic over the 12th fret is 2/1, the 1st octave.
> The harmonic over the 7th fret is 3/2, a new harmonic that is a
>Just perfect 5th.
> The harmonic over the 5th fret is 4/2 or 4/1, another octave of 1/1.
> You'll find the 5th harmonic a bit flat of the 4th fret. 5/4 is a
>Just major 3rd.
> This gives us a Just major chord = 1/1, 5/4 3/2.
> The fundamental, 1/1 with it's octaves, 2/1 & 4/2 are the same note.
> And so on....follow?
> Time for a walk and a hair cut. Later.

Yes! Got it! You explained it well.

Thanks
Walter

--- In tuning@yahoogroups.com, David Beardsley <davidbeardsley@b...>
wrote:
> ----- Original Message -----
> From: "electricwally77" <earth7@o...>
>
> >Fellow Tuning Members,
> >I'm not understanding "a major third above 1". How can 5 and 10 be
> >octaves of 386? I'm confused because (I think) we are now mixing
> >cents (386) with frequencies (5Hz and 10Hz). Please explain as
basic
> >as possible. Thanks all.
>
> Let's look at the harmonic series on a guitar string.
>
> The open string is the fundamental. 1/1
>
> The harmonic over the 12th fret is 2/1, the 1st octave.
>
> The harmonic over the 7th fret is 3/2, a new harmonic that is a
Just perfect
> 5th.
>
> The harmonic over the 5th fret is 4/2 or 4/1, another octave of 1/1.
>
> You'll find the 5th harmonic a bit flat of the 4th fret. 5/4 is a
Just major
> 3rd.
>
> This gives us a Just major chord = 1/1, 5/4 3/2.
>
> The fundamental, 1/1 with it's octaves, 2/1 & 4/2 are the same note.
>
> And so on....follow?
>
> Time for a walk and a hair cut. Later.
>
>
> * David Beardsley
> * microtonal guitar
> * http://biink.com/db

--- In tuning@yahoogroups.com, "electricwally77" <earth7@o...>
wrote:

>
> David, (or other members) can you please expand on your
reply to
> message 43043 which basically is…
>
> >>I stand corrected. You are right. 10 is a double of 5 thus 10
is an
> >> octave above 5. However when using 1 (the fundamental)
as the
> >>reference point, 5 and 10 are not octaves of 1. Correct(I
hope)?
> >>
> >> Walter
>
> >correct! they are octaves of a note a major third above the 1.
386
> >cents, to be exact.
>
> Fellow Tuning Members,
> I'm not understanding "a major third above 1". How can 5 and
10 be
> octaves of 386? I'm confused because (I think) we are now
mixing
> cents (386) with frequencies (5Hz and 10Hz). Please explain
as basic
> as possible. Thanks all.
>
> Walter

an octave is 1200 cents. so if the fundamental is 0 cents, the 4th
harmonic is 2400 cents, and the fifth harmonic is 2786 cents. the
tenth harmonic is 1200 cents higher, at 3986 cents above the
fundamental.

let me know if i can help clarify anything else.