Yahoo Tuning Groups Ultimate Backup tuning Is the harmonic series arithmatic? (Wikipedia entry)

Happy Holidays. I know you guys are strong in both math and
acoustics. Is it accurate to say the harmonic series is an arithmatic
series? Here is a quote from wikipedia entry on "harmonic series, music"

>>>>>"The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f,
5×f, ...). In terms of frequency (measured in cycles per second, or
hertz (Hz) where f is the fundamental frequency), the difference
between consecutive harmonics is therefore constant. But because our
ears respond to sound nonlinearly, we perceive higher harmonics
as "closer together" than lower ones. On the other hand, the octave
series is a geometric progression (2×f, 4×f, 8×f, 16×f, ...), and we
hear these distances as "the same" in the sense of musical interval. In
terms of what we hear, each octave in the harmonic series is divided
into increasingly "smaller" and more numerous intervals.">>>>

Aren't harmonics naturally decreasing, and not equal in spacing (the
equal spacing is just an effect of log hearing). Or is this connected
with whether we are viewing the harmonic series as frequencies vs
wavelengths?

Kelly

🔗Petr Parízek <p.parizek@...>

Kelly wrote:

> Aren't harmonics naturally decreasing, and not equal in spacing (the
> equal spacing is just an effect of log hearing).

Depends on what you mean by "equal spacing". The harmonic series, as Wikipedia sais, is equally spaced in terms of linear (i.e. arithmetic) frequency progression. However, it sounds unequal to our ears because we don't hear linear frequency distances as equal but instead we hear the exponential distances as if they were equal. So, viewed from the other side, if you také frequencies of 1Hz, 2Hz, 4Hz, 8Hz and so on, you get a set of tones which are unequally spaced in terms of linear frequency progression but are equally spaced in terms of exponential frequency progression so our ears hear the intervals between consecutive tones as equal.

> Or is this connected with whether we are viewing
> the harmonic series as frequencies vs wavelengths?

I'm not sure what you mean.

Petr

🔗djtrancendance@...

---Aren't harmonics naturally decreasing
Yes, far as the space in between them as you go up the series.
Even on a per octave (exponential) basis....as you go up there are more and more overtones/"notes" per octave going up the harmonic series.

This means the notes beat more and faster against each other going up the series and, IMVHO, after about root-frequency*22 or so (where each note is about the ratio 22/21 apart) they start getting so close your ears start objecting a bit.

-Michael

--- On Mon, 12/22/08, traktus5 <kj4321@...> wrote:

From: traktus5 <kj4321@...>
Subject: [tuning] Is the harmonic series arithmatic? (Wikipedia entry)
To: tuning@yahoogroups.com
Date: Monday, December 22,
2008, 8:45 AM

Happy Holidays. I know you guys are strong in both math and

acoustics. Is it accurate to say the harmonic series is an arithmatic

series? Here is a quote from wikipedia entry on "harmonic series, music"

>>>>>"The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f,

5×f, ...). In terms of frequency (measured in cycles per second, or

hertz (Hz) where f is the fundamental frequency), the difference

between consecutive harmonics is therefore constant. But because our

ears respond to sound nonlinearly, we perceive higher harmonics

as "closer together" than lower ones. On the other hand, the octave

series is a geometric progression (2×f, 4×f, 8×f, 16×f, ...), and we

hear these distances as "the same" in the sense of musical interval. In

terms of what we hear, each octave in the harmonic series is divided

into increasingly "smaller" and more numerous intervals."> >>>

Aren't harmonics naturally decreasing, and not equal in spacing (the

equal spacing is just an effect of log hearing). Or is this connected

with whether we are viewing the harmonic series as frequencies vs

wavelengths?

Kelly

🔗Michael Sheiman <djtrancendance@...>

Agreed...it is equal but ONLY as a linear function.
As an exponential function (IE relative to hearing, which is exponential), it decreases rapidly.
That is what is important concerning how you hear it...how it functions exponentially.

An octave, of course, is a purely exponential phenomena IE it is 2^n NOT 2n.

