Yahoo Tuning Groups Ultimate Backup tuning Harmonics and Their Equal Tempered Frequencies

Hi Members

I was investigating the harmonic series recently because I wanted to
learn exactly what harmonic we stopped at to form our western 12 tone
equal tempered scale. And where each note of our 12tEQ tempered scale
is related to what harmonic. In other words where (in the harmonic
series ) did our 12tEQ tempered intervals originate from?
I have a book that explains all the harmonics that are the closest to
equal-tempered intervals and collapsed to one octave. Starting from
the bottom with the fundamental and moving upwards to the octave they
are as follows:

Natural Intervals(cents)-------Tempered Interval(cents)

1200 Octave ------------------ 1200 Octave
15th Harmonic 1088 ----------- 1100 Major 7th
7th Harmonic 969 -------------1000 minor 7th
27 Harmonic 906 -------------900 Major 6th
13 Harmonic 841 -------------800 minor 6th
3rd Harmonic 702 -------------700 Perfect 5th
11 Harmonic 551--------------600 Tritone
21st Harmonic 471 -------------500 Perfect 4th
5th Harmonic 386 -------------400 Major 3rd
19 Harmonic 298 -------------300 minor 3rd
9th Harmonic 204 -------------200 Major 2nd
17 Harmonic 105 -------------100 minor 2nd
0 Fundamental -----------------0 Fundamental

Should I assume we stopped at the 27 harmonic to form what is todays
12tEQ scale?

Thank you
Walter Lepore

🔗Carl Lumma <ekin@lumma.org>

Hi Walter,

>Should I assume we stopped at the 27 harmonic to form what is todays
>12tEQ scale?

Finding the origin of anything involves historical inquiry, which
is always speculative. That said, I think there is a strong
consensus here that the harmonic series had no direct part in the
development of equal temperament in Western music. Rather, 12-tET
was selected because it contains the diatonic scale and allows
unlimited modulation on the Halberstadt (7-white, 5-black) keyboard
that was already entrenched in the musical culture. Had 19-tone
keyboards been the norm, we suppose 19-tET would have become the
standard tuning.

The harmonic and subharmonic series are the basis of harmony,
however. The diatonic scale and 12-tET each support many different
styles of harmony. In Jazz, I think the argument is strong that we
are seeing "7-limit" (Partch's term) approximations. But Common-
Practice music is usually considered 5-limit.

I highly recommend you get a copy of Partch's book, _Genesis
of a Music_.

-Carl

🔗Carl Lumma <ekin@lumma.org>

I wrote...

>>Natural Intervals(cents)-------Tempered Interval(cents)
>>
>>1200 Octave ------------------ 1200 Octave
>>15th Harmonic 1088 ----------- 1100 Major 7th
>>7th Harmonic 969 -------------1000 minor 7th
>>27 Harmonic 906 -------------900 Major 6th
>>13 Harmonic 841 -------------800 minor 6th
>>3rd Harmonic 702 -------------700 Perfect 5th
>>11 Harmonic 551--------------600 Tritone
>>21st Harmonic 471 -------------500 Perfect 4th
>>5th Harmonic 386 -------------400 Major 3rd
>>19 Harmonic 298 -------------300 minor 3rd
>>9th Harmonic 204 -------------200 Major 2nd
>>17 Harmonic 105 -------------100 minor 2nd
>>0 Fundamental -----------------0 Fundamental
>
>The diatonic scale and 12-tET each support many different
>styles of harmony.

And indeed, studying how close degrees of 12-tET come to
the harmonic series is a good way to start understanding
these styles!

-Carl

🔗monz <monz@attglobal.net>

----- Original Message -----
From: "Carl Lumma" <ekin@lumma.org>
To: <tuning@yahoogroups.com>
Sent: Saturday, March 29, 2003 12:47 PM
Subject: Re: [tuning] Harmonics and Their Equal Tempered Frequencies

> Hi Walter,
>
> >Should I assume we stopped at the 27 harmonic to form what is todays
> >12tEQ scale?
>
> Finding the origin of anything involves historical inquiry, which
> is always speculative. That said, I think there is a strong
> consensus here that the harmonic series had no direct part in the
> development of equal temperament in Western music. Rather, 12-tET
> was selected because it contains the diatonic scale and allows
> unlimited modulation on the Halberstadt (7-white, 5-black) keyboard
> that was already entrenched in the musical culture. Had 19-tone
> keyboards been the norm, we suppose 19-tET would have become the
> standard tuning.

