Higher resolution Octave bands

HIGHER RESOLUTION OCTAVE BANDS

Introduction

The word ‘octave’ is derived from the Latin word meaning ‘eight’. In the musical world where there are 7 notes, the 8^th note sounds twice as high as the 1^st note. The 8^th note is an octave higher than the 1^st note. Similarly in the octave band, the upper limit frequency is twice the lower limit frequency. Octave bands are very useful in engineering applications because they reveal the spectral content, meaning they represent the change in noise levels with respect to the frequency of sound. This helps in identifying which frequency is responsible for the noise which helps in nailing down the component in a machine responsible for the frequency.

Octave 1/6^th, Octave 1/12^th and Octave 1/24^th are examples of higher resolution octave bands. They are similar to the octave 1/1 band except that the spectral resolution is exorbitantly high. Octave 1/6^th band is similar to octave 1/1 band but each frequency band is divided equally into 6 parts, 12 parts for Octave 1/12^th band and 24 parts for Octave 1/24^th band. This increases the frequency resolution to an extent that pointing out tonal frequencies becomes easier but at the expense of large processing times and data consumption.

Octave 1/6th Band

Fig .1 1/6th Octave Band

Equations

The relation between the next and the previous center frequency is given by,

n = 6 for 1/6^th octave band

n = 12 for 1/12^th octave band

n = 24 for 1/24^th oc1tave band

C_F – center frequency

C_Fnext – next center frequency

C_Fprev – previous center frequency

In case of 1/6^th Octave band, the next center frequency is 1.12 times the previous. This ratio decreases with increasing spectral resolution. The ratio is 1.06 for 1/12^th Octave band and 1.03 for 1/24^th Octave band.

The relation between the upper band and lower band frequency limit for a given frequency band is given by,

C_L – lower band limit for a given center frequency

C_U – upper band limit for a given center frequency

n = 6 for Octave 1/6^th band

n = 12 for Octave 1/12^th band

n = 24 for Octave 1/24^th band

As the spectral resolution keeps increasing or in other words when ‘n’ increases, the number of upper and lower bands also increases.

Visual Representation

The picture below explains how one band of Octave1/1 is divided into 6 equal parts for Octave 1/6 bands. The center frequencies of Octave 1/6 bands are calculated and mentioned in the table below. The corresponding Octave 1/1 center frequencies are also highlighted for understanding the relationship between the two.

Fig .2 Formation of Octave 1/6th band from Octave 1/1 band

Tabular column

Octave 1/6 band	Center frequency	Corresponding Octave 1/1 band
	Hz
Band1	16	Band1
Band2	17.95939277
Band3	20.1587368
Band4	22.627417
Band5	25.39841683
Band6	28.50875898
Band7	32	Band2
Band8	35.91878555
Band9	40.3174736
Band10	45.254834
Band11	50.79683366
Band12	57.01751796
Band13	64	Band3
Band14	71.83757109
Band15	80.63494719
Band16	90.50966799
Band17	101.5936673
Band18	114.0350359
Band19	128	Band4
Band20	143.6751422
Band21	161.2698944
Band22	181.019336
Band23	203.1873347
Band24	228.0700718
Band25	256	Band5
Band26	287.3502844
Band27	322.5397888
Band28	362.038672
Band29	406.3746693
Band30	456.1401437
Band31	512	Band6
Band32	574.7005687
Band33	645.0795775
Band34	724.0773439
Band35	812.7493386
Band36	912.2802874
Band37	1024	Band7
Band38	1149.401137
Band39	1290.159155
Band40	1448.154688
Band41	1625.498677
Band42	1824.560575
Band43	2048	Band8
Band44	2298.802275
Band45	2580.31831
Band46	2896.309376
Band47	3250.997354
Band48	3649.12115
Band49	4096	Band9
Band50	4597.60455
Band51	5160.63662
Band52	5792.618751
Band53	6501.994709
Band54	7298.242299
Band55	8192	Band10
Band56	9195.2091
Band57	10321.27324
Band58	11585.2375
Band59	13003.98942
Band60	14596.4846

Octave 1/6 Band Real Time Analysis

Fig .3 Real time Octave 1/6th band