--- On Mon, 12/22/08, Petr Parízek <p.parizek@chello.cz> wrote:

From: Petr Parízek <p.parizek@...>
Subject: Re: [tuning] Is the harmonic series arithmatic? (Wikipedia entry)
To: tuning@yahoogroups.com
Date: Monday, December 22, 2008, 9:05 AM

Kelly
wrote:
>
Aren't harmonics naturally decreasing, and not equal in spacing (the
>
equal spacing is just an effect of log hearing).
Depends
on what you mean by „equal spacing“. The harmonic series, as Wikipedia sais, is
equally spaced in terms of linear (i.e. arithmetic) frequency progression.
However, it sounds unequal to our ears because we don’t hear linear frequency
distances as equal but instead we hear the exponential distances as if they were
equal. So, viewed from the other side, if you také frequencies of 1Hz, 2Hz, 4Hz,
8Hz and so on, you get a set of tones which are unequally spaced in terms of
linear frequency progression but are equally spaced in terms of exponential
frequency progression so our ears hear the intervals between consecutive tones
as equal.
>
Or is this connected with whether we are viewing
> the harmonic series as
frequencies vs wavelengths?
I’m
not sure what you mean.
Petr

🔗caleb morgan <calebmrgn@...>

not to nitpick, but the beating rate doesn't get faster, that's the
whole point of the overtone series:

the beat-rate between any two adjacent overtones is, by definition,
the fundamental frequency.

It stays exactly the same.

But, I know this is not exactly what you meant.

On Dec 22, 2008, at 2:55 PM, djtrancendance@... wrote:

>
> ---Aren't harmonics naturally decreasing
> Yes, far as the space in between them as you go up the series.
> Even on a per octave (exponential) basis....as you go up there > are more and more overtones/"notes" per octave going up the harmonic
> series.
>
> This means the notes beat more and faster against each other going
> up the series and, IMVHO, after about root-frequency*22 or so (where
> each note is about the ratio 22/21 apart) they start getting so
> close your ears start objecting a bit.
>
> -Michael
>
> --- On Mon, 12/22/08, traktus5 <kj4321@...> wrote:
>
> From: traktus5 <kj4321@hotmail.com>
> Subject: [tuning] Is the harmonic series arithmatic? (Wikipedia entry)
> To: tuning@yahoogroups.com
> Date: Monday, December 22, 2008, 8:45 AM
>
> Happy Holidays. I know you guys are strong in both math and
> acoustics. Is it accurate to say the harmonic series is an arithmatic
> series? Here is a quote from wikipedia entry on "harmonic series,
> music"
>
> >>>>>"The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f,
> 5×f, ...). In terms of frequency (measured in cycles per second, or
> hertz (Hz) where f is the fundamental frequency), the difference
> between consecutive harmonics is therefore constant. But because our
> ears respond to sound nonlinearly, we perceive higher harmonics
> as "closer together" than lower ones. On the other hand, the octave
> series is a geometric progression (2×f, 4×f, 8×f, 16×f, ...), and we
> hear these distances as "the same" in the sense of musical interval.
> In
> terms of what we hear, each octave in the harmonic series is divided
> into increasingly "smaller" and more numerous intervals."> >>>
>
> Aren't harmonics naturally decreasing, and not equal in spacing (the
> equal spacing is just an effect of log hearing). Or is this connected
> with whether we are viewing the harmonic series as frequencies vs
> wavelengths?
>
> Kelly
>
>
>
>

🔗Petr Parízek <p.parizek@...>

🔗Paul Poletti <paul@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>

> This means the notes beat more and faster against each other going
up the series and, IMVHO, after about root-frequency*22 or so (where
each note is about the ratio 22/21 apart) they start getting so close
your ears start objecting a bit.
>
The critical factor is of course appropriately named the Critical
Band, which is the interval between a unison and that point at which
we begin to hear two separate tones. The point of maximum dissonance
is at about 25% CB. Problem is, the width of the CB is not same over
the entire aural spectrum. It's incredibly wide in low frequencies,
meaning the interval of maximum dissonance can be an interval which is
normally considered a consonance. At 16 Hz, MD is a fifth, at 33 Hz,
it's a major third, at 50 Hz a minor third. By about 250 Hz it's
gotten down to a semitone, and at about 500 Hz it drops to the level
it holds to for most of the rest of the spectrum, about 1/2 semitone.
So you have to get up into the high 20's before adjacent overtones
start bumping up against each other in an annoying manner.

Ciao,

🔗Danny Wier <dawiertx@...>

I'll say it a few days early in case I'm busy in a few days - Merry
Christmas.