Carl is right. i suggest that you read some of my
Tuning Dictionary entries, which concern the historical
background behind the answers to your question.

i think these might help:

http://sonic-arts.org//dict/12-eq.htm

http://sonic-arts.org//dict/pythag.htm

http://sonic-arts.org/dict/schismatic-tuning.htm

http://sonic-arts.org//dict/just.htm

http://sonic-arts.org//dict/duodene.htm

http://sonic-arts.org//dict/meantone.htm

in my opinion, the facts that 12edo:

1) resembles Pythagorean tuning
2) acts as a schismatic temperament
3) acts as a meantone

... all conspired, chronologically in the order listed,
in favor of its acceptance. reason #3 was probably
the one that clinched it, historically, because meantone
tuning became firmly established in Western music
c. 1450-1800.

after that, 12-tone "well-temperaments" were most
favored for keyboards, because by then the 7-white/5-black
keyboard pattern of keys had become firmly entrenched.
from there it was only a small step to equalize the
error-from-JI and make it 12edo.

it should also be noted that 12edo was favored as
the tuning for fretted instruments (cheifly the lute)
from c. 1600.

as i have done more and more research into temperaments,
i've gained greater and greater respect for 12edo.

it's a very small-cardinality EDO which tempers out
a great many unison-vectors, and thus is simple to
use and is very flexible in representing several
different families of temperaments simultaneously.

as far as the influence of the harmonic series on
the acceptance of 12edo, i'd say that the only thing
there that really mattered was that prime-factor 3
was represented with little error in 12edo.

the error of the 12edo representation of 5 is small
enough that it is arguably insignificant, and that
of 7 is still small enough that 12edo *can* represent
ratios of 7, alto not very well.

for ratios of 11 and 13, 12edo breaks down completely,
as most of those ratios are very nearly "quarter-tones"
which fall between the degrees of 12edo.

for prime-factors 17 and 19, 12edo once again provides
very good representations, and for 23 it's so-so.

but none of the prime-factors above 5 were historically
important in establishing the hegemony of 12edo as a
musical tuning.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> ----- Original Message -----
> From: "Carl Lumma" <ekin@l...>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, March 29, 2003 12:47 PM
> Subject: Re: [tuning] Harmonics and Their Equal Tempered Frequencies
>

> it should also be noted that 12edo was favored as
> the tuning for fretted instruments (cheifly the lute)
> from c. 1600.

I was just listening to music for lute, 1500-1550. Any idea how that
is authentically tuned?

> the error of the 12edo representation of 5 is small
> enough that it is arguably insignificant, and that
> of 7 is still small enough that 12edo *can* represent
> ratios of 7, alto not very well.

Not very good argument; the difference is all too easy to hear.

🔗Walter Lepore <earth7@optonline.net>

Hi Carl

> Finding the origin of anything involves historical inquiry, which
> is always speculative. That said, I think there is a strong
> consensus here that the harmonic series had no direct part in the
> development of equal temperament in Western music.

I just realized today the book I was reading was merely comparing
the "just frequencies" to the "tempered frequencies" used in 12tET.
It was not an end all solution to how the 12tET diatonic intervals
were discovered. Sorry my mistake.

I already went through a rather laborious procedure by "attempting"
to crawl into the mind of Pythagoras and mathematically re-create
his "endless" cycle of fifths which come close to our current system
of 12tET.

Although it involved simple mathematics to accomplish the task it was
a well deserved cause for celebration when I finished. It's the math
I'm alittle slow on but it does eventually "sink in".

Sometimes I try to seek an answer that's simple and just go with it.
Then after awhile I'll go deeper into other sytems such as meantone
etc. Unfortunately, the books I read on tuning get harder and harder
for me to understand as I dive deeper into the subject. Sometimes I'm
stumped on a page for weeks trying to figure it out.

Some books are waaaaay over my head and others are workable.

How is Partch's book? I've heard about this book for quite some time.
Is it for advanced tuners or can a novice benefit from it?

Stay well
Walter

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Hi Walter,
>
> >Should I assume we stopped at the 27 harmonic to form what is
todays
> >12tEQ scale?
>
> Finding the origin of anything involves historical inquiry, which
> is always speculative. That said, I think there is a strong
> consensus here that the harmonic series had no direct part in the
> development of equal temperament in Western music. Rather, 12-tET
> was selected because it contains the diatonic scale and allows
> unlimited modulation on the Halberstadt (7-white, 5-black) keyboard
> that was already entrenched in the musical culture. Had 19-tone
> keyboards been the norm, we suppose 19-tET would have become the
> standard tuning.
>
> The harmonic and subharmonic series are the basis of harmony,
> however. The diatonic scale and 12-tET each support many different
> styles of harmony. In Jazz, I think the argument is strong that we
> are seeing "7-limit" (Partch's term) approximations. But Common-
> Practice music is usually considered 5-limit.
>
> I highly recommend you get a copy of Partch's book, _Genesis
> of a Music_.
>
> -Carl