And yes, I'm talking about 72-edo again. This time, instead of Miracle
and the 31- and 41-tone subsets, I'm planning on a work in kleismic,
with a generator of 316.67 cents (I mentioned it on MMM earlier). This
produces 15-, 19-, 34- and 53-tone sets; I'm most interested in 19 and
53:

(0-D) 1 3 (4-D#) 5 7 (8-Eb) 10 (11-E)
12 14 (15-E#) 16 18 (19-F) 20 22 (23-F#)
24 26 (27-Gb) 29 (30-G) 31 33 (34-G#) 35
37 (38-Ab) 39 41 (42-A) 43 (45-A#) 46
48 (49-Bb) 50 52 (53-B) 54 56 (57-Cb) 58
60 (61-C) 62 (64-C#) 65 67 (68-Db) 69 70 (72-D)

The 19-tone set with corresponding 19-et note names are in parentheses.

I've searched the [tuning] archives for anything on 19&53 kleismic, and
we've discussed it a little. I haven't vetted [tuning-math] much yet.
Since we have "blackjack", "canasta" and "studloco" for 21-, 31-,
41-tone miracle, then we need names for 19- and 53-tone kleismic. For
the latter, I came up with another card-inspired name: "joker", since
there are 52 cards in a standard deck and the two jokers make 54.

I think there might be a name for 19 already, since 19-et is so
well-worn. Ennealimmal, right?

I'm working on something using both 31&41 and 19&53, and I need an
excuse to use a chord like D-F-Ab-Cb-D#-F#-A (in 72-edo:
0-19-38-57-76-85-114).

~D.

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Danny Wier <dawiertx@...> wrote:
>
> I'll say it a few days early in case I'm busy in a few days - Merry
> Christmas.

Merry Christmas!

> I've searched the [tuning] archives for anything on 19&53
> kleismic,

This system is also called (since 2004 or so) "hanson".
There's been a great deal of discussion on the system...
not sure what searches to suggest.

> I think there might be a name for 19 already, since 19-et is so
> well-worn. Ennealimmal, right?

Ennealimmal is a different temperament. It's supported
by 72 but really comes into its own in 99.

-Carl

🔗Danny Wier <dawiertx@...>

On Mon, 2008-12-22 at 21:57 +0000, Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Danny Wier <dawiertx@...> wrote:

> > I've searched the [tuning] archives for anything on 19&53
> > kleismic,
>
> This system is also called (since 2004 or so) "hanson".
> There's been a great deal of discussion on the system...
> not sure what searches to suggest.

I forgot, I did find Hanson, and I'm interested in his keyboard layout.
It's a type of kleismic though, right?

> > I think there might be a name for 19 already, since 19-et is so
> > well-worn. Ennealimmal, right?
>
> Ennealimmal is a different temperament. It's supported
> by 72 but really comes into its own in 99.

Oh okay. I was going to suggest "Metonic" then, or some other nickname.
~D.

🔗Michael Sheiman <djtrancendance@...>

OK...you're right.
It gets more dramatic IE has more amplitude and is more pronounced...but does not get faster.

--- On Mon, 12/22/08, caleb morgan <calebmrgn@...> wrote:

From: caleb morgan <calebmrgn@...>
Subject: Re: [tuning] Is the harmonic series arithmatic? (Wikipedia entry)
To: tuning@yahoogroups.com
Date: Monday, December 22, 2008, 12:04 PM

not to nitpick, but the beating rate doesn't get faster, that's the whole point of the overtone series:
the beat-rate between any two adjacent overtones is, by definition, the fundamental frequency.
It stays exactly the same.
But, I know this is not exactly what you meant.

On Dec 22, 2008, at 2:55 PM, djtrancendance@ yahoo.com wrote:

This means the notes beat more and faster against each other going up the series and, IMVHO, after about root-frequency* 22 or so (where each note is about the ratio 22/21 apart) they start getting so close your ears start objecting a bit.

-Michael

--- On Mon, 12/22/08, traktus5 <kj4321@hotmail. com> wrote:

From: traktus5 <kj4321@hotmail. com>
Subject: [tuning] Is the harmonic series arithmatic? (Wikipedia entry)
To: tuning@yahoogroups. com
Date: Monday, December 22, 2008, 8:45 AM

Kelly

🔗Carl Lumma <carl@...>

🔗Michael Sheiman <djtrancendance@...>

Odd...I have read the critical band never completely levels out...changing all the way up to near 10khz.
http://www.music.gla.ac.uk/~george/audio/psy/psy.html

One thing is for sure...take a sine wave and another at 22/21 times its frequency with both sine waves being around 300hz (fairly typical in music)...and you'll hear pronounced, although "harmonically rhythmic" beating.
Of course, if your root note is 200hz and your test notes are at 200 * 22 and 200 * 21...that is a different story...they sound fairly good...though it seems rather unrealistic to be playing/composing only with notes that high on a piano.
____________________________________________________
I guess my point is, at least though my own testing...the harmonic series itself does have some pronounced beating if you are, say, trying to use it to make a 7-note-or-more-per-octave scale in an octave range realistic for composing music with.

-Michael

--- On Mon, 12/22/08, Paul Poletti <paul@...> wrote:

From: Paul Poletti <paul@...>
Subject: [tuning] Re: Is the harmonic series arithmatic? (Wikipedia entry)
To: tuning@yahoogroups.com
Date: Monday, December 22, 2008, 1:17 PM

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

> This means the notes beat more and faster against each other going

up the series and, IMVHO, after about root-frequency* 22 or so (where

each note is about the ratio 22/21 apart) they start getting so close

your ears start objecting a bit.

The critical factor is of course appropriately named the Critical

Band, which is the interval between a unison and that point at which

we begin to hear two separate tones. The point of maximum dissonance

is at about 25% CB. Problem is, the width of the CB is not same over

the entire aural spectrum. It's incredibly wide in low frequencies,

meaning the interval of maximum dissonance can be an interval which is

normally considered a consonance. At 16 Hz, MD is a fifth, at 33 Hz,

it's a major third, at 50 Hz a minor third. By about 250 Hz it's

gotten down to a semitone, and at about 500 Hz it drops to the level

it holds to for most of the rest of the spectrum, about 1/2 semitone.

So you have to get up into the high 20's before adjacent overtones

start bumping up against each other in an annoying manner.

Ciao,

🔗Graham Breed <gbreed@...>

2008/12/23 Danny Wier <dawiertx@...>:

> I've searched the [tuning] archives for anything on 19&53 kleismic, and
> we've discussed it a little. I haven't vetted [tuning-math] much yet.
> Since we have "blackjack", "canasta" and "studloco" for 21-, 31-,
> 41-tone miracle, then we need names for 19- and 53-tone kleismic. For
> the latter, I came up with another card-inspired name: "joker", since
> there are 52 cards in a standard deck and the two jokers make 54.

I don't see a need to follow the card game theme outside miracle.

> I think there might be a name for 19 already, since 19-et is so
> well-worn. Ennealimmal, right?

No, as pointed out elsewhere.

I'd like to bagsie the 19 note magic MOS, anyway. I'm planning to
call it, or something related to it, "Pengcheng" as part of a Han
Dynasty city theme. The kleismatic MOS needn't have anything to do
with that.

It is interesting that there's no name for meantone-19. 5 is
"pentatonic", 7 is "diatonic", 12 is "chromatic", and the 24 note
scale with "half sharps" could be called "enharmonic". Did Yasser
give the 19 note scale a name, maybe?

Fokker called his 19 note periodicity block a "scale of tritotones" IIRC.

Another thing, while I'm here, what about a cute name for the 7 prime
limit? The 3-limit is "Pythagorean". The 5-limit is sometimes called
"Ptolemaic". The 11-limit could obviously be "Partchian" if only
because the man himself would have hated such a name ;-) But for the
7-limit we only have the rather prosaic "septimal" which sounds a bit
like "septic".

Graham

🔗traktus5 <kj4321@...>

hiPetr.

> > Aren't harmonics naturally decreasing, and not equal in spacing
(the > > equal spacing is just an effect of log hearing).

> Depends on what you mean by "equal spacing". The harmonic series,
as Wikipedia sais, is equally spaced in terms of linear (i.e.
arithmetic) frequency progression.

>> but in terms of wavelength (and interval size), the size
diminishes up the series (1/2, 1/3, 1/4...). Shouldn't the
definition encompass both 'modes' 1:2:3...;1/2,1/3,1/4...)?

>However, it sounds unequal to our ears because we don't hear linear
frequency distances as equal

>>or it sounds unequal because, in terms of wavelenths(1/2, 1/3, 1/4)
the sizes are unequal, and corresponds to decreasing interval size
(octave, P5, P4, M3,m3, M2, etc)

>but instead we hear the exponential distances as if they were equal.

>>you mean, the exponential distances in the octave series. What
about the exponention distances in the harmonic series?