🔗Walter Lepore <earth7@optonline.net>

🔗Carl Lumma <ekin@lumma.org>

🔗Alison Monteith <alison.monteith3@which.net>

on 30/3/03 1:55 am, Gene Ward Smith at gwsmith@svpal.org wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>>
>> ----- Original Message -----
>> From: "Carl Lumma" <ekin@l...>
>> To: <tuning@yahoogroups.com>
>> Sent: Saturday, March 29, 2003 12:47 PM
>> Subject: Re: [tuning] Harmonics and Their Equal Tempered Frequencies
>>
>
>> it should also be noted that 12edo was favored as
>> the tuning for fretted instruments (cheifly the lute)
>> from c. 1600.
>
> I was just listening to music for lute, 1500-1550. Any idea how that
> is authentically tuned?

Most of my repertoire for lute falls into this period, for example the
vihuelistas and Francesco da Milano. The scholars agree that 12 tet was the
tuning of the day, and I'm quite happy to go along with this.

However, I was playing yesterday afternoon at an Early Music event and met
an instrument maker who told me he had attended an early music recital
recently, a German ensemble I think.

The theorbo (a long necked lute with sympathetic strings) player had some
system of secondary frets on the neck behind the main frets. My impression
was of a version of the type of neck that Dave Beardsley and other JI
guitarists might use. He also told me that the repertoire was 16th century.
This would suggest either a non-12 tuning or perhaps simply microtonal
inflections or even ornaments. I do know that the current crop of young
early music specialists are delving into Arabic influences and that might be
a factor as well. I'll try to get more details.

Regards
a.m.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

but the indirect relevance was huge!

> Rather, 12-tET
> was selected because it contains the diatonic scale and
allows
> unlimited modulation on the Halberstadt (7-white, 5-black)
keyboard
> that was already entrenched in the musical culture. Had
19-tone
> keyboards been the norm, we suppose 19-tET would have
become the
> standard tuning.

why 19? because the harmonics series through the 6th is better
represented there.

> But Common-
> Practice music is usually considered 5-limit.

exactly. so why your comment above?
>
> I highly recommend you get a copy of Partch's book, _Genesis
> of a Music_.
>
> -Carl

yup, and take everything it says with a *huge* grain of salt!

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > ----- Original Message -----
> > From: "Carl Lumma" <ekin@l...>
> > To: <tuning@yahoogroups.com>
> > Sent: Saturday, March 29, 2003 12:47 PM
> > Subject: Re: [tuning] Harmonics and Their Equal Tempered
Frequencies
> >
>
> > it should also be noted that 12edo was favored as
> > the tuning for fretted instruments (cheifly the lute)
> > from c. 1600.
>
> I was just listening to music for lute, 1500-1550. Any idea how
that
> is authentically tuned?

fumbly meantone, or 12-equal.

> > the error of the 12edo representation of 5 is small
> > enough that it is arguably insignificant, and that
> > of 7 is still small enough that 12edo *can* represent
> > ratios of 7, alto not very well.
>
> Not very good argument; the difference is all too easy to hear.

same is true for the 5 and 5:3. but the difference doesn't stop the
representation from happening.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Walter Lepore" <earth7@o...>
wrote:
> Hi Members
>
> I was investigating the harmonic series recently because I
wanted to
> learn exactly what harmonic we stopped at to form our western
12 tone
> equal tempered scale. And where each note of our 12tEQ
tempered scale
> is related to what harmonic. In other words where (in the
harmonic
> series ) did our 12tEQ tempered intervals originate from?
> I have a book that explains all the harmonics that are the
closest to
> equal-tempered intervals and collapsed to one octave. Starting
from
> the bottom with the fundamental and moving upwards to the
octave they
> are as follows:
>
> Natural Intervals(cents)-------Tempered Interval(cents)
>
> 1200 Octave ------------------ 1200 Octave
> 15th Harmonic 1088 ----------- 1100 Major 7th
> 7th Harmonic 969 -------------1000 minor 7th
> 27 Harmonic 906 -------------900 Major 6th
> 13 Harmonic 841 -------------800 minor 6th
> 3rd Harmonic 702 -------------700 Perfect 5th
> 11 Harmonic 551--------------600 Tritone
> 21st Harmonic 471 -------------500 Perfect 4th
> 5th Harmonic 386 -------------400 Major 3rd
> 19 Harmonic 298 -------------300 minor 3rd
> 9th Harmonic 204 -------------200 Major 2nd
> 17 Harmonic 105 -------------100 minor 2nd
> 0 Fundamental -----------------0 Fundamental
>
> Should I assume we stopped at the 27 harmonic to form what
is todays
> 12tEQ scale?
>
> Thank you
> Walter Lepore

hi walter. the 12-equal scale can indeed be viewed as having
derived from any of a number of theoretical 12-tone just
intonation scales, but the above is not one of them. i prefer to
view 12-equal as having derived from meantone temperament,
which in turn can be said to derive from any of a number of just
frameworks. but take a look at the "gentle introduction to fokker
periodicity blocks" and see if it makes any sense to you.