>So, viewed from the other side, if you také frequencies of 1Hz, 2Hz,
4Hz, 8Hz and so on, you get a set of tones which are unequally spaced
in terms of linear frequency progression but are equally spaced in
terms of exponential frequency progression so our ears hear the
intervals between consecutive tones as equal.

>>And for the harm series, you have unequal distances as measured by
wavelenthgs(1/2, 1/3, 1/4...), and as measured, intervallically, by
scale steps, but equal spacing as measured by frequency.

🔗Killian-O'Callaghan Residence <gottharddanae@...>

🔗Mohajeri Shahin <shahinm@...>

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My microtonal web site <http://www.96edo.com/>

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Telephone: (9821) 88072501-9

Fax: (9821) 88072500
Email: shahinm@... <mailto:shahinm@...>

P Please consider the environment before printing this mail note.

🔗Petr Parízek <p.parizek@...>

Kelly wrote:

>>or it sounds unequal because, in terms of wavelenths(1/2, 1/3, 1/4)
> the sizes are unequal, and corresponds to decreasing interval size
> (octave, P5, P4, M3,m3, M2, etc)

Sorry, but this is NOT the reason. It's a generally confirmed fact that we hear exponential distances as equal. And since the exponential distances between consecutive harmonic ratios are decreasing, we hear the intervals as of decreasing size.

> you mean, the exponential distances in the octave series. What
> about the exponention distances in the harmonic series?

Have you ever tried to convert the intervals of the harmonic series to some logarithmically measured units like, for example, cents? What you get is 1200 cents for 2/1, ~701.955 cents for 3/2, ~498.045 cents for 4/3, ~386.314 cents for 5/4 and so on. On the other hand, if you do this with a subharmonic series (i.e. the one which has equally spaced wavelengths), you'll get -1200 cents when comparing 2/1 length, -~701.955 cents for 3/2 length, -~498.045 cents for 4/3 length and so on.

> And for the harm series, you have unequal distances as measured by
> wavelenthgs(1/2, 1/3, 1/4...), and as measured, intervallically, by
> scale steps, but equal spacing as measured by frequency.

Yes, but wavelengths don't matter here unless you want to care about guide tones.

Petr

🔗Petr Pařízek <p.parizek@...>

Graham wrote:

> It is interesting that there's no name for meantone-19. 5 is
> "pentatonic", 7 is "diatonic", 12 is "chromatic", and the 24 note
> scale with "half sharps" could be called "enharmonic". Did Yasser
> give the 19 note scale a name, maybe?

Isn’t there really? Some years ago, I was thinking about „enharmonic“ for 19 and something like „diachromatic“ or similar for 31. But I’m still not decided. The thing is that in 5-equal you temper out the diatonic step, in 7-equal you temper out the chromatic step, in 12-equal you temper out the enharmonic step, and in 19-equal you temper out this strange step which I was calling „diachromatic“ at that time.

> The 3-limit is "Pythagorean". The 5-limit is sometimes called
> "Ptolemaic". The 11-limit could obviously be "Partchian" if only
> because the man himself would have hated such a name ;-) But for the
> 7-limit we only have the rather prosaic "septimal" which sounds a bit
> like "septic".

Very often, I’ve seen 5-limit tuning called „Didymic“ and 81/80 called „Didymus comma“. Even in some materials that mention tuning only very briefly, the distinction is made between „Pythagorean“ major thirds and „Didymic“ major thirds, those being 81/64 and 5/4, respectively. As to the 7-limit systém, I could think of something like „Tartinian“ (Rapoport said that Tartini was among the first inventors of the semi-flat/sharp symbols used for 7-limit intervals) or Fokkerian (which I myself am unable to comfortably pronounce).

Petr

🔗Graham Breed <gbreed@...>

2008/12/23 Petr Pařízek <p.parizek@chello.cz>:
>
> Graham wrote:
>
>> It is interesting that there's no name for meantone-19. 5 is
>> "pentatonic", 7 is "diatonic", 12 is "chromatic", and the 24 note
>> scale with "half sharps" could be called "enharmonic". Did Yasser
>> give the 19 note scale a name, maybe?
>
> Isn't there really? Some years ago, I was thinking about „enharmonic" for 19
> and something like „diachromatic" or similar for 31. But I'm still not
> decided. The thing is that in 5-equal you temper out the diatonic step, in
> 7-equal you temper out the chromatic step, in 12-equal you temper out the
> enharmonic step, and in 19-equal you temper out this strange step which I
> was calling „diachromatic" at that time.

None are established that I know of. I could well be wrong and maybe
somebody'll join in and tell us so. Google tells me another Graham is
calling it "hyper-chromatic" which is also a real word in a different
context.

"Enharmonic" originally referred to the Greek tetrachord, with
something like a quarter-tone. Vicentino then used it for his
24-from-31 scale. He called one step on that scale an "enharmonic
diesis". "Enharmonic equivalents" differ by such a diesis. The word
may have attracted all kinds of other meanings over the centuries.

>> The 3-limit is "Pythagorean". The 5-limit is sometimes called
>> "Ptolemaic". The 11-limit could obviously be "Partchian" if only
>> because the man himself would have hated such a name ;-) But for the
>> 7-limit we only have the rather prosaic "septimal" which sounds a bit
>> like "septic".
>
> Very often, I've seen 5-limit tuning called „Didymic" and 81/80 called
> „Didymus comma". Even in some materials that mention tuning only very
> briefly, the distinction is made between „Pythagorean" major thirds and
> „Didymic" major thirds, those being 81/64 and 5/4, respectively. As to the
> 7-limit systém, I could think of something like „Tartinian" (Rapoport said
> that Tartini was among the first inventors of the semi-flat/sharp symbols
> used for 7-limit intervals) or Fokkerian (which I myself am unable to
> comfortably pronounce).

Yes, "Didymic" does as well. It has the advantage that it starts with
a different letter to "Pythagorean". If anybody's setting up a
microtonal prog rock band I suggest "The Commas of Didymus" for the
name, BTW.

I believe Tartini, Euler and some other mathematician all advocated
the 7-limit but I forget where I read it. Huygens certainly did. Did
Tartini actually use it to make music though?

"Fokkian" would be easier to say. But from what I've heard you have
to be careful about pronouncing "Fokker" too authentically when you're
speaking English. "Can you change that to a Fokkian chord?" "It
already is a chord!" And so on.

Graham

🔗Petr Pařízek <p.parizek@...>

Graham wrote:

> "Enharmonic" originally referred to the Greek tetrachord, with
> something like a quarter-tone. Vicentino then used it for his
> 24-from-31 scale. He called one step on that scale an "enharmonic
> diesis". "Enharmonic equivalents" differ by such a diesis. The word
> may have attracted all kinds of other meanings over the centuries.

The meaning I had in mind was a distance of 12 fifths in the meantone chain, similarly as tones which differ by a chromatic step are 7 fifths apart in the chain. I was taking it just from a „numerical“ point of view -- 7-equal has the diatonic step as the smallest interval but tempers out the chromatic one. 12-equal has the chromatic step as the smallest one but tempers out the enharmonic one (which is the reason for enharmonic equivalence). And, one level higher, 19-equal has the enharmonic step as the smallest one and tempers out the (let’s stay with the word) „diachromatic“ step.

> I believe Tartini, Euler and some other mathematician all advocated
> the 7-limit but I forget where I read it. Huygens certainly did. Did
> Tartini actually use it to make music though?

In fact, I’m not sure if Tartini had ever used 7-limit intervals himself. And I’m not sure, either, if Euler had actually taken 7-limit into account. His „genera musica“ were nothing else than layering powers of 3 and 5 which were, if necessary, divided or multiplied by powers of 2 for good musical use. The term „Euler-Fokker genus“ originally meant, if I’m well informed, a genus containing primes up to 7, but it later became used to mean 5-limit genera as well - i.e. as a collective term for this kind of scales.

Petr

🔗Danny Wier <dawiertx@...>

On Tue, 2008-12-23 at 11:44 +0800, Graham Breed wrote:

I second Petr's vote for "enharmonic"; I've been using the term myself
for pitches a diesis or quarter tone apart in many tunings including 19
and 24. And 31.

> Another thing, while I'm here, what about a cute name for the 7 prime
> limit? The 3-limit is "Pythagorean". The 5-limit is sometimes called
> "Ptolemaic". The 11-limit could obviously be "Partchian" if only
> because the man himself would have hated such a name ;-) But for the
> 7-limit we only have the rather prosaic "septimal" which sounds a bit
> like "septic".