🔗Walter Lepore <earth7@optonline.net>

Hi Wallyesterpaulrus,

>the 12-equal scale can indeed be viewed as having
> derived from any of a number of theoretical 12-tone just
> intonation scales, but the above is not one of them.

I stand corrected. I believe the chart I listed earlier which
included Just harmonics and Tempered frequencies was actually a
comparison to show the actual deviation of 12tET frequencies from
their actual Just harmonics. It was not an attempt to explain the
origin of the diatonic scale.

I studied the Pythagorean cycle of fifths awhile back and was quite
satisfied that this could have been one potential event which made
inroads to the origin of the diatonic scale.It even explains how and
why half steps occured between 3, 4 and 7, 8 in the major scale.

I agree meantone is another avenue but I haven't quite nailed that
one day yet to present an educated opinion. I'm working on it.
Meantone is harder for me to follow than the Pythagorean cycle of
fifths.

>but take a look at the "gentle introduction to fokker
> periodicity blocks" and see if it makes any sense to you.

Where do I read on the fokker periodicity blocks?

Thank you
Walter

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Walter Lepore" <earth7@o...>
> wrote:
> > Hi Members
> >
> > I was investigating the harmonic series recently because I
> wanted to
> > learn exactly what harmonic we stopped at to form our western
> 12 tone
> > equal tempered scale. And where each note of our 12tEQ
> tempered scale
> > is related to what harmonic. In other words where (in the
> harmonic
> > series ) did our 12tEQ tempered intervals originate from?
> > I have a book that explains all the harmonics that are the
> closest to
> > equal-tempered intervals and collapsed to one octave. Starting
> from
> > the bottom with the fundamental and moving upwards to the
> octave they
> > are as follows:
> >
> > Natural Intervals(cents)-------Tempered Interval(cents)
> >
> > 1200 Octave ------------------ 1200 Octave
> > 15th Harmonic 1088 ----------- 1100 Major 7th
> > 7th Harmonic 969 -------------1000 minor 7th
> > 27 Harmonic 906 -------------900 Major 6th
> > 13 Harmonic 841 -------------800 minor 6th
> > 3rd Harmonic 702 -------------700 Perfect 5th
> > 11 Harmonic 551--------------600 Tritone
> > 21st Harmonic 471 -------------500 Perfect 4th
> > 5th Harmonic 386 -------------400 Major 3rd
> > 19 Harmonic 298 -------------300 minor 3rd
> > 9th Harmonic 204 -------------200 Major 2nd
> > 17 Harmonic 105 -------------100 minor 2nd
> > 0 Fundamental -----------------0 Fundamental
> >
> > Should I assume we stopped at the 27 harmonic to form what
> is todays
> > 12tEQ scale?
> >
> > Thank you
> > Walter Lepore
>
> hi walter. the 12-equal scale can indeed be viewed as having
> derived from any of a number of theoretical 12-tone just
> intonation scales, but the above is not one of them. i prefer to
> view 12-equal as having derived from meantone temperament,
> which in turn can be said to derive from any of a number of just
> frameworks. but take a look at the "gentle introduction to fokker
> periodicity blocks" and see if it makes any sense to you.

🔗monz <monz@attglobal.net>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Walter Lepore <earth7@optonline.net>

🔗Dave Keenan <d.keenan@uq.net.au>

--- In tuning@yahoogroups.com, Alison Monteith wrote:
> on 30/3/03 1:55 am, Gene Ward Smith wrote:
> >> From: "Carl Lumma"
> >> it should also be noted that 12edo was favored as
> >> the tuning for fretted instruments (cheifly the lute)
> >> from c. 1600.
> >
> > I was just listening to music for lute, 1500-1550. Any idea how
> > that is authentically tuned?
>
> Most of my repertoire for lute falls into this period, for example
> the vihuelistas and Francesco da Milano. The scholars agree that
> 12 tet was the tuning of the day, and I'm quite happy to go along
> with this.
>
> However ...

In regard to the possibility of historical meantone lute tunings see
this post by Wim Hoogewerf
/tuning/topicId_26665.html#26665
The post of mine that it refers to is
/tuning/topicId_26652.html#26652

Regards,
-- Dave Keenan