On p. 92 of _Genesis of a Music_, Harry Partch lays out a chronology of
when intervals and prime limits were "recognized" in music theory. It
mentions ancient figures like Ling Lun, who discovered Pythagorean
tuning long before Pythagoras, Archytas and Euclid. In the medieval era,
Odington is credited with 5-limit, and in the early modern era, Mersenne
with 7-limit. In the 20th century, Partch credits himself with 11-limit,
and Kathleen Schlesinger with 13-limit. However, he also gives Ptolemy a
nod for using 7 and 11, not just 5 and 9.

So if 11-limit is "Partchian", can 7-limit be "Mersennian" and 13-limit
"Schlesingerian"? And what about 17?

~D.

🔗traktus5 <kj4321@...>

🔗Petr Parízek <p.parizek@...>

Danny wrote:

> I second Petr's vote for "enharmonic"; I've been using the term myself

Thanks.

> for pitches a diesis or quarter tone apart in many tunings including 19
> and 24. And 31.

31 is debatable since Cb-B# (the smallest step) is a distance of 19 fifths, not 12, and if I should také Kornerup's "Golden meantone" as a "model" example, then this step is smaller than the enharmonic diesis similarly to the diesis being smaller than the chroma (i.e. augmented prime).

Petr

🔗Herman Miller <hmiller@...>

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Danny Wier <dawiertx@...> wrote:
>>> This system is also called (since 2004 or so) "hanson".
>>> There's been a great deal of discussion on the system...
>>> not sure what searches to suggest.
>> I forgot, I did find Hanson, and I'm interested in his
>> keyboard layout. It's a type of kleismic though, right?
> > That's right, 5-limit kleismic. I forgot about that
> distinction. Keemun is 7-limit kleismic.

Technically, keemun is one 7-limit kleismic, and catakleismic is another (they differ in the mapping of 7). The best 7/4 in 72-ET is 58 steps above 1/1, which fits with the catakleismic mapping but not keemun. I've played around with some kleismic tunings, but I usually stop around 15 notes, and I don't know that any of the MOS scales have any specific names.

🔗Graham Breed <gbreed@...>

2008/12/24 Danny Wier <dawiertx@...>:
> On Tue, 2008-12-23 at 11:44 +0800, Graham Breed wrote:
>
>> It is interesting that there's no name for meantone-19. 5 is
>> "pentatonic", 7 is "diatonic", 12 is "chromatic", and the 24 note
>> scale with "half sharps" could be called "enharmonic". Did Yasser
>> give the 19 note scale a name, maybe?
>
> I second Petr's vote for "enharmonic"; I've been using the term myself
> for pitches a diesis or quarter tone apart in many tunings including 19
> and 24. And 31.

So that's a load of different scales. It doesn't uniquely suggest 19
from meantone.

> So if 11-limit is "Partchian", can 7-limit be "Mersennian" and 13-limit
> "Schlesingerian"? And what about 17?

Partch has a good claim for the 11-limit because he wrote compelling
music in it. I don't know who really took hold of the 7-limit :-S
Mersenne, yes, he's the other mathematician who suggested it. It
tends to be something mathematicians say should work but musicians
don't find a use for. Elsie Hamilton wrote music around Schlesinger's
scales, but nobody cared about it at the time. There are plenty of
theorists throughout history who use ratios with 13 in them.

It's probably too soon to say for 17.

Graham

🔗Graham Breed <gbreed@...>

🔗Daniel Wolf <djwolf@...>

🔗Bruce R. Gilson <brgster@...>

--- In tuning@yahoogroups.com, Petr PaÅÃzek <p.parizek@...> wrote:
>
> Graham wrote:
>
> > It is interesting that there's no name for meantone-19. 5 is
> > "pentatonic", 7 is "diatonic", 12 is "chromatic", and the 24 note
> > scale with "half sharps" could be called "enharmonic". Did Yasser
> > give the 19 note scale a name, maybe?
>
> Isn't there really? Some years ago, I was thinking
about "enharmonic" for 19 and something like "diachromatic" or
similar for 31. But I'm still not decided. The thing is that in 5-
equal you temper out the diatonic step, in 7-equal you temper out the
chromatic step, in 12-equal you temper out the enharmonic step, and
in 19-equal you temper out this strange step which I was
calling "diachromatic" at that time.

Well, in Woolhouse's book, a 19-equal scale is advocated, and he
considers it an enharmonic scale, so on that you and he are in
agreement. In fact, however, elsewhere in Woolhouse's book it appears
that an enharmonic scale is any that distinguishes (e. g.) C# from Db.